To get more out of life, become an intellectual! Being intellectual is fun.
Try to learn the truth. Dig deeper! Mark Twain said:
It ain’t what you don’t know that gets you into trouble. It’s what you know for sure that just ain’t so.
He also said:
To begin, God made idiots. That was for practice. Then he made school boards. I’ve never let my school interfere with my education.
There are 3 kinds of people:
intellectuals, average people, small-minded people
President Franklin Roosevelt’s wife (Eleanor Roosevelt) said:
Great minds discuss ideas.
Average minds discuss events.
Small minds discuss people.
You can become a professor. Though professors get low pay, they enjoy short hours and long vacations (for summer, Christmas, and “spring break”). They can use their free time to soak up more cultural experiences or to moonlight as consultants or writers.
How many hours?
There’s the tale of the farmer who asked the professor how many hours of class he taught. The professor said “14 hours.” The farmer said, “Well, that’s a long day, but at least the work’s easy.” The farmer didn’t realize the professor meant 14 hours per week.
Being a professor is not a total joyride: you must spend lots of time grading papers, going to faculty meetings, preparing & researching your lectures, and doing other administrative crap. But compared to many other jobs, it’s a piece of cake. And you get lots of free benefits, such as medical plans, campus events, and other entertainment, such as the joy of laughing at your students.
If you’re a successful professor, you’ll be promoted to “dean” or “president,” which will make your life more miserable, since you’ll have to spend lots of time administering instead of “fooling around” (I mean “doing research”). “Administering” means “dealing with headaches and trying to embarrass people into donating money.”
Back in the 1960’s, when students were protesting for more freedom, Stanford University’s president gave this description of his job:
A university president has 3 responsibilities: provide sex for the students, athletics for the alumni, and parking for the faculty.
Advice for students
What colleges teach is overpriced. Instead of paying many
thousands of dollars per year to enroll, you can just go to a bookstore, buy
the textbooks, and read them yourself, for a total cost of a few hundred
dollars instead of thousands. But you won’t take that shortcut, because nobody
will motivate you.
The main reason for going to college is social: to chat with other students and professors who’ll motivate you, argue with you, and encourage you to move yourself ahead.
The average professor spends just a small percentage of his day in front of a big class; he spends most of his day helping individuals or tiny groups. But most students spend most of their days in the big classes; just a few take the opportunity to chat with the professor one-to-one or in small groups. That’s why the typical student says “most of the classes I take are big” while the typical professor says “most of the classes I teach are small.” Example:
At Dartmouth College I did statistics proving the average student spent most of his time in huge classes, while the average professor spent most of his time in tiny classes, leading to wildly different perceptions of what the “average” student-faculty ratio was.
In many colleges, students complain the professors are cold and unapproachable. On the other hand, the professors complain that not enough students come visit the professors during the professors’ office hours. When students fail, the students therefore blame the professors (for being unapproachable), while the professors blame the students (for not approaching).
If you’re a student, remember that you (or your parents) spend lots of money on college: make sure you get your money’s worth!
Ask professors lots of questions (during class or privately), interact with your classmates, take advantage of the many cultural events on campus, and do whatever else you can to make your experience more worthwhile than just reading textbooks you could have bought for a tenth of the price of a college education.
Groucho Marx said this in Horsefeathers:
Let’s tear down the dormitories!
The students can sleep where they’ve always slept: in the classroom.
W.H. Auden said:
A professor is a person who talks in someone else’s sleep.
Dave Barry gave this advice to students:
Memorize things, then write them down in little exam books, then forget them. If you fail to forget them, you become a professor and must stay in college the rest of your life.
To get good grades on your English papers, never say what anybody with common sense would say.
Anybody with common sense would say Moby Dick’s a big white whale, since book’s characters call it a big white whale many times. So in your paper, say Moby Dick is actually the Republic of Ireland. Your professor, who’s sick to death of reading papers and never liked Moby Dick anyway, will think you’re enormously creative. If you can regularly come up with lunatic interpretations of simple stories, major in English.
If philosophers were honest, they’d call themselves “fullosophers” — since when they give their arguments, the audience usually thinks, “You’re full of it!”
Will philosophy disappear?
The British philosopher Bertrand Russell was being interviewed by the BBC (British Broadcasting Corporation), when he made the comment that most “philosophical” problems eventually become “scientific” problems. Examples:
The question of whether matter is infinitely divisible (able to be divided into smaller and smaller particles, without reaching any limit) was originally a “philosophical” problem argued by Greek philosophers but eventually became a “scientific” problem analyzed by physicists.
The question “What is happiness” used to be a philosophical problem but has become a question of psychology, psychiatry, and biochemistry.
The interviewer asked him, “Does that mean philosophy will disappear?” Bertrand Russell replied, “Yes.”
Why become a philosopher?
When Bertrand Russell was young, he was a mathematician and the world’s most famous logician. But when he saw dead bodies come back from World War 1, he switched his career to philosophy, because he felt math wasn’t relevant to the most important problems of living. He said:
The “timelessness” of mathematics consists just in the fact that mathematicians don’t talk about time.
Back in the 1970’s, the basements of Wesleyan University’s dorms were connected by tunnels, upon whose walls the students wrote philosophy. Sample:
“To do is to be.” — Socrates
“To be is to do.” — Sartre
“Do be do be do.” — Sinatra
There’s nothing to do on a rainy day in Kansas;
but it never rains, so you never get the chance.
Don’t let your failures discourage you. Learn from them. They’ll also help you appreciate your later successes more. Truman Capote said:
Failure is the condiment that gives success its flavor.
Remember this famous saying:
If at first you don’t succeed? Try, try again!
But also heed W.C. Field’s elaboration:
If at first you don’t succeed? Try, try again!
Then stop. No use being a damn fool about it!
Success versus happiness
Don’t confuse “success” with “happiness.” Actress Ingrid Bergman said:
Success is getting what you want.
Happiness is wanting what you get.
My philosophy of life can be summarized in 3 sentences:
Life’s an adventure.
Enjoy the ride.
Watch out for the curves.
Here it is in one sentence:
Have fun, but be careful.
The Internet offers this inspiring tale:
A farmer’s donkey fell into a well. The animal cried piteously for hours as the farmer tried to figure out what to do.
Finally, he decided that since the donkey was old and the well needed to be covered up anyway, it wasn’t worth the trouble to retrieve the donkey.
He invited his neighbors to come help him. They all grabbed shovels and began to throw dirt into the well.
The donkey realized what was happening and whined horribly. But then he suddenly quieted down. A few shovelfuls later, the farmer looked down the well and was astonished to see that for every shovelful of dirt hitting the donkey’s back, the donkey would shake it off and step up onto it. Soon everyone was amazed as the donkey stepped up over the well’s edge and trotted off.
Life is going to shovel dirt on you, all kinds of dirt. The trick to getting out of the well is to shake off the dirt and take a step up.
Each of our troubles is a steppingstone. We can emerge from the deepest wells just by persevering. Never giving up! Shake it off and take a step up!
Remember these 5 simple rules to be happy:
Free your heart from hatred. Free your mind from worries. Live simply. Give more. Expect less.
By the way, the donkey kicked the shit out of the bastard who tried to bury him. Moral:
When you try to cover your ass, it always comes back to get you.
Why did the chicken cross the road?
According to the Internet, these thinkers would give straight answers.…
Traditional answer: To get to the other side.
Ernest Hemingway: To die. In the rain. Alone.
Walt Whitman: To cluck the song of itself.
Robert Frost: To cross the road less traveled by.
Mae West: I invited it to come up and see me sometime.
Captain Kirk: To boldly go where no chicken has gone before.
Jack Nicholson: ’Cause it fucking wanted to. That’s the fucking reason.
Timothy Leary: That’s the only kind of trip the Establishment would let it take.
Jerry Falwell: The chicken was gay, going to the “other side.” If you eat it, you’ll get gay.
Moses: God told the chicken, “Thou shalt cross the road.” There was much rejoicing.
Zsa Zsa Gabor: To get a better look at my legs, which — thank goodness — are good, dahling.
Martin Luther King: It had a dream where all chickens can freely cross without their motives questioned.
Sigmund Freud: The chicken was female and envied the crosswalk-sign pole as a phallic symbol.
So would these scientists.…
Sir Isaac Newton: Chickens at rest tend to stay at rest. Chickens in motion tend to cross the road.
Darwin: Chickens, over centuries, have been naturally selected to cross roads.
Hippocrates: Because of an excess of light pink gooey stuff in its pancreas.
Gregor Mendel: To get various strains of roads.
These thinkers would deny that the chicken simply crossed the road:
Joseph Conrad: Mistah Chicken, he dead.
Emerson: It didn’t cross the road: it transcended the road.
Mark Twain: The news of its crossing has been greatly exaggerated.
John Cleese: This chicken is no more. It’s a stiff, an ex-chicken. Ergo, it didn’t cross the road.
Saddam Hussein: Its rebellion was unprovoked, so we justifiably dropped 50 tons of nerve gas on it.
Albert Einstein: Did the chicken really cross the road, or did the road move beneath the chicken?
These thinkers would investigate further:
Jerry Seinfeld: Why the heck was this chicken walking around all over the place anyway?
George W. Bush: We just want to know whether the chicken is on our side of the road or not.
Sherlock Holmes: Ignore the chicken that crossed; the answer lies with the chicken that didn’t.
Oliver Stone: Who else was crossing and overlooked, in our haste to observe the chicken?
These thinkers would raise questions.…
Bob Dylan: How many roads must one chicken cross?
Shakespeare: To cross or not to cross, that is the question.
John Lennon: Imagine all chickens crossing roads in peace.
Dr. Seuss: Did the chicken cross the road? Did he cross it with a toad?
Voltaire: I may not agree with the chicken, but I’ll defend to death its right to cross.”
These thinkers would brag about technology:
Al Gore: I invented the chicken and the road. The crossing serves the American people.
Bill Gates: My eChicken 2.0 also lays eggs, files documents, and balances your checkbook.
These thinkers think the others are too long-winded:
Grandpa: In my day, we didn’t ask why. We were told the chicken crossed. That was that!
Fox Mulder: You saw it with your own eyes! How many must cross before you believe?
Alfred E. Neumann: What? Me worry?
Colonel Sanders: I missed one?
Which of those thinkers is closest to your own philosophy?
The most misspelled word in the English language is “psychology.” That’s how most people spell it, but that spelling is wrong! You should spell it “sighcology,” since it’s the study of why people sigh.
It studies what makes people sad or glad (the meaning of happiness!) and what motivates people to do things and keep on living.
It also studies why people act crazy. At Dartmouth College, the course in “Abnormal Psychology” is nicknamed “Nuts & Sluts.”
Many psychology experiments are performed on rats before being tried on people. That’s why at Northwestern University, the course in “Psychology” is nicknamed “Ratology.”
Trick the professor
According to psychology, if you make your victim happy when he’s performing an activity, he’ll do that activity more often. That’s called reinforcement.
At Dartmouth College, a psychology professor was giving a lecture about that, but his lecture was too effective: his students secretly decided to make him the victim! They decided on a goal: make him teach while standing next to the window instead of the blackboard. Whenever he moved toward the window, they purposely looked more interested in what he was saying; whenever he returned to the blackboard, they purposely looked more bored. Sure enough, they finally got him to give all lectures from the window! They’d trained their human animal: the classroom was his cage; his class became a circus. When the students finally told him what they’d done, he was so embarrassed!
Okay, kids, try this with your teachers! Pick a goal (“Let’s make the teacher lecture from the back of the room while he does somersaults”) and see how close you can come to success!
But actually, with an experiment like this, everybody wins, since the students have to keep watching the teacher to find out when to pretend to look interested. That means the students can’t fall asleep in class. If one of the students secretly snitches to the teacher about what’s going on, the teacher should play along with it, because the teacher knows that the students will be watching the teacher’s every move while the game continues. A rapt, excited audience is exactly what the teacher wants!
If you want to do experiments on humans, to determine which social settings and drugs are most effective, make sure that neither the experimenters nor the patients know which patients got which treatments, until after the experiment is over. If the experimenters or patients know too much too soon, they’ll bias the results of the tests.
The most accurate kind of experiment is called double-blind: neither the experimenters nor the patients know who gets which treatment; the experimenters & patients are both blind to what’s going on, until after the test. For example, to accurately test whether a pill is effective, it’s important that neither the experimenters nor the patients know which patients got the real pills and which patients got the placebos (fake pills) until after the experiment is over.
Here are 3 famous examples proving that double-blindness can be essential to accuracy.…
Clever Hans In the late 1800’s, a Berlin math professor named Wilhelm Von Osten believed animals could become as smart as humans. He tried to teach a cat and and bear to do arithmetic but failed. Then he tried to teach a horse to do arithmetic and seemed to succeed, after training the horse for just 2 years. He called the horse “Clever Hans.”
The horse correctly answered questions about arithmetic — and also about advanced math, German, political history, and classical music. Whenever Wilhelm asked the horse a question whose answer was a small integer (1, 2, 3, 4, 5, etc.), the horse would tap his foot the correct number of times, even if the question was complicated, such as:
“What’s the square root of 16?” (The answer is 4.)
“If you add 2/5 to ½, what’s the total’s numerator?” (The answer is 9.)
“How many people in the audience are wearing hats?”
Wilhelm really believed he’d taught the horse to do advanced thinking. He and his horse became famous celebrities.
In 1904, Germany created a scientific committee to determine whether the horse was really smart or whether the whole thing was just a hoax. The committee included two zoologists, a psychologist (Carl Stumpf), a horse trainer, and a circus manager. The committee concluded that the horse really was smart, since it could answer questions asked by audience members (who’d never seen the horse before) even when Wilhelm Von Osten and his staff weren’t present.
But one of Carl Stumpf’s students, Oskar Pfunkst, experimented on the horse further. Oskar discovered that if the interrogator (the person interrogating the horse) didn’t know the right answer himself, the horse didn’t know the answer either. He finally discovered how the horse got the right answer: the horse looked at the interrogator’s body language. After an interrogator asked the horse a question, the interrogator had a natural human tendency to look intensely at the horse’s leg, lean forward to look at it, and be tense until horse tapped the correct number of times. Then the interrogator relaxed a bit, unconsciously. The horse noticed that relaxation and stopped tapping.
Moral: when testing the intelligence of a horse — or anything else — it’s important that the experimenter (interrogator) not be biased by expecting an outcome, since the patient (horse) can be influenced by that bias.
Hawthorne In the 1920’s and 1930’s, psychologists tried some experiments in Western Electric’s “Hawthorne” factory in Chicago.
First, psychologists tried improving the lighting, by making the place brighter. As expected, the workers’ productivity increased.
But then, after a while, the psychologists tried another experiment: they lowered the lighting. Strange as it seems, lowering the lighting made productivity increase further!
It turned out that what made the workers productive wasn’t “more lighting”; it was “attention and variety.” Anything that made the workers’ life more interesting and less monotonous made productivity increase. Also, perhaps more important, workers work harder when they know they’re being watched!
The same thing happened when the “rest breaks” and pay were changed: the act of change itself made productivity increase, regardless of whether the change was intended for better or worse.
That’s called the Hawthorne Experiment. Moral: workers (and patients) do better when they know they’re watched and cared about, even if the conditions are worse. So if you try a new technique (or pill) that seems to be successful, the success might be just because the patients know they’re being watched, not because your technique itself is really good.
Bloomers In the 1960’s, Robert Rosenthal and Lenore Jacobson had psychologists sit in the back of 18 elementary-school classrooms, watch the students, and then tell the teachers that certain kids were “intellectual bloomers” who would probably do better and improve a lot. Then the psychologists left. At the end of the year, the psychologists came back, gave the kids IQ tests and and, sure enough, the kids that had been called “intellectual bloomers” improved more than the other kids and were also “better liked,” even though those kids had actually been picked at random! That’s because the teacher treated those kids differently, after hearing they were “intellectual bloomers.”
They repeated the experiment with a welding class: they told the teacher that certain students in the welding class were “high aptitude.” Sure enough, those students scored higher on welding exams, learned welding skills in about half as much time as their classmates, and were absent less often than classmates, even though those students had actually been picked at random.
In an earlier test, they told psychology students that certain rats were “bright.” Sure enough, the “bright” rats learned to run through mazes faster, even though those rats had actually been picked at random.
Moral: if you expect more of a person (or rat), you’ll tend to give that individual more helpful attention, so the individual will live up to those expectations. Second moral: if you (or teachers) expect a certain outcome, it will happen, just because you expected it.
Whenever you feel bummed out, take a trip — for a month or a week or a day — or at least take a walk around the block or watch TV or read a newspaper or book. When you see other people acting out their own lives and ignoring yours, you’ll realize that your momentary personal crisis is unimportant in the grand scheme of life.
So what if a close acquaintance thinks badly of you? There are billions of other people in the world who don’t care, who don’t have any opinion of you at all, know nothing about what you’ve done, and don’t care about it. All they care about is that you act like a nice person now.
Act nice, and the world will grow to love you. If your little world temporarily hates you and you don’t want to deal with it, explore a new world: take a trip!
More suicides occur on Sunday than any other day of the week. That’s because Sunday’s the only day when Americans have enough time to ponder how meaningless their lives are.
The best cure for suicidal thoughts is: Monday! Go back to work, get reinforced every hour for your accomplishments, and keep yourself busy enough to avoid introspection.
Every day, I think about killing myself, but the main thing stopping me is curiosity. I’m a news junkie with a sci-fi bent: I want to know what will happen to the world tomorrow, and if I kill myself I won’t find out!
The old news anchors — Peter Jennings, Tom Brokaw, and Dan Rather — saved my life. They gave me a reason for living: to find out what stupid things they’d be forced to say the next day. Now that they’re gone, along with the relevance of broadcast TV news, I get my life force by reading The Wall Street Journal and the Reuters news feed on Yahoo’s Website.
When I see the daily newsreels of horrors around the world, I remember why God created evil: to make us feel better, by knowing that other people are even worse off, and we’re so lucky not to be them!
Learn from your miseries and become a better person.
If your travails are long and tough
And your rewards are few,
Remember that the mighty oak
Was once a nut like you.
But if you nevertheless decide to kill yourself, here’s a suggestion about the best way to do it:
A local newspaper here ran an article whose headline said “Police kill suicidal man.” The police in Henniker NH got a call saying a relative (a man in his 40’s) was depressed (because he was fired from a bookstore) and seemed suicidal (judging from what he phoned to his 5-year-old estranged daughter), so the police went to his house. Nobody responded to their knocks, so they forcibly entered and found him. They asked him if he was okay. Instead of replying, he walked near a rifle, picked it up, and aimed it at a policeman, so they shot him in self-defense. Since his gun was loaded, the police were exonerated.
Hey, that’s a clever way to commit suicide: get the police to do the killing for you! But plan carefully, to make sure you don’t accidentally shoot the police when they shoot you.
When you get old, your brain might have trouble working properly: you’ll lose your memory, be senile, act demented. The most common form of dementia is Alzheimer’s disease, where you forget the purpose of things.
Elderly people are scared that they might be getting demented. Here are some quick tests:
If you forget where your keys are, that’s normal; but if you forget what your keys are for, you’re demented.
If you were ironing your clothes but forget where you put your iron, that could be normal; but if you put your iron in the freezer, that’s demented.
If you put clean dishes into the dishwasher, you’re probably either demented or Chinese. (The Chinese often use their dishwashers just as storage racks.)
British researchers have discovered this quick test for pre-Alzheimer’s (having an Alzheimer-damaged brain even through you don’t act crazy yet): within one minute, name as many fruits & vegetables as you can think of. (You can name fruits or vegetables or a mix.) If you’re normal, you’ll name at least 20; if you have pre-Alzheimer’s (or Alzheimer’s), you’ll name no more than 15 (because your mind will repeatedly mull over the first 15 and have difficulty breaking loose to go beyond them). As for myself, I score about 17, so I guess I’d better be careful!
One reason why the elderly seem demented is that they have trouble focusing on the task at hand. My crazy relative passed me this e-mail from the Internet:
Do you have AAADD?
They’ve finally found a diagnosis for my condition. Hooray! I’ve recently been diagnosed with AAADD — Age-Activated Attention-Deficit Disorder. Here’s how it goes.…
I decide to wash the car; I start toward the garage and notice the mail on the table. Yeah, I’m going to wash the car; but first I’d better go through the mail. I lay the car keys on the desk, discard the junk mail, and notice the garbage can is full. I’d better take it out; but since I’m going to be near the mailbox anyway, I should pay these bills first. Where’s my checkbook? Oops, there’s just one check left. My extra checks are in my desk. I’d better get them.
Oh, there’s the Coke I was drinking. I’ll look for those checks; but first I must put my Coke farther from the computer — or maybe I’ll pop it into the fridge to keep it cold awhile. As I head towards the kitchen, flowers catch my eye: they need water. I set the Coke on the counter and… Oh! There are my glasses! I was looking for them all morning! I’d better put them away first. I fill a container with water, head for the flowerpots, and… Aaaaaagh! Someone left the TV remote in the kitchen. We’ll never think to look in the kitchen tonight when we want to watch TV, so I’d better put it back in the family room where it belongs. I splash some water into the pots and onto the floor, throw the remote onto a cushion on the sofa, head back down the hall, and try to figure out what I was going to do.
End of day: the car isn’t washed, the bills are unpaid, the Coke is still on the kitchen counter, the flowers are half-watered, the checkbook still has just one check in it, and I can’t find my car keys!
When I try to figure out why nothing got done today, I’m baffled because I know I was busy, all day long! I realize this is a serious condition and I’ll get help, but first I think I’ll check my e-mail.…
Please send this to
everyone you know because
I don’t remember whom I’ve sent this to!
But please don’t send it back to me or I might send it to you again!
Here are quick thoughts on several psych topics.
The 2/3 solution During the 1960’s, when I was learning to be a clinical psychologist, the professor told us that 2/3 of all psychological problems resolve themselves, without help — though a nudge sure helps!
Bored people grow up. Fascinating people grow down: they reconnect with their inner child.
Just because you’re paranoid doesn’t mean they’re not out to get you.
Habits In a psychology lecture about habits, the professor said he knew a bishop who dispensed advice to priests. To the question, “Is it okay to kiss a nun?” the bishop replied:
It’s okay to kiss a nun once in a while,
but don’t get in the habit.
Now yesterday is history.
Tomorrow is a mystery.
Today is God’s great gift to you:
That’s why it’s called “the present,” too!
That’s my edited version of the closing poem at a one-woman show/seminar: a PBS special called “The Joy of Stress” by humorous therapist Loretta LaRoche. The poem means this:
Don’t fret about the past, for you can’t change it.
Don’t fret about the future: can’t explain it!
So calm down and savor
The moment you’re in.
It’s God’s little favor:
Come taste every flavor!
Now Loretta has a new presentation, called “Stop Global Whining.”
Test about life
Here’s a multiple-choice test about life.
Laugh, and the world laughs with you.
Which completion is most correct?
Cry, and you cry alone.
Cry, and you get a loan.
Cry, and the world laughs at you.
Cry, and your dad says to shut up.
Cry, and you win the Academy Award.
Cry, and you get on a Jerry Springer talk show.
Cry, and your lover pities you and marries you.
Why do people act strangely? Sometimes it’s because their strangeness makes them feel unique & powerful.
They call me Mr. Stupid
Because I am so cool!
I put my pants on backwards —
Just love to break the rules!
I fall in love with any girl
Who dares to tell me “no,”
Since any girl who dislikes me
Must really be a show!
Though I’m called Mr. Stupid,
I never really mind,
Since I know how behind my back
They whisper I’m so fine!
Sticks and stones may break my bones
But names will never hurt.
Though maybe stupid, I’m unique.
The other folks are dirt.
Folks do not mind my joyous brags.
In fact, they even laugh.
Each time I tell a dirty joke,
They offer me a bath.
Stupidity is wonderful
When I am in control.
I may be just a character,
But on my bridge, the troll!
During the Christmas season, many people feel stressed. The Internet recommends these Christmas carols for the psychologically challenged:
Diagnosis Song title
Muliple-personality disorder We 3 queens disoriented are
Amnesia I (think) I’ll be home for Christmas?
Narcissist Hark the herald angels sing about me
Paranoid Santa Claus is coming to town to get me
Tourette’s syndrome Chestnuts… grrr! roasting on… bite me!
Seasonal-affective disorder Oh the weather outside is frightful, so frightful
Schizophrenic Do you hear what I hear: the voices, the voices?
Depressed Silent night, holy night, all is calm, all is pretty lonely
Agoraphobic I heard the bells on Christmas Day but wouldn’t leave my house
Alzheimer’s disease Walking in a winter wonderland, miles from my house, in my bathrobe
Social-anxiety disorder Have yourself a merry little Christmas while I sit here and hyperventilate
Passive/aggressive On the first day of Christmas, my true love gave to me then took it all away, so I pouted for a week to teach that ass a lesson
Bipolar disorder, manic episode Deck the halls and walls and house and lawn and streets and stores and office and town and cars and buses and trucks and trees and fire hydrants…
Obsessive-compulsive disorder Jingle bells, jingle bells,
jingle bells, jingle bells, jingle bells,
jingle bells, jingle bells, jingle bells, jingle bells, jingle bells…
Autistic Jingle bell rock and rock and rock and rock…
Borderline personality disorder You better watch out, I’m gonna cry, I’m gonna pout, maybe say why
Borderline personality disorder 2 Thoughts of roasting in an open fire
Antisocial-personality disorder Thoughts of roasting you on an open fire
Oppositional-defiant disorder “You better not cry” “Oh yes, I will” “You better not shout” “I can if I want to” “You better not pout” “Can if I want to” “I’m telling you why” “Not listening” “Santa Claus is coming to town” “No, he’s not!”
Oppositional-defiant disorder 2 I saw Mommy kissing Santa Claus, so I burned down the house
Attention-deficit disorder We wish you… hey look! It’s snowing!
Attention-deficit disorder 2 Silent night, holy… oooh, look at the froggy! Can I have a chocolate? Why is France so far away?
Attention-deficit/hyperactivity All I want for Christmas is everything, and I want it now!
Psychologists invent ways to test your personality. Here’s my own test: are you more like me (Russ) or my wife (Donna)? Are you a “Donna” type (emotional) or a “Russ” type (logical)?
Donna eats whatever tastes good.
At home, Russ eats just what’s “healthy” (but he indulges at restaurants).
When offered chicken, Donna chooses dark meat (because it’s tastier).
Russ chooses white meat (because it’s healthier, since it has less fat).
To figure out how to install and use a new product, Donna guesses.
Russ reads the instructions.
Donna likes to take photos (to preserve the memories).
Russ doesn’t bother.
Donna is warm to relatives and loves to spend time with them.
Russ has less time for relatives; he’s under time pressure from work.
Donna takes her shower in the evening, to feel better while dreaming.
Russ takes his shower in the morning, to feel better while working.
In the summer, Donna likes to turn the air conditioner on, for comfort.
Russ likes to turn the air conditioner off, to save money.
In the winter, Donna likes to turn the furnace on, for comfort.
Russ likes the turn the furnace off, to save money.
Donna sees doctors and dentists just when things hurt.
Russ gets regular checkups (though just occasionally, to reduce expense).
Donna takes cars to repair shops just when cars break.
Russ maintains cars regularly (according to schedule).
Donna believes the elderly should dye their hair (to look younger).
Russ believes in letting the gray show (to look natural and truthful).
Donna rushes through most tasks, to dispose of them quickly.
Russ does things more carefully — and finishes them too late.
Donna gets up early, to start her day energetically.
Russ stays up late to finish things, because he’s always behind.
Donna believes in being tactful, even if that means fibbing a little.
Russ believes in being frank, even if that means breaking a secret.
Donna says doctors should hide bad news from patients, to preserve hope.
Russ says doctors should tell the truth, so patients can act wisely.
When driving alone, Donna turns the radio on, to create fun or learn.
Russ turns the radio off, so he can concentrate on driving and planning.
Donna believes in alternative medicine, such as (herbs.
Russ believes in traditional medicine, just pills approved by the A.M.A.
Donna throws out newspapers immediately, to reduce clutter.
Russ hoards newspapers, to avoid losing information.
Donna worries about security after retirement.
Russ believes life is unpredictable, so he focuses on just this year.
Decide whether you’re more like Donna or Russ. Then invent your own test, containing your own name and a friend’s.
According to the Donna-versus-Russ test, Donna differs from me (Russ) in many ways. We stay married because our differences are smaller than what we have in common:
similar tastes in music, movies, furniture, and clothing
enjoy keyboard instruments more than guitar
skilled at math, logical reasoning, and teaching
love reading & studying, like to explore different cultures
like to spend more time in cultural cities than quiet countryside
kind of cheap, don’t pursue luxury or name brands
like to eat at inexpensive restaurants
naively trust other people, get surprised and upset at cheating
sex is not a priority
not very optimistic; a little stubborn
What do you and your friends have in common? List the reasons you stay friends. Share that list with your friends: you’ll appreciate each other even more!
Mental illness strikes us all, eventually. During one of my bouts, I wrote this ditty to cheer myself up:
I’m mentally ill.
My mind’s made of swill.
I’m king of the hill
I hope that someday
Life turns out okay,
But now I’m in bed
Extract me from here.
Have you got some beer?
Can you give me cheer,
Just wish I were dead.
Come please shoot my head.
What happens? I dread
Take me away
The most famous song about mental illness is They’re coming to take me away, recorded in 1966 by Jerry Samuels (whose stage name is Napoleon XIV). I’ve recast it here as a poem:
Remember when you ran away?
Upon my knees, I begged “Don’t leave
Or else I’ll go beserk.”
You left me anyhow, and then
The days got worse and worse, and now
I’ve lost my mind. You jerk!
So now they’re taking me away
To farms (with beauty all the time
And men in clean white coats).
When I said losing you would make
Me flip my lid, you thought it was
A harmless joke. You laughed.
You know you laughed. I heard you laugh.
You laughed and laughed, and then you left;
And now I’ve gone quite mad.
So now they’re taking me away
To Happy Home with trees and birds,
Where people twiddle thumbs.
In movie-making courses, students create movies using Jerry’s original recording as the scary soundtrack. Here’s an example:
Chemists are mixed up.
To discover how good a chemistry you are, see how long you take to solve these chemistry puzzles:
1.. A chemist noticed a certain reaction took 80 minutes whenever he was wearing a green necktie, and the same reaction took an hour and twenty minutes whenever he was wearing a purple necktie. Why?
2. If you drop a steel ball, would it fall faster through water at 20 degrees Fahrenheit or water at 60 degrees?
To torture kids, ask them those puzzles. If you can’t solve those problems yourself, ask your friends, until you find a friend who’s smart — and kind enough to tell you the answers.
Or, if you’re lazy, read the answers here:
1. 80 minutes is the same as an hour and twenty minutes.
2. At 20 degrees Fahrenheit, water is ice, which would slow the ball.
The first puzzle comes from Martin Gardner’s book, Mathematical Puzzles. The second puzzle can be found in many sources, such as S. Harold Collins’ book, Mastering the Art of Substitute Teaching.
To have more fun, get those books!
Many people worry that our food contains too many chemicals. They say our food should contain no chemicals at all.
With that worry in mind, concerned chemists have created a Website called DHMO.org, which warns about the dangers of a chemical called DHMO, which is dihydrogen monoxide. Examples of DHMO’s dangers:
Many people have died from imbibing too much DHMO. Even just a thimbleful, up your nose, can kill you!
Unfortunately, DHMO is very prevalent. It’s the main component in acid rain. DHMO spreads very easily. Many evil industries pour DHMO into rivers & streams.
DHMO is used in the distribution of pesticides. Trying to wash off your fruits & vegetables doesn’t remove the DHMO. The cells of most plants and animals are now full of DHMO — and so is your food! Horribly, DHMO is added to many junk foods!
DHMO can be a solid, liquid, or gas. Your skin can get badly burned by contact with solid or gaseous DHMO. Your whole life can disappear — you die! — when you’re immersed in liquid DHMO.
DHMO can destroy electrical circuits. It can even render ineffective your car’s brakes!
DHMO is used by many criminals, for many purposes. To make matters worse, DHMO is highly addictive: to get access to a hit of DHMO, cultures around the world have gotten so desperate that they’ve even resorted to violence & wars. Whole communities have been destroyed by being flooded with DHMO.
DHMO can sneak up to you without warning, since it’s odorless and colorless. The atomic chemicals that make up DHMO are in many other deadly substances, such as explosive nitroglycerin and poisonous cyanide.
Few laws limit DHMO. In 2002, a radio news show reported that Atlanta’s water system was contaminated with DHMO, but Atlanta’s water department replied that Atlanta’s water contained no more DHMO than permitted by law.
When told of DHMO’s dangers, 86% of Americans believe the U.S. government should ban DHMO.
DHMO (dihydrogen monoxide) is also known as dihydrogen oxide, hydrogen hydroxide, hydronium hydroxide, and hydric acid. Dihydrogen monoxide’s chemical symbol is H2O. That chemical is also called water.
The site’s purpose is to laugh at Americans who fear anything that sounds chemical. Look again at those examples of DHMO’s dangers, and see how they’re true about the dangers of… water!
In April 1988, William DeBuvitz wrote about the discovery of administratium. Here’s a summary of what he and later researchers have reported:
Chemists have finally discovered the heaviest element known to science. The element, administratium, has no protons or electrons, so its atomic number is 0; but it has 1 neutron, 125 assistant neutrons, 75 vice-neutrons, and 111 assistant vice-neutrons, giving it an atomic mass of 312. These 312 particles are held together by a force involving the continuous exchange of meson-like particles (called morons) and surrounded by vast quantities of lepton-like particles (called peons).
Administratium is inert (since it has no electrons) but can be detected chemically, since it impedes every reaction it contacts: a tiny amount of administratium can make a reaction take 4 days that would normally take less than a second.
Administratium has a half-life of 3 years, after which it doesn’t decay but instead undergoes a reorganization in which assistant neutrons, vice-neutrons, and assistant vice-neutrons exchange places. Administratium’s mass increases over time, since each reorganization makes some morons become neutrons, forming new isotopes, called isodopes. The moron promotion makes chemists think administratium forms spontaneously whenever morons reach a certain concentration, called a critical morass.
Administratium occurs naturally in the atmosphere but concentrates at certain points (such as government agencies, large corporations, and universities). It usually appears in buildings that are new, fancy, and well-maintained.
Since administratium is toxic at any concentration level, it destroys any productive reaction. We’re trying to control administratium, to prevent irreversible damage. Help stop this deadly element from spreading!
Back around 1950, chemists tried to prove heaven’s hotter than hell. The proofs gradually got more sophisticated. A 1972 article in Applied Optics gave this argument:
Revelations 21:8 says hell is a “lake burning with fire & brimstone,” so hell’s temperature is below the boiling point of brimstone (sulfur), which is 444.6°C.
Isaiah 30:26 says heaven is full of intense light, which generates lots of heat energy, 525°C according to our calculations.
So heaven is hotter than hell.
The full article is at LhuP.edu/~dsimanek/hell.htm.
This bonus question appeared on a chemistry test:
Is hell exothermic (giving off heat) or endothermic (absorbing heat)?
Prove your answer.
The professor expected the students to use Boyle’s law (which says compressing a gas makes it hotter). According to the tale, the top student gave this answer:
First, we must discover how hell’s mass is changing, so we need to know how fast souls enter hell and how fast they leave.
Once a soul gets to hell it won’t leave, but how many souls enter hell? According to most religions, if you’re not a member of that religion, you go to hell. Since there are many religions but no single person belongs to more than one, all people and their souls go to hell; so in light of birth and death rates, the number of souls in hell will increase exponentially.
Boyle’s Law says that for hell’s temperature and pressure to remain constant, hell’s volume must expand proportionately as souls are added. That gives two possibilities.…
#1: If hell expands slower than souls enter hell, hell’s temperature and pressure will increase until all hell breaks loose.
#2: If hell expands faster than souls enter hell, hell’s temperature and pressure will drop until hell freezes over.
So which is it?
If we accept the postulate given me by Teresa during my freshman year that “It will be a cold day in hell before I sleep with you” and realize I slept with her last night, hell’s already frozen over, so hell is exothermic and #2 is true. Since hell’s frozen over, it isn’t accepting more souls and is extinct, leaving just heaven, thereby proving the existence of a divine being, which explains why last night Teresa kept shouting “Oh my God!”
In 1959 Tom Lehrer wrote a song called The Elements, where he sang the names of the 102 chemical elements discovered so far, to the tune of the Major-General’s Song from Gilbert & Sullivan’s Pirates of Pinzance. Here are 3 videos about it:
Tom sings, with element photos: YouTube.com/watch?v=SmwlzwGMMwc
Tom sings, with periodic table: YouTube.com/watch?v=zGM-wSKFBpo
Harry Potter’s Daniel Radcliffe sings: YouTube.com/watch?v=rSAaiYKF0cs
Warning: for the first video’s Web address, the letter after w is a lower-case L.
An improved song, called The New Periodic Table Song, gives 118 elements listed in correct order (by atomic number), sung to the tune of Jacques Offenbach’s Gaîté Parisienne. It’s at:
Fast version: YouTube.com/watch?v=VgVQKCcfwnU
Slow version: DailyMotion.com/video/x2q1nnr
Those singers also made a song about which scientist to become: is it better to be a physicist, chemist, biologist, or mathematician? Here them sing their arguments at:
Physics is phunny.
Physics for poets
To help liberal-arts students understand physicists such as Newton and Einstein, physicists teach a course called “Physics for Poets.” The whole course is summarized in 4 sentences:
Physics rule Poetic meaning
Newton’s theory of gravitation The earth sucks.
Newton’s third law of motion Every jerk creates his equal opponent.
Einstein’s E=MC² A small matter can mushroom into a big whoopee.
Einstein’s theory of relativity Your views are influenced by your relatives.
Back in 1958, Reader’s Digest published a tale by Alexander Calandra about a barometer test. Over the years, he and others embellished the tale. These new fancier versions are fictional but fun. Here’s an example:
A physics test said to “Find a height of a tall building by using a barometer.” The professor considered the correct answer to be “Use the barometer to measure the air pressure at the building’s top and the building’s bottom, then analyze the difference.”
But one student gave this cleverer answer: “Put the barometer at the end of a rope, lower the rope from the top of the building, and measure the rope’s length plus the barometer’s length. Or throw the barometer from the top of the building and measure how long the barometer takes to fall. Or compare the length of the building’s shadow to the length of the barometer’s shadow. Or walk up the stairs while you mark, on the walls, how many barometer-heights you had to climb. Or attach the barometer to a rope, swing it like a pendulum, and measure how the swing time at the building’s top differs from the bottom.”
The professor demanded, “Don’t you know the simplest answer?”
The student replied, “Sure! Tell the building’s superintendent you’ll give him the barometer if he tells you the building’s height! That’s the simplest answer. I’m fed up with you professors telling me how I should think!”
In my former life — before I tried to be a writer or a computer guy — I was a mathematician.
Torture your friends by giving them these puzzles about arithmetic.
Apples If you have 5 apples and eat all but 3, how many are left? Kids are tempted to say “2,” but the correct answer is 3.
Birds If you have 10 birds in a tree and shoot 1, how many remain in the tree? Kids are tempted to say “9,” but the correct answer is 0.
Corners If you have a 4-sided table and chop off 1 of the corners, how many corners are left on the table? Kids are tempted to say “3,” but the correct answer is 5.
Lily pads In a lake, a patch of lily pads doubles in size every day. It takes 48 days for the patch to cover the lake. How long would it take for the patch to cover half the lake? Kids are tempted to say “24 days,” but the correct answer is 47 days.
Baseball A bat and a ball cost a total of $1.10. The bat costs $1 more than the ball. How much does the ball cost? Kids are tempted to say “10¢,” but the correct answer is 5¢.
Seven How do you make seven an even number? Remove the “s”.
Eggs Carl Sandberg, in his poem Arithmetic, asks this question:
If you ask your mother for one fried egg for breakfast, but she gives you two fried eggs and you eat both of them, who’s better in arithmetic: you or your mother?
Missing dollar Now that you’ve mastered the easy puzzles, try this harder one:
On a nice day in the 1940’s, three girls go into a hotel and ask for a triple. The manager says sorry, no triples are available, so he puts them in three singles, at $10 each. The girls go up to their rooms.
A few minutes later, a triple frees up, which costs just $25. So the manager, to be a nice guy, decides to move the girls into the triple and refund the $5 difference. He sends the bellboy up to tell the girls of their good fortune and move them into the triple.
While riding up in the elevator, the bellboy thinks to himself, “How can the girls split the $5? $5 doesn’t divide by 3 evenly. I’ll make it easier for them: I’ll give them just $3 — and keep $2 for myself.” So he gave the girls $3 and moved them into the triple.
Everybody was happy. The girls were happy to get refunds. The manager was happy to be a nice guy. And the bellboy was happy to keep $2.
Now here’s the problem: each girl spent $10 and got $1 back, so each girl spent $9. Altogether, the girls spent $9+$9+$9, which is $27, and the bellboy got $2. That makes $29. But we started with $30. What happened to the missing dollar?
Ask your friends that question and see how many crazy answers you get!
Here’s the correct answer:
At the end of the story, who has the $30?
The manager has $25, the bellboy has $2, and the girls have $3.
Adding what the girls spent ($27) to what the bellboy got ($2) doesn’t give a meaningful number. But that nonsense total, $29, is close enough to $30 to be intriguing.
Here’s an alternative analysis:
The girls spent a net of $9+$9+$9, which is $27.
$25 of that went to the manager, and $2 went to the bellboy.
Coins Try this task:
Arrange 10 coins so they form 5 rows, each containing 4 coins.
“5 rows of 4 coins” would normally require a total of 20 coins, but if you arrange properly you can solve the puzzle. Hint: the rows must be straight but don’t have to be horizontal or vertical. Ask your friends that puzzle to drive them nuts.
Here’s the solution:
Draw a 5-pointed star. Put the coins at the 10 corners.
Which type are you?
Here’s Warren Buffet’s favorite saying about math.
There are 3 types of people: those who can count, and those who can’t.
Courses in statistics can be difficult. That’s why they’re called “sadistics.”
Lies Statisticians give misleading answers.
For example, suppose you’ve paid one person a salary of $1000, another person a salary of $100, another person a salary of $10, and two other people a salary of $1 each. What’s the “typical” salary you paid? If you ask that question to three different statisticians, they’ll give you three different answers!
One statistician will claim that the “typical” salary is $1, because it’s the most popular salary: more people received $1 than any other amount. Another statistician will claim that the “typical” salary is $10, because it’s the middle salary: as many people were paid more than $10 as were paid less. The third statistician will claim that the “typical” salary is $222.40, because it’s the average: it’s the sum of all the salaries divided by the number of people.
Which statistician is right? According to the Association for Defending Statisticians (started by my friends), the three statisticians are all right! The most common salary ($1) is called the mode; the middle salary ($10) is called the median; the average salary ($222.40) is called the mean.
But which is the “typical” salary, really? Is it the mode ($1), the median ($10), or the mean ($222.40)? That’s up to you!
If you leave the decision up to the statistician, the statistician’s answer will depend on who hired him.
If the topic is a wage dispute between labor and management, a statistician paid by the laborers will claim that the typical salary is low (just $1); a statistician paid by the management will claim that the typical salary is high ($222.40); and a statistician paid by the arbitrator will claim that the typical salary is reasonable ($10).
Which statistician is telling the whole truth? None of them!
A century ago, Benjamin Disraeli, England’s prime minister, summarized the whole situation in one sentence. He said:
There are 3 kinds of lies:
lies, damned lies, and statistics.
A course in “logic” is a blend of math and philosophy. It can be lots of fun — and also help you become a lawyer.
Beating your wife There’s the old logic question about how to answer this question:
Have you stopped beating your wife?
Regardless of whether you answer that question by saying “yes” or “no,” you’re implying that you did indeed beat your wife in the past.
Interesting number Some numbers are interesting. For example, some people think 128 is interesting because it’s “2 times 2 times 2 times 2 times 2 times 2 times 2.” Here’s a proof that all numbers are interesting:
Suppose some numbers are not interesting. For example, suppose 17 is the first number that’s not interesting. Then people would say, “Hey, that’s interesting! 17 has the very interesting property of being the first boring number!” But then 17 has become interesting! So you can’t have a first “boring” number, and all numbers are interesting!
Surprise test When I took a logic course at Dartmouth College, the professor began by warning me and my classmates:
I’ll give a surprise test sometime during the semester.
Then he told the class to analyze that sentence and try to deduce when the surprise test would be.
He pointed out that the test can’t be on the semester’s last day — because if the test didn’t happen before then, the students would be expecting the test when they walk into class on that last day, and it wouldn’t be a surprise anymore. So cross “the semester’s last day” off the list of possibilities.
Then he continued his argument:
But once you cross “the semester’s last day” off the list of possibilities, you realize the surprise test can’t be “the day before the semester’s last day” either, because the test would be expected then (since the test hadn’t happened already and couldn’t happen on the semester’s last day). Since the test would be expected then, it wouldn’t be a surprise. So cross “the day before last” off the list of possibilities.
Continuing in that fashion, he said, more and more days would be crossed off, until eventually all days would be crossed off the list of possibilities, meaning there couldn’t be a surprise test.
Then he continued:
But I assure you, there will be a test, and it will be a surprise when it comes.
Think about it.
Mathematicians versus engineers
The typical mathematician finds abstract concepts beautiful, and doesn’t care whether they have any “practical” applications. The typical engineer is exactly the opposite: the engineer cares just about practical applications.
Engineers complain that mathematicians are ivory-tower daydreamers who are divorced from reality. Mathematicians complain that engineers are too worldly and also too stupid to appreciate the higher beauties of the mathematical arts.
To illustrate those differences, mathematicians tell 3 tales.…
Boil water Suppose you’re in a room that has a sink, stove, table, and chair. A kettle is on the table. Problem: boil some water.
An engineer would carry the kettle from the table to the sink, fill the kettle with water, put the kettle onto the stove, and wait for the water to boil. So would a mathematician.
But suppose you change the problem, so the kettle’s on the chair instead of the table. The engineer would carry the kettle from the chair to the sink, fill the kettle with water, put the kettle onto the stove, and wait for the water to boil. But the mathematician would not! Instead, the mathematician would carry the kettle from the chair to the table, yell “now the problem’s been reduced to the previous problem,” and walk away.
Analyze tennis Suppose 1024 people are in a tennis tournament. The players are paired, to form 512 tennis matches; then the winners of those matches are paired against each other, to form 256 play-off matches; then the winners of the play-off matches are paired against each other, to form 128 further play-off matches; etc.; until finally just 2 players remain — the finalists — who play against each other to determine the 1 person who wins the entire tournament. Problem: compute how many matches are played in the entire tournament.
The layman would add 512+256+128+64+32+16+8+4+2+1, to arrive at the correct answer, 1023.
The engineer, too lazy to add all those numbers, would realize that the numbers 512, 256, etc., form a series whose sum can be obtained by a simple, magic formula! Just take the first number (512), double it, and then subtract 1, giving a final result of 1023!
But the true mathematician spurns the formula and searches instead for the problem’s underlying meaning. Suddenly it dawns on him! Since the problem said there are “1024 people” but just 1 final winner, the number of people who must be eliminated is “1024 minus 1,” which is 1023, and so there must be 1023 matches!
The mathematician’s calculation (1024-1) is faster than the engineer’s. But best of all, the mathematician’s reasoning applies to any tournament, even if the number of players isn’t a magical number such as 1024. No matter how many people play, just subtract 1 to get the number of matches!
Prime numbers Mathematicians are precise, physicists somewhat less so, chemists even less so. Engineers are even less precise and sometimes less intellectual. To illustrate that view, mathematicians tell the tale of prime numbers.
First, let me explain some math jargon. The counting numbers are 1, 2, 3, etc. A counting number is called composite if you can get it from multiplying a pair of other counting numbers. For example:
6 is composite because you can get it from multiplying 2 by 3.
9 is composite because you can get it from multiplying 3 by 3.
15 is composite because you can get it from multiplying 3 by 5.
A counting number that’s not composite is called prime. For example, 7 is prime because you can’t make 7 from multiplying a pair of other counting numbers. Whether 1 is “prime” depends on how you define “prime,” but for the purpose of this discussion let’s consider 1 to be prime.
Here’s how scientists would try to prove this theorem:
All odd numbers are prime.
Actually, that theorem is false! All odd numbers are not prime! For example, 9 is an odd number that’s not prime. But although 9 isn’t prime, the physicists, chemists, and engineers would still say the theorem is true.
The physicist would say, slowly and carefully:
1 is prime. 3 prime. 5 is prime. 7 is prime.
9? — no.
11 is prime. 13 is prime.
9 must be just experimental error, so we can ignore it. All odd numbers are prime.
The chemist would rush for results and say just this:
1 is prime, 3 is prime. 5 is prime. 7 is prime.
That’s enough evidence. All odd numbers are prime.
The engineer would be the crudest and stupidest of them all. He’d say the following as fast as possible (to meet the next deadline for building his rocket, which will accidentally blow up):
Sure, 1 is prime, 3 is prime, 5 is prime, 7 is prime, 9 is prime, 11 is prime, 13 is prime, 15 is prime, 17 is prime, 19 is prime, all odd numbers are prime!
Every few years, authors of math textbooks come out with new
editions, to reflect the latest fads. Here’s an example, as reported (and
elaborated on) by Reader’s Digest (in February 1996), Recreational
& Educational Computing (issue #91),
John Funk (and his daughter), ABC News Radio WTKS 1290 (in Savannah), and others:
Teaching math in 1960: traditional math
A logger sells a truckload of lumber for $100. His cost of production is 4/5 of the price. What’s his profit?
Teaching math in 1965: simplified math
A logger sells a truckload of lumber for $100. His cost of production is 4/5 of the price, or $80. What’s his profit?
Teaching math in 1970: new math
A logger exchanges a set L of
lumber for a set M of money. The cardinality of set M is 100. Each element is
worth $1. Make 100 dots representing the elements of M. The set C (cost of
production) contains 20 fewer points than set M. Represent the set C as a
subset of set M and answer this question:
what’s the cardinality of the set P of profits?
Teaching math in 1975: feminist-empowerment math
A logger sells a truckload of lumber for $100. Her cost is $80, and her profit is $20. Your assignment: underline the number 20.
Teaching math in 1980: environmentally conscious math
An unenlightened logger cuts down beautiful trees, desecrating the precious forest for $20. Write an essay explaining how you feel about that way to make money. How did the forest’s birds and squirrels feel?
Teaching math in 1985: computer-based math
A logger sells a truckload of lumber for $100. His production costs are 80% of his revenue. On your calculator, graph revenue versus costs. On your computer, run the LOGGER program to determine the profit.
Teaching math in 1990: Wall Street math
By laying off 40% of its loggers, a company improves its stock price from $80 to $100. How much capital gain per share does the CEO make by exercising his options at $80? Assume capital gains have become untaxed to encourage investment.
Teaching math in 1995: managerial math
A company outsources all its loggers. The firm saves on benefits; and whenever demand for its products is down, the logging workforce can be cut back easily. The average logger employed by the company earned $50,000 and had a 3-week vacation, nice retirement plan, and medical insurance. The contracted logger charges $30 per hour. Based on that data, was outsourcing a good move? If a laid-off logger comes into the logging company’s corporate headquarters and goes postal, mowing down 16 executives and a couple of secretaries, was outsourcing the loggers still a good move?
Teaching math in 2000: tax-based math
A logger sells a truckload of lumber for $100. His cost of production is 4/5 of the price. After taxes, why did he bother?
Teaching math in 2005: profit-pumping math
A logger sells a truckload of lumber for $100. His production cost is $120. How did Arthur Anderson determine that his profit margin is $60?
Teaching math in 2010: multicultural math
Un maderero vende un camión de madera para $100. Su coste de producción es $80….
Winston Churchill (who was England’s prime minister) said:
I had a feeling once about Mathematics — that I saw it all. Depth beyond Depth was revealed to me: the Byss and the Abyss. I saw — as one might see the transit of Venus or even the Lord Mayor’s Show — a quantity passing through infinity and changing its sign from plus to minus. I saw exactly why it happened and why the tergiversation was inevitable — but it was after dinner and I let it go.
A colleague passed me this e-mail, forwarded anonymously:
A teacher was arrested because he attempted to board a flight while possessing a ruler, protractor, and calculator. Attorney General Alberto Gonzales believes the man’s a member of the notorious Al-gebra movement. The man’s been charged with carrying weapons of math instruction.
“Al-gebra is a problem for us,” Gonzales said. “Its followers desire solutions by means & extremes and sometimes go off on tangents in search of absolute values. They use secret code names like ‘x’ and ‘y’ and refer to themselves as “unknowns,’ but we’ve determined they belong to a common denominator of the axis of medieval, with coordinates in every country.”
When asked to comment on
the arrest, President George W. Bush said,
“If God had wanted us to have better weapons of math instruction, He’d have given us more fingers and toes.” Aides told reporters they couldn’t recall a more intelligent or profound statement by the President.
In math, the most famous constant is pi, which is roughly 3.14. But another famous math constant is 1089. It’s the favorite constant among math magicians because it creates this trick.…
Write down any three-digit number “whose first digit differs from the last digit by more than 1.” For example:
852 is okay, since its first digit (8) differs from the last digit (2) by 6, which is more than 1.
479 is okay, since its first digit (4) differs from the last digit (9) by 5, which is more than 1.
282 is not okay, since the difference between 2 and 2 is 0.
Take your three-digit number, and write it backwards. For example, if you picked 852, you have on your paper:
You have two numbers on your paper. One is smaller than the other. Subtract the small one from the big one:
Take your answer, and write it backward:
Add the last two numbers you wrote:
Notice the final answer is 1089.
1089 is the final answer, no matter what three-digit number you started with (if the first and last digits differ by more than 1).
Here’s another example:
Take a number: 724
Write it backward & subtract: -427
Write it backward & add: +792
Here’s another example:
Take a number: 365
Write it backward & subtract: 563
Write it backward & add: +891
Yes, you always get 1089!
Proof To prove you always get 1089, use algebra: make letters represent the digits, like this.…
Hundreds Tens Ones
Take a number: A B C
Write backwards: C B A
To subtract the bottom (C B A) from the top (A B C), the top must be bigger. So in the hundreds column, A must be bigger than C. Since A is bigger than C, you can’t subtract A from C in the ones column, so you must borrow from the B in the tens column, to produce this:
Hundreds Tens Ones
A B-1 C+10
C B A
Now you can subtract A from C+10:
Hundreds Tens Ones
A B-1 C+10
C B A
In the tens column, you can’t subtract B from B-1, so you must borrow from the A in the hundreds column, to produce this:
Hundreds Tens Ones
A-1 B-1+10 C+10
C B A
Complete the calculation:
Hundreds Tens Ones
Start with this: A-1 B-1+10 C+10
Subtract this: C B A
Get this result: A-1-C 9 C+10-A
Backwards: C+10-A 9 A-1-C
Get this total: 10 8 9
Don’t burn your arm I call 1089 the “don’t burn your arm” number, because of this trick suggested by Irving Adler in The Magic House of Numbers:
Tell a friend to write a 3-digit number whose first & last digits differ by more than 1. Tell him to write the number backwards, subtract, write that backwards, and add. Tell him to burn the paper he did the figuring on. Put your arm in the ashes. When you take your arm out, the number 1089 will be mysteriously written on your arm in black. (The way you get 1089 to appear is to write “1089” on your arm with wet soap before you begin the trick. When you put your arm in the ashes, the answer will stick to the soap.) The trick works — if you don’t burn your arm.
Variants That procedure (reverse then subtract, reverse then add) gives 1089 if you begin with an appropriate 3-digit number. If you begin with a 2-digit number instead, you get 99.
If you begin with a 4-digit number instead, you get 10989 or
10890 or 9999, depending on which of the 4 digits are the biggest. If you begin
with a 5-digit number, you get 109989 or 109890 or 99099. Notice that the
4-digit and 5-digit numbers — 10989, 10890, 9999, 109989, 109890, and 99099 — are all formed from the number 99 and 1089.
The most amazing math discovery made by Greeks is the Pythagorean theorem. It says that in a right triangle (a triangle including a 90° angle), a²+b²=c², where c is the length of the hypotenuse (the longest side) and a&b are the lengths of the legs (the other two sides). It says that in this diagram —
c’s square is exactly as big (has the same area) as a’s square and b’s square combined.
The Chinese discovered the same truth, perhaps earlier.
Why is the Pythagorean theorem true? How do you prove it?
You can prove it in many ways. The 2nd edition of a book called The Pythagorean Proposition contains many proofs (256 of them!), collected in 1940 by Elisha Scott Loomis when he was 87 years old. Here are the 5 most amazing proofs.…
3-gap proof Draw a square, where each side has length a+b. In each corner of that square, put a copy of the triangle you want to analyze, like this:
Now the square contains those 4 copied triangles, plus 1 huge gap in the middle. That gap is a square where each side has length c, so its area is c².
Now move the bottom 2 triangles up, so you get this:
The whole picture is still “a square where each side has length a+b,” and you still have 4 triangles in it; but instead of a big gap whose area is c², you have two small gaps, of sizes a² and b². So c² is the same size as a²+b².
1-gap proof Draw the same picture that the 3-gap proof began with. You see the whole picture’s area is (a+b)². You can also see that the picture is cut into 4 triangles (each having an area of ab/2) plus the gap in the middle (whose area is c²). Since the whole picture’s area must equal the sum of its parts, you get:
(a+b)² = ab/2 + ab/2 + ab/2 + ab/2 + c²
In this proof, instead of “moving the bottom 2 triangles,” we use algebra. According to algebra’s rules, that equation’s left side becomes a² + 2ab + b², and the right side becomes 2ab + c², so the equation becomes:
a² + 2ab + b² = 2ab + c²
Subtracting 2ab from both sides of that equation, you’re left with:
a² + b² = c²
1-little-gap proof Draw a square, where each side has length c. In each corner of that square, put a copy of the triangle you want to analyze, like this:
The whole picture’s area is c². The picture is cut into 4 triangles (each having an area of ab/2) plus the little gap in the middle, whose area is (b-a)². Since the whole picture’s area must equal the sum of its parts, you get:
c² = ab/2 + ab/2 + ab/2 + ab/2 + (b-a)²
According to algebra’s rules, that equation’s right side becomes 2ab + (b² - 2ba + a²). Then the 2ab and the -2ba cancel each other, leaving you with a² + b², so the equation becomes:
c² = a² + b²
1-segment proof Draw the triangle you’re interested in, like this:
Unlike the earlier proofs, which make you draw many extra segments (short lines), this proof makes you draw just one extra segment! Make it perpendicular to the hypotenuse and go to the right angle:
The original big triangle (whose sides have lengths a, b, and c) has the same-size angles as the tiny triangle (whose sides have lengths x and a), so it’s “similar to” the tiny triangle, and so the big triangle’s ratio of “shortest side to hypotenuse” (a/c) is the same as the tiny triangle’s ratio of “shortest side to hypotenuse” (x/a). Write that equation:
a/c = x/a
Multiplying both sides of that equation by ac, you discover what a² is:
a² = xc
Using similar reasoning, you discover what b² is:
b² = yc
Adding those two equations together, you get:
a² + b² = (x+y)c
Since x+y is c, that equation becomes:
a² + b² = c²
1-segment general proof Draw the triangle you’re interested in, like this:
As in the previous proof, draw one extra segment, perpendicular to the hypotenuse and going to the right angle:
Now you have 3 triangles: the left one, the rightmost one, and the big one.
Since the left triangle’s area plus the rightmost triangle’s area equals the big triangle’s area, and since the 3 triangles are similar to each other (“stretched” versions of each other, as you can prove by looking at their angles), any area constructed from “parts of the left triangle” plus the area constructed from “corresponding parts of the rightmost triangle” equals the area constructed from “corresponding parts of the big triangle.” For example, the area constructed by drawing a square on the left triangle’s hypotenuse (a²) plus the area constructed by drawing a square on the rightmost triangle’s hypotenuse (b²) equals the area constructed by drawing a square on the big triangle’s hypotenuse (c²).
Which proof is the best? The 3-gap proof is the most visually appealing, but it bothers mathematicians who are too lazy to draw (construct) so many segments. (It also requires you to prove the gap is indeed a square, whose angles are right angles, but that’s easy.)
The 1-gap proof uses fewer lines by relying on algebra instead. It’s fine if you like algebra, awkward if you don’t. The 1-little-gap proof uses algebra slightly differently.
The 1-segment proof appeals to mathematicians because it requires constructing just 1 segment, but you can’t understand it until you’ve learned the laws of similar triangles. This proof was invented by Davis Legendre in 1858.
The 1-segment general proof is the most powerful because its thinking generalizes to any area created from the 3 triangles, not just square areas. In any right triangle:
The area of a square drawn on the hypotenuse (c²) is the sum of the areas of squares drawn on the legs (a² + b²).
The area of a circle drawn on the hypotenuse (using the hypotenuse as the diameter) is the sum of the areas of circles drawn on the legs.
The area of any blob (such as a square or circle or clown’s head) drawn on the hypotenuse is the sum of the areas of similarly-shaped blobs drawn on the legs.
That proof was invented by a 19-year-old kid (Stanley Jashemski in Youngstown, Ohio) in 1934.
To understand the concept of math ugliness, remember these math definitions:
The numbers 0, 1, 2, 3, etc.,
Those numbers and their negatives (-1, -2, -3, etc.) are all called integers.
The integers and fractions made from them (1/4, 2/3, -7/5, etc.) are all called rational numbers (because they’re all simple fractions, simple ratios).
All numbers on the number
line are called
real numbers: they include all the rational numbers but also include irrational numbers (such as “pi” and “the square root of 2”), which can’t be expressed accurately as fractions made of integers.
Now you can tackle the 3 rules of ugliness:
1. Most things are ugly.
2. Most things you’ll see are nice.
3. Every ugly thing is almost nice.
Suppose you have a big set of numbers (such as the set of all real numbers), and you consider a certain subset of those numbers to be “nice” (such as the set of all rational numbers). The 3 rules of ugliness say:
1. Most members of the big set aren’t in the nice subset. (For example, most real numbers aren’t rational.)
2. When you operate on most members of the nice subset, you stay in the nice subset. (For example, if you add, subtract, multiply, or divide rational numbers, you get another rational number, if you don’t divide by 0.)
3. Ever member of the big set can be approximated by members of the nice subset. (For example, every irrational number can be approximated by rational numbers.)
In different branches of math, those same 3 rules keep cropping up, using different definitions of what’s “ugly” and “nice.”
The rules apply to people, too:
1. Most people aren’t like you. You’ll tend to think their behaviors are ugly.
2. Most people you’ll meet will appeal to you, because you’ll tend to move to a neighborhood or career composed of people like you.
3. The “ugly” people are actually almost like you: once you make an attempt to understand them, you’ll discover they really aren’t as different from you as you thought!
How math should be taught
I have complaints about how math is taught. Here’s a list of my main complaints. If you’re a mathematician, math teacher, or top math student, read the list and phone me at 603-666-6644 if you want to chat about details or hear about my other complaints, most of which result from research I did in the 1960’s and 1970’s. (On the other hand, if you don’t know about math and don’t care, skip these comments.)
Percentages Middle-school students should learn how to compute percentages (such as “What is 40% of 200?”); but advanced percentage questions (such as “80 is 40% of what?” and “80 is what percent of 200?”) should be delayed until after algebra, because the easiest way to solve an advanced percentage question is to turn the question into an algebraic equation by using these tricks:
change “what” to “x”
change “is” to “=”
change “percent” to “/100”
change “of” to “·”
Graphing a line To graph a line (such as “y = 5 + 2x”), students should be told to use this formula:
the graph of the equation y = h + sx
is a line whose height (above the origin) is h
and whose slope is s
So to graph y = 5 + 2x, put a dot that’s a distance of 5 above the origin; then draw a line that goes through that dot and has a slope of 2.
The formula “y = h + sx” is called the “hot sex” formula (since it includes h + sx). It’s easier to remember than the traditional formula, which has the wrong letters and wrong order and looks like this:
the graph of the equation y = mx + b
is a line whose height (above the origin) is b
and whose slope is m
Imaginary numbers Imaginary numbers (such as “i”) should be explained before the quadratic formula, so the quadratic formula can be stated simply (without having to say “if the determinant is non-negative”).
Factoring Students should be told that every quadratic expression (such as x² + 6x + 8) can be factored by this formula:
the factorization of x² + 2ax + c is
(x+a+d)(x+a-d), where d=Öa²-c
to factor x² + 6x + 8,
realize that a=3 and c=8,
so d=1 and the factorization is (x+3+1)(x+3-1),
which is (x+4)(x+2)
As you can see from that example, the a (which in the example is 3) is the average of the two final numbers (4 and 2). That’s why it’s called a.
The d (which is 1) is how much each final number differs from a (4 and 2 each differ from 3 by 1). That’s why it’s called d. You can call d the difference or divergence or displacement.
Here’s another reason why it’s called d: it’s the determinant, since it determines what kind of final answer you’ll get (rational, irrational, imaginary, or single-root). You can also call d the discriminant, since it lets you discriminate among different kinds of answers.
Quadratic equations To solve any quadratic equation (such as “x² + 6x + 8 = 0”), you can use that short factoring formula. For example:
to solve “x² + 6x + 8 = 0,”
factor it to get “(x+4)(x+2) = 0,”
whose solutions are -4 and -2
Another way to solve a quadratic equation is to use “Russ’s quadratic formula,” which is:
the solution of “x² = 2bx+c” is b ± Öb²+c
That’s much shorter and easier to remember than the traditional quadratic formula, though forcing an equation into the form “x2 = 2bx+c” can sometimes be challenging. Here’s an application:
to solve x²=6x+16,
realize that b=3 and c=16,
so the solution is 3±Ö25, which is 3±5,
which is 8 or -2
Prismoid formula Students should be told that the volume of any reasonable solid (such as a prism, cylinder, pyramid, cone, or sphere) can be computed from this prismoid formula:
height • (area of the typical cross-section)
where “area of the typical cross-section” means (top + bottom + 4 • middle)/6, where
“top” means “area of top cross-section”
“bottom” means “area of bottom cross-section”
“middle” means “area of halfway-up cross-section”
That formula can be written more briefly, like this:
V = H (T + B + 4M)/6,
where V means volume,
H means height,
T means top cross-section’s area
B means bottom cross-section’s area
M means middle cross-section’s area
For example, the volume of a pyramid (whose height is H and whose base area is L times W) is:
H (0 + LW + 4(L/2)(W/2))/6, which is
H (LW + 4LW/4)/6, which is
H (LW + LW)/6, which is
H (2LW)/6, which is
The volume of a cone (whose height is H and whose base area is πr²) is:
H (0 + πr² + 4π(r/2)²)/6, which is
H (πr² + 4πr²/4)/6, which is
H (πr² + πr²)/6, which is
H (2πr²)/6, which is
The volume of a sphere (whose radius is r) is:
(2r) (0 + 0 + 4πr²)/6, which is
2r (4πr²)/6, which is
In the prismoid formula, V = H (T + B + 4M)/6, the “4” is the same “4” that appears in Simpson’s rule (which is used in calculus to find the area under a curve). The formula gives exactly the right answer for any 3-D shape whose sides are “smooth” (so you can express the cross-sectional areas as a quadratic or cubic function of the distance above the base). To prove the prismoid formula works for all such shapes, you must study calculus.
Balanced curriculum Math consists of many topics. Schools should reevaluate which topics are most important.
All students, before graduating from high school, should taste what statistics and calculus are about, since they’re used in many fields. For example, economists often talk about “marginal profit,” which is a concept from calculus. Students should also be exposed to other branches of math, such as matrices, logic, topology, and infinite numbers.
The explanation of Euclidean geometry should be abridged, to make room for other topics that are more important, such as coordinate geometry, which leads to calculus.
Like Shakespeare, Euclid’s work is a classic that should be shown to students so they can savor it and enjoy geometric examples of what “proofs” are; but after half a year of that, let high-school students move on to other topics that are more modern and more useful, to see examples of how proofs are used in other branches of math.
Too much time is spent analyzing triangles.
For example, consider the experience of John Kemeny, who headed Dartmouth College’s math department (and also invented the Basic programming language and later became Dartmouth College’s president). When he was a high-school student, his teacher told him to master “trigonometry, the study of analyzing triangles”; but for the next 20 years, he never had to analyze another triangle, even though he was a mathematician. That trigonometry course was totally useless!
Finally, one day, he bought a plot of land that was advertised as being “an acre, more or less.” He wanted to discover whether it was more or less, so he had survey it and analyze triangles. (The plot turned out to be more than an acre.)
When he told that tale to me and my classmates at Dartmouth, he then went on to make his point: mathematicians don’t have much use for analyzing triangles, though they do have use for how trigonometric functions (such as sine and cosine) help analyze circles (and circular motion and periodic motion). So let’s spend less time on triangles and more time on other topics!
Math can be frustrating.
Pick any number from 1 up to 10.
Double that number. Then double again.
Multiply that by the square root of pi.
If you can do that, go pluck out your eye.
Pluck it out faster and faster and faster.
If you can’t do that, kid, you’re a disaster.
Fry it with roots of the old mango tree.
God is in heaven. A math guy is He.
Algebra 2 is like algebra 1:
Double the trouble. So go get your gun,
Fill it with methods you need to remember.
If you forget them, repeat next September.
Calculus, next, can be really a hoot.
Infinitesimals crawl in your boot,
Climb up your leg and go into your crotch,
Go to the limit and then up a notch,
All while your calculus prof says that fate
Makes your life hell when you go integrate.
Kid, if you don’t feel such vectors amusing,
Switch to biology. It’s much more soothing.
High-school algebra axioms
Here are the best definitions, axioms, and theorems for formalizing the elementary part of high-school algebra.
Equality The undefined symbol “=” leads to these definitions:
a = b = c means a=b and b=c
a ¹ b means it is false that a=b
Here are the axioms (fundamental properties):
Reflexive: a = a
Substitution: if a=b, you can switch “a” to “b”
Those definitions and axioms lead to these theorems (consequences that can be proved):
a=b iff b=a
if a=b=c then a=c
a=b or a¹b
In that first theorem, the “iff” is pronounced “if and only if”
“is equivalent to”.
Most other books have two more
(“a=b iff b=a” and “if a=b=c then a=c”), but I prove those statements and make them theorems.
Addition The undefined symbol “+” leads to this definition:
a+b+c means (a+b)+c
Here’s the axiom:
Backwards: a+b+c = c+b+a
One The undefined symbol “1” leads to these definitions —
2 means 1+1 6 means 5+1
3 means 2+1 7 means 6+1
4 means 3+1 8 means 7+1
5 means 4+1 9 means 8+1
Negative The undefined symbol “-” leads to these definitions:
-a + b means (-a) + b
a - b means a + -b
0 means 1-1
Most other books leave 0 undefined, but I define 0 to be 1-1.
Here’s the axiom:
Disappearing: a+(b-b) = a
Those definitions and axioms lead to these theorems:
a+0 = a a-a = 0
a+b = b+a a + -a = 0
0+a = a -0 = 0
0-a = -a a-0 = a
You also get these theorems involving the associative law:
a+(b+c) = a+b+c 3+3 = 6
2+2 = 4 4+3 = 7
3+2 = 5 5+3 = 8
4+2 = 6 6+3 = 9
5+2 = 7 4+4 = 8
6+2 = 8 5+4 = 9
7+2 = 9
Most other books give 4 axioms about addition: a+b=b+a, a+(b+c)=(a+b)+c, a+0=a, and a+-a=0. But I prove all 4 of those statements from the backwards and disappearing axioms (which I invented), so my 2 axioms replace the traditional 4.
You also get these theorems about solving equations:
a=b iff a+c=b+c a-b=x iff x+b=a
a=b iff a-c=b-c a+x=0 iff x=-a
x+a=b iff x=b-a a+x=0 iff -a=x
Theorems about double negatives:
--a = a
a - -b = a + b
Theorems involving three negatives:
-(a+b) = -a + -b
-(a-b) = b-a
Theorems about negating both sides:
a=b iff -a=-b
-x=a iff x=-a
Theorems about simultaneous equations:
(a=b and c=d) iff (a=b and a+c=b+d)
(a=b and c=d) iff (a=b and a-c=b-d)
Positivity The undefined phrase “is positive” leads to these definitions:
a < b means b-a is positive
a < b < c means a<b and b<c
a > b means b < a
a > b > c means a>b and b>c
a £ b means a<b or a=b
a ³ b means a>b or a=b
a is negative means -a is positive
a is real means a is positive or negative or 0
a is full means a³1 or a£-1 or a=0
Most other books pronounce “a £ b” as
“a is less than or equal to b”, but I pronounce it as “a lequals b”, which is shorter and lets you pronounce theorems faster.
Most other books pronounce
“a ³ b” as
“a is greater than or equal to b”, but I pronounce it as “a grequals b”, which is shorter.
Most other books make “a<b” undefined and write axioms about “a<b”, but I define “a<b” to mean “b-a is positive” and write axioms about “is positive” instead. My approach leads to fewer axioms.
Here are the axioms:
One positive: 1 is positive
Sum positive: if a and b are positive, so is a+b
Zero not positive: 0 is not positive
Sum real: if a and b are real, so is a+b
Those axioms lead to these theorems about “positive”:
2 is positive 6 is positive
3 is positive 7 is positive
4 is positive 8 is positive
5 is positive 9 is positive
Theorems about “not”:
if a is positive then a¹0
1 ¹ 0
1 ¹ 2
if a is positive then -a is not positive
Theorems about “<”:
0<a iff a is positive
a<b iff a+c<b+c
a<b iff a-c<b-c
if a<b<c then a<c
if a<b and c<d then a+c<b+d
if a<b and c is positive then a<b+c
“a<a” is false
if a<b then “b<a” is false
Theorems about “>”:
a>0 iff a is positive
a>b iff a+c>b+c
a>b iff a-c>b-c
if a>b>c then a>c
if a>b and c>d then a+c>b+d
“a>a” is false
if a>b then “b>a” is false
a<b iff -a>-b
Theorems about “£”:
0£a iff a is 0 or positive
a£b iff a+c£b+c
a£b iff a-c£b-c
if a<b£c then a<c
if a£b<c then a<c
if a£b£c then a£c
if a<b and c£d then a+c<b+d
if a£b and c£d then a+c£b+d
a £ a
if a£b then “b<a” is false
Theorem about “³”:
a£b iff -a³-b
Theorems about “negative”:
a is positive iff -a is negative
-1 is negative
if a and b are negative, so is a+b
a is negative iff a<0
Theorems about “real”:
if a is real, so is -a
a is real iff (a<0 or a=0 or a>0)
if a and b are real, so is a-b
if a and b are real then (a<b or a=b or a>b)
Multiplication The undefined symbol “•” is pronounced “multiplied by” or “times” or “of”. Mathematicians are often lazy and don’t bother writing that symbol. For example, instead of writing “a•b” they often write just “ab” to be brief.
Here are the definitions:
abc means (ab)c
a + bc means a + (bc)
-ab means -(ab)
Here are the axioms:
Multiplication backwards: abc = cba
Distributive: a(b+c) = ab + ac
Product positive: if a and b are positive, so is ab
You get these theorems (about multiples of simultaneous equations), which you can prove without using the multiplication axioms:
(a=b and c=d) iff (a=b and c+ea=d+eb)
(a=b and c=d) iff (a=b and c-ea=d-eb)
Exponents The undefined symbol “xa” (pronounced “x raised to the a power” or “x exponent a” or “x to the a”) leads to these definitions:
x + ya means x + (ya) /xa means /(xa)
-xa means -(xa) Öx means x/2
axb means a(xb) Öa + b means (Öa) + b
/a means a-1 i meansÖ-1
/a + b means (/a) + b
Most other books insist that you write the reciprocal of a as either a-1 or 1/a. They don’t let you write just /a.
Most other books agree with
me that -xa means
-(xa), but some software (such as Excel) accidentally defines -xa to be (-x)a instead.
Here are the axioms:
First power: x1 = x
Add exponents: xaxb = xa+b (if x¹0 or b¹-a)
Zero power: x0 = 1
Real power: if x is positive and a is real,
xa is positive
Beyond one: if x > 1 then xa > 1
(assuming a is positive)
Multiply exponents: (xa)b = xab (if b is full or
(x³0 and a is real))
Most other books have a crazy
rule, saying you’re not allowed to raise 0 to a negative power. So in those
books, the add-exponents
axiom is restricted, by making its “if” clause say “if x¹0 or (a³0
and b³0)”. That long-winded “if” clause
makes more theorems have long “if” clauses. My approach makes theorems shorter
and easier to prove. My approach leads to surprising theorems saying 0 is the
answer to most computations about 0. For example, 0 is the answer to 0-1
and 1/0 and 0/0 and 5/0. Most other books say such expressions should never be
written or uttered (as if they were the Devil or Lord Voldemort or passwords
for setting off nuclear bombs) or say such expressions are “undefined” or
“infinity” or “plus or minus infinity” or “complex infinity” or “unsigned
infinity”. Since those books are scared of dealing with zero, I call those
Those books restrict the
multiply-exponents axiom also.
Most mathematicians, calculus teachers, and college textbooks
agree with my zero-power
axiom, which says x0 is always 1, so 00 is
1, which simplifies calculus and the binomial theorem. But stupid high-school
teachers and most high-school textbooks say
00 is “undefined”; they restrict the zero-power axiom by
“if x¹0”, creating another case of zerophobia.
Most other books don’t
multiply-exponents axiom’s “if” clause correctly. The equation “(xa)b = xab” is sometimes false (such as when x=-1 and a=2 and b=1/2), but most books don’t notice that or assume x is positive (though later they assume x is not positive when they talk about the square root of -1 being i).
Those definitions and axioms lead to these theorems about exponent notation:
x2 = xx
xa+1 = xax (if x¹0 or a¹-1)
x3 = xxx
x4 = xxxx
xa = xa-1x (if x¹0 or a¹0)
00 = 1
Those definitions and axioms also lead to these theorems about multiplying:
a1 = a 2·3 = 6
ab = ba 2·4 = 8
1a = a 3a = 2a + a
(a+b)c = ac + bc 3a = a+a+a
2a = a + a 3·3 = 9
2·2 = 4 a(bc) = abc
Most other books have an axiom about multiplying by 1, but I use the first three exponent axioms to prove “a1 = a.”
Most other books have an
“ab = ba,” but I prove that from the other axioms.
Most other books have an axiom saying “a+(b+c)=(a+b)+c,” but I prove that from the multiplication-backwards axiom, which I invented.
Theorems about exponent computation:
32 = 9
22 = 4
23 = 8
xax-a = 1 (if x¹0)
xx-1 = 1 (if x¹0)
Theorems about 0:
0a = 0
a0 = 0
0a = 0 (if a¹0)
0-1 = 0
xa = 0 iff (x=0 and a¹0)
Theorems about multiplying negatives:
(-a)b = -(ab)
(-1)a = -a
a(-b) = -(ab)
(-a)(-b) = ab
a(b-c) = ab - ac
Theorems about multiplying negativity:
if a and b are negative, ab is positive
if a is negative and b is positive, ab is negative
if a and b are real, so is ab
The FOIL theorem:
(a+b)(c+d) = ac + ad + bc + bd
Advanced theorems about squaring:
(-x)2 = x2
if x is positive or negative, x2 is positive
if x is real, x2 ³ 0
(x+y)2 = x2 + 2xy + y2
(x+y)2 > x2 + y2 (if x and y are positive)
(x-y)2 = x2 - 2xy + y2
(x+y)(x-y) = x2 - y2
(x+u)(x+v) = x2 + (u+v)x + uv
Advanced theorems about cubing:
x3 - y3 = (x-y)(x2 + xy + y2)
x3 + y3 = (x+y)(x2-xy+y2)
(x+y)3 = x3 + 3x2y + 3xy2 + y3
Theorems about “/”:
1/a = /a 6/2 = 3
0/a = 0 8/2 = 4
0/0 = 0 6/3 = 2
a/a = 1 (if a¹0) 9/3 = 3
/1 = 1 8/4 = 2
a/1 = a (-a)/b = -(a/b)
(ab)/a = b (if a¹0) a(b/c) = (ab)/c
4/2 = 2 a/x + b/x = (a+b)/x
Theorems about solving equations:
a=b iff ac=bc (assuming c¹0)
ac=bc iff (a=b or c=0)
a=b iff a/c=b/c (assuming c¹0)
ab=0 iff (a=0 or b=0)
ab¹0 iff (a¹0 and b¹0)
(x-r)(x-s)=0 iff (x=r or x=s)
x2=y2 iff x=±y
if ax=1 then x=/a
ax=b iff x=b/a (assuming a¹0)
ax+b=c iff x=(c-b)/a (assuming a¹0)
ax+b=cx+d iff x=(d-b)/(a-c) (assuming a¹c)
Theorems relating exponents to “/”:
x-a = /(xa)
(xa)/(xb) = xa-b (if x¹0 or a¹b)
Theorem about advanced factoring:
(ax+u)(ax+v)/a = ax2 + (u+v)x + uv/a (if a¹0)
Theorems about “/0”:
/0 = 0
a/0 = 0
Theorems relating /a to 0:
a=0 iff /a=0
a¹0 iff /a¹0
Theorems about slashing different numbers:
//a = a /(a/b) = b/a
/-a = -/a a/(ab) = /b (if a¹0)
/(ab) = (/a)(/b) a=b iff /a=/b
Theorems about changing a fraction’s denominator:
a/-b = -(a/b)
(-a)/(-b) = a/b
(a/b)(c/d) = (ac)/(bd)
a/b = (ac)/(bc) (if c¹0)
a/b + c/d = (ad+bc)/(bd) (if b¹0 and d¹0)
a/(b/c) = a(c/b)
a/b=c/d iff b/a=d/c
a/b=c/d iff ad=bc (assuming b¹0 and d¹0)
a/b=c/d iff a/c=b/d (assuming b¹0 and c¹0)
Theorems about positivity:
if x and a are positive, so is xa
if a is positive, so is /a
if a and b are positive, so is a/b
if a is negative, so is /a
if a is real, so is /a
if a and b are real, so is a/b
a<b iff ac<bc (assuming c is positive)
if 0<a<b then /a>/b
a<b iff xa<xb (assuming a and b real and x>1)
Theorems using the multiply-exponents axiom:
(xa)b = (xb)a
(if a and b are full or (x³0 and a and b are real))
1a = 1
x=y iff xa=ya
(assuming x³0 and y³0 and a is positive or negative)
xa=ya iff (x=y or a=0)
(assuming x³0 and y³0 and a is real)
a=b iff xa=xb
(assuming a and b are real and x is positive but not 1)
Theorems about square roots:
Ö0 = 0
Ö1 = 1
(Öx)2 = x
Ö(x2) = x (if x ³ 0)
Ö4 = 2
Ö9 = 3
x is positive iff Öx is positive
x<y iff Öx<Öy (assuming x³0 and y³0)
Ö(x2+y2) < x + y (if x and y are positive)
Theorems about solving quadratic equations:
x2 = a iff x = ±Öa
x2 + 2bx = c iff x = ±Ö(c+b2) - b
x2 + 2bx = c iff x = -b ± Ö(b2+c)
ax2 + bx +c = 0 iff x = (-b ± Ö(b2-4ac))/(2a)
Theorems about i:
i2 = -1
i3 = -i
i4 = 1
(i+1)2 = 2i
(i+Ö3)3 = 8i
i is not real
i ¹ 0
/i = -i
(x+yi)(x-yi) = x2 + y2
a = 0 iff a and ai are real
a+bi = c+di iff a=c and b=d
(assuming a, b, c, and d are real)
Logarithms The symbol “logx a” (pronounced “the logarithm, base x, of a” or “log, base x, of a”) leads to these definitions:
logx a + b means (logx a) + b
logx ab means logx (ab)
logx ab means logx (ab)
Here are the axioms:
Log: xlogx a = a (if a¹0 and x is neither 0 nor 1)
Log real: if x and a are positive, logx a is real
Most other books require x to be positive if you write “logx a”. My log axiom is more permissive: it lets x be any number that’s neither 0 nor 1, so x can even be negative or imaginary.
Those definitions and axioms lead to these theorems about logarithms:
logx xa = a (if a is real and x is positive but not 1)
log2 8 = 3
log3 9 = 2
log2 4 = 2
logx x = 1 (if x is positive but not 1)
logx 1 = 0 (if x is positive but not 1)
logx /x = -1 (if x is positive but not 1)
logx /a = -logx a (if a and x are positive and x¹1)
logx a = 0 iff a = 1
(assuming a¹0 and x is positive but not 1)
Those definitions and axioms also lead to these theorems about exponents:
(xy)a = xaya (if a is full or x³0 or y³0)
Ö(xy) = (Öx)(Öy) (if x³0 or y³0)
Ö-x = iÖx (if x³0)
Ö-4 = 2i
Ö-9 = 3i
(/x)a = /(xa) (if a is full or x³0)
Ö/x = /Öx (if x³ 0)
Ö(x/y) = (Öx)/Öy (if x³0 or y³0)
(x/y)a = xa/ya (if a is full or y³0)
if 0£x<y then xa < ya (assuming a is positive)
x<y iff xa<ya
(assuming a is positive and x³0 and y³0)
Theorems about the logarithm of 2 variables:
logx ab = logx a + logx b
(if a, b, and x are positive and x¹1)
logx a/b = logx a - logx b
(if a, b, and x are positive, and x¹1)
logx ab = b logx a
(if b is real, a and x are positive, and x¹1)
Theorems about changing the log base:
(logx a)(loga b) = logx b
(if a, b, and x are positive and neither x nor a is 1)
loga b = (logx b)/(logx a)
(if a, b, and x are positive and neither x nor a is 1)
log4 8 = 3/2
loga b = /logb a
(if a and b are positive and neither is 1)
No bell prize
I’ve invented several new ideas. I figure I should get a Nobel prize for them, except the ideas are half-baked: they need further research to make them fleshed out, complete, and fully useful. So I beg you: improve on these ideas, so you can get a Nobel prize. If you mention me in a footnote, I’d appreciate that. We can split the Nobel prize: you get the Bell prize, and I get No prize.
There’s just one little hitch in our plan to split a Nobel prize:
The Nobel prize was invented by Alfred Nobel, who decided to award prizes just to achievements that are “practical.”
He thought math wasn’t practical, so there’s no “Nobel prize” in math. To get a Nobel prize, your achievement must fit into one of these 6 Nobel-prize categories: physics, chemistry, medicine, economics, peace, or literature.
Although my ideas are mathy, we must pretend they aren’t. We must pretend my first idea, “derived happiness,” is about economics, not math or psychology. We must pretend my other ideas, about infinity & infinitesimals, are about physics (infinite blasts!), not math.
… or else we must create our own “No” and “Bell” prizes for ourselves!
What makes people happy? Several centuries ago, the “meaning of happiness” was considered a philosophical problem. Nowadays, it’s considered a psychiatric problem: happiness is whatever makes your happiness hormones increase. In the future, it will become a math problem; here’s why....
To begin our fancy-schmancy math analysis, let’s do the same thing physicists do when analyzing motion: oversimplify! Later, we’ll discuss all the complications of the “real world,” such as friction.
Physicists begin by assuming objects move in a vacuum, then later add the effects of friction. We’ll begin by assuming happiness consists of having lots of money, then later add the effects of interpersonal friction (good & bad relationships with other people) and God friction (good & bad relationships with the desire to have a meaningful life). I’ll start with money, rather than frictions, because money is easier to measure.
Zeroth-derivative happiness Let’s start with the simplest situation:
Joe has $200.
Tim has $100.
That’s all we know about Joe & Tim so far. They’re both American males, so we don’t know any cultural differences between Joe & Tim yet. On the basis of what we know so far, Joe is probably happier than Tim, since Joe is wealthier.
This explanation is going to get mathy, and I’m even going to say jargon from calculus! But to avoid scaring the anti-math part of your brain, I promise to explain all math jargon simply.
Using math jargon, we say that Joe is higher up on the “wealth function” than Tim. That stupidly simple explanation is called the zeroth-derivative function.
First-derivative happiness Now let’s complicate the situation slightly, by peeking at the past:
Joe had $400 yesterday — but now has $200.
Tim had $50 yesterday — but now has $100.
Now the happiness seems different. Tim is happy because his money doubled. Joe is unhappy because Joe’s money halved. Even though Joe still has more money than Tim, Joe feels unhappy because Joe’s “life is going downhill,” so his future looks grim, whereas Tim is thrilled because Jim’s “life is going uphill” so his future looks bright.
Compared to yesterday, Tim gained $50, whereas Joe lost $200. In calculus jargon, we say:
Tim’s slope (gain divided by time) is $50 per day.
Joe’s slope (gain divided by time) is minus $200 per day.
So Tim’s slope is better than Joe’s slope. Slope is also called the derivative. More precisely, it’s called the first derivative. So to figure out a person’s happiness, you should look at the person’s slope (first derivative).
Second-derivative happiness Now let’s complicate the situation further, by peeking further into the past:
Ann had $200 then $300 but now has $305.
Sue had $100 then $60 but now has $55.
Who’s happier: Ann or Sue?
Ann has more money than Sue (since Ann has $305 while Sue has just $55). Ann’s recent slope is also better than Sue’s recent slope (since Ann’s recent slope was $5 per day, while Sue’s recent slope was minus $5 per day).
But in spite of all that good news for Ann, she probably feels depressed, because her recent raise (the $5 raise from $300 to $305) is worse than her previous raise (the $100 raise from $200 to $300). Her raise decreased by $95 (since the $100 raise dropped to $5). She feels her life isn’t improving as much as it used to. She fears her life will, in the future, improve less and less and finally go downhill. She’s depressed that she has less pride now (going from $300 to $305) than she had before (going from $200 to $300). She feels she’s no longer a star on the rise. She’s a has-been with probably a depressing future. She wants to commit suicide, because the great part of her life is over.
Sue, by contrast, is feeling relieved. Although her money dropped recently (a $5 drop, since $60 became $55), the drop wasn’t as dramatically bad as the period before (a $40 drop, from $100 to $60). She’s happy she didn’t drop $40 again. She’s happy her drop this time was just slight, almost insignificant, so her losses are “stemming” (becoming less significant). She feels her life is “turning the corner” and might soon rise. Her slope improved: it was minus $40 per day previously but became minus $5 per day for the recent day.
Comparing old slopes against new slopes is called
computing the second derivative. Since Ann’s slope got worse (decreased), her second derivative is negative, and Ann feels depressed; since Sue’s slope got better (not as bad as before), her second derivative is positive, and Sue feels relieved.
So according to that theory, happiness is the second derivative of the wealth function.
If you graph the history of Ann’s money and Sue’s money, you see that Ann’s graph looks like the left half of a cap (which has no visor); Sue’s graph looks like the left half of a cup (which has no handle). A cap graph means the second derivative is negative; a cup graph means the second derivative is positive. So according to that happiness theory, happiness is a cup.
To improve that theory further, we should make modifications....
Logarithms The first improvement is to use logarithms. Here are the details.
Compare these two people:
Bud had $100 yesterday — but now has $115.
Sam had $10 yesterday — but now has $20.
We don’t know enough of the past to compute a second derivative. According to the previous theory, Bud should be happier than Sam, since Bud has more money ($115) and a bigger slope ($15 per day). But in reality, Sam is more thrilled than Bud, since Sam’s money doubled (from $10 to $20), whereas Bud’s money went up by just a small percentage of what Bud had before (15%). Sam can brag to himself & friends that his money doubled, whereas Bud hasn’t much to brag about. Bud is happy (since Bud’s money went up, not down), but Sam is thrilled.
So to measure happiness, we should measure the percentage by which money increased. To do that, we can choose two methods, each giving the same result:
Percentage method Instead of computing the simple slope (the money increase per day), compute the “slope as a percentage (or fraction) of the money”: take the slope and divide it by the amount of money. In calculus, the wealth function is written as f(t), its slope is written as f'(t), and this method is written as “f'(t) divided by f(t).”
Logarithm method Instead of using the simple wealth, use the wealth’s logarithm (base 2 or e or 10 or whatever you please), by using a calculator or by graphing the wealth on log-graph paper. When you do that, you see the distance up from $10 to $20 is the same as the distance up from $20 to $40, which is the same as the distance up from $40 to $80, which is the same as the distance up from $80 to $160. That’s because going from $10 to $20 feels as good as going from $20 to $40, since each means your wealth has doubled. Then find the slope of that vertical distance. In calculus, that can be written as “the derivative of log f(t).”
The two methods give the same result because, according to calculus, “the derivative of log f(t)” equals “f'(t) divided by f(t).”
Use the percentage method (or the equivalent logarithm method) to compute first-derivative happiness and second-derivative happiness.
Blended derivatives If your second derivative and first derivative are both negative, you might feel depressed. But if you start whining about them, your friends might remind you that you shouldn’t feel so bad, because you still have enough money to live on. For example, if you had 4 billion dollars but then had just 3 billion and then just 1 billion, your second and first derivatives are both negative; but your friends might remind you that you still have a billion dollars left and you’re still better off than most other people, so cheer up!
How important to your happiness are the first and second derivatives in relation to the amount of money you actually have? Your happiness is actually a blend of all that data. Your happiness might even be affected by the third derivative (which measures how much your second derivative is better than it was before). Maybe the happiness of people (and other animals) having impaired memory isn’t influenced much by derivatives, second derivatives, and third derivatives. Experiments should be done to determine how much the various derivatives contribute to the happiness of various kinds of people.
Beyond money Besides money in your pocket, these other things can give you happiness: investments, things you own, food, shelter, health (and being pain-free), beauty, intelligence, good relationships (with people, pets, and the environment), love, sex, feeling useful (in your career or by volunteering or by helping friends & family), feeling powerful, feeling moral, and — alas! — taking mood-enhancing drugs (alcohol, nicotine, marijuana, heroin, and beyond). Your happiness is affected by how much you have of all those things, how much more you have than your neighbors, and how much fame you have for what you do. Your happiness is a blend of all those factors. Experiments should be done to determine how important those factors are in the blend.
Focus Maybe most factors in your life are okay, but one factor is bugging you at the moment. Maybe it’s a test you must take tomorrow (and you haven’t studied for yet), or a friend who’s dying, or a lover you’re in the middle of breaking up with, or you’re being arrested and transported in a paddy wagon to the police station, or you’re having a medical emergency and need help fast.
Or maybe one factor is thrilling you at the moment. For example, maybe you’ve just won an award, or won a lottery, or had an orgasm.
During those especially bad or good moments, your attention focuses on one thing and nearly ignores everything else; but those other things still have some effect on your happiness then, though maybe just slightly. To compute your overall happiness in that situation, we must invent a formula that’s a weighted average of your feelings about everything: that formula must emphasize (give more weight to) the extreme feelings (feelings that are extremely positive or extremely negative) and de-emphasize the feelings that are closer to neutral (and therefore nearly ignored).
Please finish this explanation and get a Nobel prize.
In elementary school, you learned how to count: 1, 2, 3, etc. Later, you learned about other kinds of numbers: zero, negative numbers, and fractions. If you took 2 years of high-school algebra, you also learned about “imaginary” numbers, such as “i”, which is the square root of minus one.
During the last 3,000 years, whenever new kinds of numbers were invented, critics laughed at the inventors:
When zero was invented, the critics laughed and said “How can you have zero? If you have zero, you don’t have anything at all, so you don’t have zero.”
When negative numbers were invented, the critics laughed and said, “How can you have less than nothing?”
When “imaginary” numbers were invented, critics laughed and said, “How can minus one have a square root, really?”
The critics got silenced when inventors drew pictures:
Zero is the height of an Egyptian pyramid before you start putting the bricks on it. Zero is also how much money you have before you start getting some.
Negative numbers are what you see on a thermometer when the temperature is colder than zero degrees. When you draw a vertical number line that shows how far up something went, negative numbers represent going down instead of up. When you draw a horizontal number line that shows how far something went toward the right, negative numbers represent traveling to the left instead.
Imaginary numbers became
believable when Caspar Wessel and Jean-Robert Argand drew pictures including
them. Those pictures, called
Argand diagrams, are drawn on graph paper, with the “real” numbers on the horizontal x axis and “i” on the vertical y axis, so the “i” sits above 0.
When Germany’s Gottfried Leibniz and England’s Isaac Newton invented calculus in the 1600’s, they thought about an “infinitesimal number,” which is a number so tiny that it’s less than every fraction of integers (less than ½, less than 1/10, less than 1/100, less than a millionth, less than a trillionth, etc.) but is still more than zero. But since an “infinitesimal number” was hard to picture, it was hard to discuss confidently, so mathematicians later did calculus a different way, involving “limits” and awkward phrases such as “for every epsilon there exists a delta such that….” Those long-winded phrases make students want to cry, or give up and just sleep through the calculus lectures, or snore.
Mathematicians wish there were an easy, confident, pictorial,
accurate way to mention infinitesimals, but that goal has eluded them. In 1966
at Yale University, Professor Abraham Robinson became famous for inventing what
non-standard analysis, which is his own way to do calculus by using infinitesimals, but it’s hard to understand. In the year 2000 at the University of Wisconsin, Professor H. Jerome Keisler invented a simpler way to explain Robinson’s work, but mathematicians complain that Keisler’s explanation seems sloppy.
Here are my own 2 ways to explain infinitesimals: the
zillions method and the minimal method. Each has its own advantages and disadvantages. Neither is completely satisfactory. I hope someday you or your friends can improve on what I’ve done and get a Nobel prize.
Zillions method This way to start doing calculus is understandable even to kids in elementary school. Just use the word “zillion.” As most elementary kids already know, “a zillion” means “a lot of,” “ridiculously many,” as in “I have a zillion chores to do.”
The word “zillion” has been popular for many years. According
to the Merriam-Webster Dictionary
(at merriam-webster.com/dictionary/zillion), the word “zillion” has been used for many decades, even back in 1934, and some folks have been saying “jillion” instead, beginning in 1942.
To do calculus, consider a zillion to be more than a million, more than a billion, more than a trillion, more than every other “illion” you ever heard of. Make the symbol for a zillion be ∞. You can call that number “infinity” if you like, but people get scared about the word “infinity,” whereas kids use the word “zillion” all the time.
Like a trillion, a zillion is a number that obeys all the normal rules of arithmetic and algebra. It pleases mathematicians because, like normal numbers, it all obeys the commutative and associative laws and all the other laws of an “ordered field.” It just happens to be even bigger than a trillion.
The only “law” a zillion doesn’t obey is the “Archimedes principle,” since you can’t reach a zillion by counting 1, 2, 3, etc. in a finite amount of time, though you can reach it in a zillion amount of time. In other words, a zillion can’t be generated by starting at 0 and then adding 1 repeatedly in a finite amount of time; it can’t be generated by multiplying two finite numbers together. But that disappointment about zillion doesn’t affect any computations used in high-school algebra or calculus, so don’t worry about it.
A zillion is not the biggest number, since “a zillion plus one” is even bigger (and written “∞+1”), and “two zillion” is bigger yet (and written “2∞”), and “a zillion times a zillion” is bigger than those (and written “∞∞” or “∞2”), and “a zillion to the zillionth power” is bigger than all those (and written “∞∞”).
An example of an infinite number that’s slightly smaller than a zillion is “a zillion minus one” (written “∞-1”). An even smaller infinite number is “the square root of a zillion.”
Just like a “million” has a reciprocal called “a millionth,” a zillion has a reciprocal called a zillionth, which is the fraction 1/∞. That fraction is an example of an infinitesimal, since it’s tinier than any normal fraction but still bigger than 0. Mathematicians like to call that fraction “epsilon” (which is the Greek letter for “e” and written “є”), but that Greek jargon confuses young kids and makes them complain “It’s Greek to me!” so obey the warning of AIDS advisors: don’t do Greek.
A zillionth isn’t the only infinitesimal number. A slightly bigger infinitesimal number is “two zillionths” (which is twice as big as a zillionth and written “2/∞”).
In elementary school, kids learn how to round numbers. Examples:
7.1 rounded to the nearest integer is 7.
7.9 rounded to the nearest integer is 8.
7.19 rounded to the nearest tenth is 7.2.
In calculus, mathematicians round using a method I call calculus round (cRound).
If a number is positive and infinite, its cRound is a zillion. Examples:
The cRound of “a zillion plus one” is a zillion, so cRound(∞+1) = ∞.
The cRound of “a zillion minus one” is a zillion, so cRound(∞-1) = ∞.
The cRound or “two zillion” is a zillion, so cRound(2∞) = ∞.
If a number is negative infinite, its cRound is “minus a zillion.” Example:
cRound(-∞+1) = -∞.
If a number is finite, its cRound is the closest number that’s normal (doesn’t involve infinitesimals). Examples (using є to mean 1/∞, assuming kids are old enough to do Greek):
cRound(7+є) = 7
cRound(7-є) = 7
cRound(7+2є) = 7
cRound(є) = 0
cRound(2є) = 0
cRound(є2) = 0
In old-fashioned calculus, the word “limit” is defined in a long-winded way, starting with “for every epsilon there exists a delta such that.” But in my zillion calculus, we can define “limit” to mean just cRound. More precisely, define “the limit, as x approaches p, of f(x)” to mean the result of performing this 3-step procedure:
Step 1: write f(x).
Step 2: switch the x to p+є, so you have f(p+є).
Step 3: cRound the result of step 2, so you have cRound(f(p+є)).
So here’s the definition:
limx®p f(x) = cRound(f(p+є))
That definition requires no “delta”! That definition works if p is ∞ or -∞ or a normal number (such as 7).
In my zillion calculus, we can define “the derivative of f(x)” to mean just the cRound of “f(x+є)-f(x), all that divided by є,” like this:
f'(x) = cRound( (f(x+є)-f(x))/є )
That definition involves no “delta,” no “limit,” and no “p,” so it lets you compute the derivative much faster than old-fashioned methods.
Minimal method Gee, infinity can be scary: so many kinds of infinite numbers! To do elementary calculus simply, fuck infinity: let’s have no infinite numbers at all! Let’s have just the minimal necessary to do elementary calculus: a special number, called epsilon (written “є”).
Epsilon is tiny. It’s tinier than any fraction you encountered in elementary school: it’s tinier than 1/10, tinier than 1/100, tinier than 1/1000, etc. It’s so tiny that when you multiply it by itself, it disappears, poof! Here’s the equation: є 2=0. Physicists brag about “black holes,” where things seem to disappear, but we mathematicians have epsilon, whose square really does disappear!
So how do you make a number system that includes epsilon and lets you do calculus, all in a reasonable way? It’s easy! It’s even easier than the crap they teach in high school’s algebra 2 class about “imaginary numbers.” In algebra 2, they teach you to draw a horizontal ruler (an x axis) labeled 0, 1, 2, etc., and draw a vertical ruler (a y axis) labeled 0, 1i, 2i, 3i, etc. Do the same thing for my minimum method, but write “є” instead of “i"”, so the vertical ruler is labeled 0, 1є, 2є, 3є, etc. In algebra 2, they teach you to invent numbers of the form x+yi, such as 3+7i; in my minimal method, invent numbers of the form x+yє, such as 3+7є. In algebra 2, they teach you to add, subtract, and multiply numbers in the obvious way, but remembering that i2=-1; in my minimal method, you can add, subtract, and multiply numbers in the obvious way, but remember that є2=0.
Inventing “i” simplified algebra, by making the quadratic formula more understandable. Inventing є simplifies calculus, by making derivatives more understandable.
For you math nerds, here’s a formal explanation….
To use є, construct the extended real numbers, which consist of numbers of the form a + bє (where “a” and “b” are ordinary “real” numbers). Add and multiply extended real numbers as you’d expect (bearing in mind that є² is 0), like this:
(a + bє) + (c + dє) = (a+c) + (b+d)є
(a + bє) • (c + dє) = ac + (ad+bc)є
(9+12є) + (2+4є) = 11+16є
(9+12є) • (2+4є) = 18 + (36+24)є, which is 18+60є
You can define order:
“a+bє < c+dє” means “a<c or (a=c and b<d)”
Those definitions of addition, subtraction, multiplication, and order obey the traditional “rules of algebra” except for one rule: in traditional algebra, every non-zero number has a reciprocal (a number you can multiply it by to get 1), but unfortunately є has no reciprocal.
If x is an extended real number, it has the form a + bє, where a and b are each real. The a is called the real part of x. For example, the real part of 3 + 7є is 3.
A number is called infinitesimal if its real part is 0. For example, є and 2є are infinitesimal; so is 0.
Infinitesimals are useful because they let you define the “derivative” of f(x) easily, by computing f(x+є):
Define the differential of f(x), which is written d f(x), to mean f(x+є) - f(x). For example, dx² is (x+є)²-x², which is (x²+2xє+є²)-x², which is 2xє (since є²=0), which is 2x dx (since dx turns out to be є).
Define the derivative of f(x) to mean (d f(x)) divided by є. For example, the derivative of x² is (2xє)/є, which is 2x. The definition of the derivative of f(x) can also be written as (d f(x))/dx, since dx is є.
Define the limit, as x approaches p, of f(x) to mean the real part of f(p+є). For example, the limit, as x approaches 0, of x/x is the real part of (0+є)/(0+є), which is the real part of є/є, which is the real part of 1, which is 1.
Define f(x) is continuous at p to mean:
for all b, f(p+bє) – f(p) is infinitesimal.
For example, the function “2 if x£=9, 3 if x>9” isn’t continuous at 9, since f(9+1є)-f(9) is 3-2, which is 1, which isn’t infinitesimal.
Define f(x) is differentiable at p to mean:
for all b, f(p+bє) = f(p) + b (the derivative of f(x) at p).
Then calculations & proofs about derivatives and limits become easy, especially when you define sin є to be є and define cos є to be 1.
If you want to chat about any of that stuff, call my cell phone (603-666-6644) anytime (24 hours). I’ll be glad to give more details, explain more clearly, or listen to your objections.