To get more out of life, become an intellectual! Being intellectual is fun.
Try to learn the truth. Dig deeper! Mark Twain said:
He also said:
There are 3 kinds of people:
President Franklin Roosevelt’s wife (Eleanor Roosevelt) said:
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 and 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:
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.” For 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) are spending lots of money on college: make sure you get your money’s worth! Ask the professors lots of questions (during class or privately), interact with your classmates too, 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:
W.H. Auden said:
Dave Barry gave this advice to students:
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.
For example, 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:
Back in the 1970’s, the basements of Wesleyan University’s dorms were connected by tunnels, upon whose walls the students wrote philosophy. Sample:
Don’t let your failures discourage you. Learn from them. They’ll also help you appreciate your later successes more. Truman Capote said:
Remember this famous saying:
But also heed W.C. Field’s elaboration:
Success versus happiness
Don’t confuse “success” with “happiness.” Actress Ingrid Bergman said:
The Internet offers this inspiring tale:
Why did the chicken cross the road?
According to the Internet, these thinkers would give straight answers.…
So would these scientists.…
These thinkers would deny that the chicken simply crossed the road:
These thinkers would investigate further:
These thinkers would raise questions.…
These thinkers would brag about technology:
These thinkers think the others are too long-winded:
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:
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.
But if you nevertheless decide to kill yourself, here’s a suggestion about the best way to do it:
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!
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:
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:
Now Loretta has a new presentation, called “Stop Global Whining.”
Test about life
Here’s a multiple-choice test about life.
Which completion is most correct?
Why do people act strange? This poem explains:
During the Christmas season, many people feel stressed. The Internet recommends these Christmas carols for the psychologically challenged:
Psychologists like to invent ways to test your personality. Here’s a crazy test I invented: are you more like me (Russ) or my wife (Donna)? Are you a “Donna” type (emotional) or a “Russ” type (logical)?
After deciding whether you’re more like “Donna” or “Russ,” 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:
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:
Take me away
The most famous song about mental illness is called “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:
In movie-making courses, students create movies using Jerry’s original recording as the scary soundtrack. Here are two examples:
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 are left 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.
Eggs Carl Sandberg, in his poem Arithmetic, asks this question:
Missing dollar Now that you’ve mastered the easy puzzles, try this harder one:
Ask your friends that question and see how many crazy answers you get!
Here’s the correct answer:
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:
Coins Try this task:
“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:
Which type are you?
Here’s Warren Buffet’s favorite saying about math.
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!
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.
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:
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:
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:
Surprise test When I took a logic course at Dartmouth College, the professor began by warning me and my classmates:
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:
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:
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:
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:
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:
The chemist would rush for results and say just this:
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):
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:
Winston Churchill (who was England’s prime minister) said:
A colleague passed me this e-mail, forwarded anonymously:
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:
Take your three-digit number, and write it backwards. For example, if you picked 852, you now 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 that 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:
Here’s another example:
Yes, you always get 1089!
Proof To prove you always get 1089, use algebra: make letters represent the digits, like this.…
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:
Now you can subtract A from C+10:
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:
Complete the calculation:
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:
April Fools Irving Adler also suggested this trick:
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 answers for 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:
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:
Subtracting 2ab from both sides of that equation, you’re left with:
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:
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:
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:
Multiplying both sides of that equation by ac, you discover what a² is:
Using similar reasoning, you discover what b² is:
Adding those two equations together, you get:
Since x+y is c, that equation becomes:
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:
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:
Now you can tackle the 3 rules of ugliness:
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:
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:
Graphing a line To graph a line (such as “y = 5 + 2x”), students should be told to use this formula:
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:
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:
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:
Another way to solve a quadratic equation is to use “Russ’s quadratic formula,” which is:
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:
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:
That formula can be written more briefly, like this:
For example, the volume of a pyramid (whose height is H and whose base area is L times W) is:
The volume of a cone (whose height is H and whose base area is πr²) is:
The volume of a sphere (whose radius is r) 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.
Infinitesimals Students should be told about infinitesimals, because they make calculus easier to understand.
Specifically, there’s an infinitesimal number, called epsilon (or є or simply e), which is positive (greater than zero) but so tiny that its square is 0:
You might say “there’s no such number,” but we can invent it, just like mathematicians invented the “imaginary” number i whose square is -1. The invention of i simplified algebra, by making the quadratic formula more understandable. The invention of є simplifies calculus, by making derivatives more understandable.
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:
You can define order:
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):
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.
High-school algebra axioms
Here are the best definitions, axioms, and theorems for formalizing the elementary part of high-school algebra.
Equality The symbol “=” (pronounced “equals” or “is”) leads to these definitions:
Here are the axioms (fundamental properties):
Those definitions and axioms lead to these theorems (consequences that can be proved):
In that first theorem, the “iff” is pronounced “if and only if” or “is equivalent to”.
Addition The symbols “+” (pronounced “added to” or “plus”) and “1” (pronounced “one”) lead to these definitions:
Here are the axioms:
Those definitions and axioms lead to this theorem:
Zero The symbol “0” (pronounced “zero”) has this axiom:
That axiom leads to this theorem:
Negatives The symbol “-” (pronounced “negative” or “minus”) has this axiom:
That axiom lead to these theorems:
Subtraction Here’s another definition:
That definition leads to these theorems:
Multiplication The symbol “•” is a raised dot. It’s pronounced “multiplied by” or “times.” 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 axioms:
Those axioms lead to these theorems —
and these theorems about multiplying negatives:
Positivity The phrase “is positive” has these axioms:
Those axioms lead to these theorems:
Reciprocals The symbol “/” is pronounced “the reciprocal of” or “slash.” For example, “/b” is pronounced “the reciprocal of b” or “slash b”. Mathematicians are lazy: instead of writing “a•/b” they write just “a/b” (which they pronounce “a divided by b” or “a slash b”) or do this: write a little “a” over a little “b” and put a horizontal line between them, to form a fraction (which they pronounce “a over b”), where the top number (“a”) is called the “numerator” and the bottom number (“b”) is called the “denominator”.
Here’s the main axiom:
That axiom leads to this theorem —
and these theorems about fractions —
and these theorems about solving equations:
What’s the reciprocal of 0? The reciprocal axiom doesn’t answer that question. Some books say the reciprocal of 0 is “undefined”; other books say the reciprocal of 0 is “infinity”; but those approaches awkwardly force many theorems to say “assuming a¹0.” We’ll use a smarter approach: we’ll define the reciprocal of 0 to be 0, by adding this axiom:
That axiom saves us from having to say “assuming a¹0” so often, though we’ll still have to say “assuming a¹0” occasionally. That axiom leads to these theorems:
Here are theorems about changing the denominator:
Here’s another definition:
Here’s another axiom about reciprocals:
That axiom leads to this theorem:
Order Here are more definitions:
Those definitions lead to these theorems:
The opposite of “>” is “<”. Here’s the definition:
Each theorem about “greater” leads to a theorem about “less”:
These theorems relate “<” to “>”:
Here are more definitions:
Each theorem about “<” leads to a theorem about “£”:
Exponents We’ve discussed addition (a+b), subtraction (a-b), multiplication (a•b), and division (a/b). Now we’ll discuss exponentiation (xa). In the operation “xa,” the “a” is called the “exponent” (or “power”); the “x” is called the “base”; the “xa” is called the “base x raised to the a power” or just “x to the a.” For example, “x2” is called “x to the 2” (or “x squared”); “x3” is called “x to the 3” (or “x cubed”).
Here are the basic axioms about exponents (powers):
Those axioms lead to these theorems:
Advanced multiplication Here are advanced theorems about multiplication:
Zero power This axiom is more advanced:
That axiom leads to these theorems:
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:
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:
Chemists are mixed up.
Are you a chemist?
To discover how good a chemist you are, see how long you take to solve this puzzle:
If you can’t solve that problem yourself, ask your friends, until you find a friend who’s smart — and kind enough to tell you the answer.
That puzzle comes from Martin Gardner’s book, Mathematical Puzzles. To have more fun, get that book!
Back around 1950, chemists tried to prove heaven’s hotter than hell. The proofs gradually got more sophisticated. A 1972 article in Applied Optics argues this way:
The full article is at www.lhup.edu/~dsimanek/hell.htm.
This bonus question appeared on a chemistry test:
The professor expected the students to argue, one way or the other, by using Boyle’s law (which says compressing a gas makes it hotter). According to the tale, the top student gave this answer:
In April 1988, William DeBuvitz wrote about the discovery of administratium. Here’s a summary of what he and later researchers have reported:
The names of the chemical elements might seem boring, but in 1959 Tom Lehrer made them fun: he wrote a song called The Elements, where he sang the names of the 102 chemical elements that had been discovered so far, to the tune of the Major-General’s Song from Gilbert & Sullivan’s Pirates of Pinzance.
Here are 4 videos about it:
Warning: for the first video’s Web address, the letter after w is a lower-case L.