Contents

Here are the contents:

Fundamentals

1. Equality

2. Addition

3. Negatives

4. Subtraction

5. Multiplication

6. Negative multiplication

7. Positive numbers

 

Reciprocals

8. Reciprocals

9. Fractions

10. Solving equations

11 Zero’s reciprocal

12. Change denominators

13. Percent

14. Positive reciprocals

 

Order

15. Exceeds

16. Undercuts

17. Flip the sign

18. Or equals

 

Exponents

19. Exponents

20. Multiply binomials

21. Cubes

22. Zero power

 

Types of numbers

23. Negative numbers

24. Real numbers

25. Integers

 

Exponent tricks

26. Multiply powers

27. Copy powers

28. Increasing bases

29. Roots

30. Square roots

31. Imaginary numbers

32. Logs

33. Logs of multiplied

 

Imaginary tricks

34. Deconstructed numbers

35. Conjugates

36. Conjugate tricks

37. Norm

38. Absolute value

39. Distance

40. Triangle inequality

 

Trigonometry

41. Rad

42. Fractions of angles

43. Degrees

44. Cosine and sine

45. Cis

46. Tangent

47. Reciprocal functions

48. Arc functions


 

Fundamentals

In high-school and college, the math courses are based on algebra. Here’s how to do it. If you have any questions while reading this stuff, phone me at 603-666-6644 for free help!

 

1. Equality

The symbol “=” (pronounced “equals” or “is”) has two axioms (fundamental properties):

Axiom 1.1 (reflexive):              

Axiom 1.2 (substitution):      if , you can switch “ ” to “

The symbol “=” also leads to two definitions:

Definition 1.1 (double equality):       iff  and

Definition 1.2 (inequality):                   iff it is false that

Those axioms and definitions lead to three theorems (consequences):

Theorem 1.1 (symmetry):     iff

Theorem 1.2 (transitive):          if  then

Theorem 1.3 (dichotomy):    or

Here are the proofs:

Proof of theorem 1.1

If , axiom 1.2 says you can switch “ ” to “ ” in axiom 1.1 (which says ), giving . Conversely, if , axiom 1.2 says you can switch “ ” to “ ” in axiom 1.1 (which says ), giving .

Proof of theorem 1.2

Suppose . Since , axiom 1.2 says you can switch “ ” to “ ” in “ ,” giving .

Proof of theorem 1.3

Either  or it’s false that .

Exercises

To practice what you learned, try these exercises. While you try them, cover up the answers, then see if you got them right!

Solve (rewrite the equation, so the equation’s left side is ):

Problem                                     Answer

                                                                                                   (by theorem 1.1)

                                                                                                   (by theorem 1.1)

                                    (by theorem 1.1)

                                       (by theorem 1.1)

                                                     (by theorem 1.1)


2. Addition

The symbol “+” (pronounced “plus”) has two axioms:

Axiom 2.1 (commutative):   

Axiom 2.2 (associative):        

Omitting parentheses means this:

Grouping 2.1 (add left to right): 

The symbol “1” (pronounced “one”) has no axioms but leads to these definitions:

Definition 2.2 (two):      

Definition 2.3 (three):   

Definition 2.4 (four):      

Definition 2.5 (five):       

Definition 2.6 (six):              

Definition 2.7 (seven):  

Definition 2.8 (eight):         

Definition 2.9 (nine):      

The symbol “0” (pronounced “zero”) has one axiom:

Axiom 2.3 (zero):   

That symbol leads to this theorem:

Theorem 2.1 (zero on left): 

Here’s the proof:

Proof of theorem 2.1

.

Exercises

Simplify (write as briefly as possible):

Problem                                                   Answer (and reason)

                                                                               (by grouping 2.1)

                                                                                         (by definition 2.4)

                                                                                              (by definition 2.4)

                                                                                                                  (by axiom 2.3)

                                                                                                                  (by axiom 2.3)

                                                                                                                  (by axiom 2.3)

                                                                                         (by axiom 2.3)

                                                                                                                  (by theorem 2.1)

                                                                                                                  (by theorem 2.1)

Evaluate (put numbers in for letters, then simplify):

Problem                                                   Answer (and reason)

If , what is ?                                          4                           (by definition 2.4)

If , what is ?                                                   (by axiom 2.3)

Prove the following. (That means give a reason why it’s true, using the axioms, definitions, theorems, and groupings you’ve learned so far.)

To prove                                            Answer (proof)

                                                                  

                                                                                                                (by axiom 2.1)

                                                                                                                          (by definition 2.8)

                                                                  

                                                                                                 (by definition 2.2)

                                                                                                 (by axiom 2.2)

                                                                                                                (by definition 2.3)

                                                                                                                          (by definition 2.4)

3. Negatives

The symbol “ ” (pronounced “negative” or “minus”) has one axiom:

Axiom 3.1 (negative):    

Omitting parentheses means this:

Grouping 3.1 (negate before summing):   

The axiom leads to 6 theorems:

Theorem 3.1 (negative zero):                

Theorem 3.2 (add to both sides):          iff

Theorem 3.3 (negative test):                    iff

Theorem 3.4 (double negative):           

Theorem 3.5 (distribute negative):   

Theorem 3.6 (negate both sides):         iff

Here are the proofs:

Proof of theorem 3.1

.

Proof of theorem 3.2

If , axiom 1.2 says you can switch “ ” to “ ” in the equation “ ,” giving . Conversely, if , we have ,” which simplifies to “ .”

Proof of theorem 3.3

 

iff          (by theorem 3.2)

iff                                                 (by simplifying both sides)

Proof of theorem 3.4

                                           (by axiom 3.1)

so                                           (by axiom 2.1)

so                                              (by theorem 3.3)

Proof of theorem 3.5

(  

so               (by theorem 3.3)

Proof of theorem 3.6

If , axiom 1.2 says you can switch “ ” to “ ” in the equation “ ,” giving . Conversely, if , we have “ ,” which simplifies to “ .”

Exercises

Simplify:

Problem                                           How to begin         Final answer

                                                                                                    2

                                                                                                   

                                                                                           

                                                                                           

                                                                                 

                                                                           

                                                                                                  

                                                                                                

                                                                                   

                                                                          

                                                                                                  5

Evaluate:

Problem                                                             Answer (and reason)

If , what is ?                                                     0                 (axiom 3.1)

If , what is ?                                                        5                 (theorem 3.4)

If , what is ?                                                               0                 (theorem 3.1)

If , what is ?                                                                 (theorem 3.4)

If  and , what is ?                                       (theorem 3.5)


Prove:

To prove                                        Answer (proof)

                                                         

                                                                                             (by axiom 2.1)

                                                                                     0                                          (by axiom 3.1)

 iff                                        

                                                                                     iff           (by theorem 3.6)

                                                                                     iff                   (by theorem 3.4)

                                                                                     iff                   (by theorem 1.1)

 iff                                    

                                                                                     iff                   (by theorem 3.3)

                                                                                     iff                   (by previous proof)

Solve (rewrite so the equation’s left side is , then simplify):

Problem           Solution

                 To make the  be alone on the left side, get rid of the 7,

                                        by adding  to both sides of the equation, like this:

                                       

                                        Then simplify each side of the equation, so you get:

                                       

                 To make the  be alone on the left side, get rid of the ,

                                        by adding  to both sides of the equation, like this:

                                       

                                        Then simplify each side of the equation, so you get:

                                       

             To make  be alone on the left side, get rid of the ,

                                        by adding 1 to both sides, so you get:

                                       

                                        Then simplify both sides, so you get:

                                       

                       To make  be alone on the left side, get rid of the ,

                                        by taking the negative of both sides, so you get:

                                       

                                        Then simplify the left side, so you get:

                                       

         To make  be alone on the left side, get rid of the ,

                                        by adding 1 to both sides, so you get:

                                       

                                        Then simplify both sides, so you get:

                                       

                                        To make  be alone on the left side, get rid of the ,

                                        by taking the negative of both sides, so you get:

                                       

                                        Then simplify the left side, so you get:

                                       

             To make  be alone on the left side, get rid of the ,

                                        by adding  to both sides, so you get:

                                       

                                        Then simplify the left side, so you get:

                                       

             To make  be alone on its side, get rid of the ,

                                        by adding 1 to both sides, so you get:

                                       

                                        Then simplify both sides, so you get:

                                       

                                        Then rewrite the equation so  is on the left:

                                       


 

Answer the questions below. To do so, rewrite each question so it looks like an equation:

change “what” to

change “is” or “gives” or “results in” to =

change “plus” or “added to” or “increased by” or “more than” to +

change “negative” or “minus” to

Then solve the equation, to find out what  is.

Question                                                        Solution

What number plus 7 is 0?                                              

                                                                                                             

                                                                                                             

                                                                                                              The answer is

 

The negative of what number is ?                

                                                                                                             

                                                                                                              The answer is 2

 

 is 1 more than what number?                       

                                                                                                        

                                                                                                        

                                                                                                             

                                                                                                             

                                                                                                             

                                                                                                              The answer is

 

What number, when added to , gives 1?   

                                                                                                             

                                                                                                             

                                                                                                             

                                                                                                             

                                                                                                              The answer is 4

 

4. Subtraction

The symbol “ ” leads to this definition:

Definition 3.1 (subtract): 

That definition leads to 7 theorems:

Theorem 4.1 (subtract from itself):                 

Theorem 4.2 (subtract from zero):                  

Theorem 4.3 (subtract zero):                               

Theorem 4.4 (subtract a negative):                  

Theorem 4.5 (reverse subtraction):                

Theorem 4.6 (subtract from both sides):       iff

Theorem 4.7 (solve simple equation):              iff

Theorem 4.8 (difference is solution):                iff

Here are the proofs:

Proof of theorem 4.1

.

Proof of theorem 4.2

.

Proof of theorem 4.3

.

Proof of theorem 4.4

.

Proof of theorem 4.5

.

Proof of theorem 4.6

 

iff  

iff .

Proof of theorem 4.7

 

iff  

iff .

Proof of theorem 4.8

 

iff  

iff  

iff .

Exercises

Simplify:

Problem                        Solution

                                          

                                          

                                          

                                      

Simplify (write as briefly as possible, and try to write the letters before the numbers):

Problem                               How to begin         Final answer

                                                                                     

                                                             

                                                             

                                                             

                                                             

                                                             

                                                         

                                                    9

                                     

                                

Solve (rewrite so  is alone on the equation’s left side, then simplify):

Problem           Solution

                 To make the  be alone on the left side, get rid of the ,

                                        by adding  to both sides of the equation, like this:

                                       

                                        Then simplify each side of the equation, so you get:

                                       

 

                 To make the  be alone on the left side, get rid of the ,

                                        by adding  to both sides of the equation, like this:

                                       

                                        Then simplify the equation’s left side, so you get:

                                       

                                        If you wish, alphabetize the equation’s right side, to get:

                                       

 

             To make the  be alone on the left side, get rid of its ,

                                        by taking the negative of both sides, so you get:

                                       

                                        Then simplify both sides, so you get:

                                       

 

  To make the  be alone on the left side, get rid of the 5,

                                        by adding  to both sides of the equation, like this:

                                       

                                        Then simplify both sides, so you get:

                                       

                                        To make the  be alone on the left side, get rid of its ,

                                        by taking the negative of both sides, so you get:

                                       

                                        Then simplify both sides, so you get:

                                       

                                        Put the letter before the number:

                                       


Answer these questions (by turning each question into an equation, then solving the equation):

Question                                                        Solution

2 minus what number is ?                                      

                                                                                                             

                                                                                                             

                                                                                                             

                                                                                                             

                                                                                                              The answer is 7

 

What is  subtracted from ?                                     

                                                                                                              The answer is

                                                                                                              (The answer is not )

 

 is 4 less than what number?                                

                                                                                                             

                                                                                                             

                                                                                                             

                                                                                                             

                                                                                                              The answer is

 


5. Multiplication

The multiplication symbol “ ” is a raised dot and pronounced “times.” For example, “ ” is pronounced “3 times 2.”

Mathematicians are usually too lazy to write the raised dot. Instead of writing “ ,” they write just “ .” Instead of writing “ ,” they write just “ .” You can take that shortcut also, and omit writing dots in those examples.

But you must write the raised dot in “ ,” because if you omit that dot the number will look too much like thirty-two. You must write the raised dot in “ ,” because if you omit that dot the number will look too much like 3 minus 2.

Omitting parentheses means this:

Grouping 5.1 (multiply before adding):         

Grouping 5.2 (multiply before negating):

Multiplication has 4 axioms:

Axiom 5.1 (one):                                                                 

Axiom 5.2 (multiplication commutative):         

Axiom 5.3 (multiplication associative):              

Axiom 5.4 (distributive):                                              

Multiplication leads to this definition:

Definition 5.1 (triple multiplication):                   

The axioms lead to these theorems:

Theorem 5.1 (one on right):         

Theorem 5.2 (sum multiplied):  

Theorem 5.3 (double):                     

Theorem 5.4 (triple):                             

Theorem 5.5 (zero multiplied)   

Here are the proofs:

Proof of theorem 5.1

.

Proof of theorem 5.2

.

Proof of theorem 5.3

.

Proof of theorem 5.4

.

Proof of theorem 5.5

.

Subtracting  from both sides of that equation, we get .


6. Negative multiplication

Here are theorems about multiplying negatives:

Theorem 6.1 (negative multiplication):         

Theorem 6.2 (minus one):                                      

Theorem 6.3 (multiply by negative):               

Theorem 6.4 (negative times negative):        

Theorem 6.5 (multiply by difference):           

Here are the proofs:

Proof of theorem 6.1

.

So by the negative-test theorem, we get ( .

Proof of theorem 6.2

.

Proof of theorem 6.3

).

Proof of theorem 6.4

.

Proof of theorem 6.5

.

 

7. Positive numbers

Some numbers are called positive. Here are axioms about being positive:

Axiom 7.1 (one positive):                1 is positive

Axiom 7.2 (zero not positive):          0 is not positive

Axiom 7.3 (sum positive):                    if  and  are positive, so is

Axiom 7.4 (product positive):           if  and  are positive, so is

Those axioms lead to these theorems:

Theorem 7.1 (two positive):                        2 is positive

Theorem 7.2 (positive not zero):              if  is positive then

Theorem 7.3 (one not zero):                       

Theorem 7.4 (negative not positive):     if  is positive,  is not positive

Here are the proofs:

Proof of theorem 7.1

Since 1 is positive, axiom 7.3 says  is positive, so 2 is positive.

Proof of theorem 7.2

Suppose  is positive. If  were 0, then 0 would be positive also, contradicting axiom 7.2. So  is not 0.

Proof of theorem 7.3

Since 1 is positive, theorem 7.2 says .

Proof of theorem 7.4

If  and  were both positive, axiom 7.3 says would be positive, so 0 would be positive, contradicting axiom 7.2.


Reciprocals

Let’s explore reciprocals and their consequences.

 

8. Reciprocals

Every number has a reciprocal. For example, since there’s a number called 4, there’s also a number called
the reciprocal of 4, which is written . (Later, we’ll prove that it can also be written as  and as .) It’s called
the reciprocal of 4 or just reciprocal 4 or bar over 4 or
bar 4 or over 4 or a 4th. More generally,  is called
the reciprocal of  or just reciprocal  or bar over  or
bar  or over .

Here’s another way to pronounce reciprocals:

 is pronounced “a half”

 is pronounced “a third”

 is pronounced “a fourth”

 is pronounced “a fifth”

 is pronounced “a sixth”

 is pronounced “a seventh”

 is pronounced “an eighth”

 is pronounced “a ninth”

Warning: I use the symbol  to mean the reciprocal of , but other mathematicians sometimes use the symbol  for something different (the complex conjugate of .)

Reciprocals have this axiom:

Axiom 8.1 (reciprocal):                                (if )

That axiom leads to this theorem:

Theorem 8.1 (reciprocal of one):   

Here’s the proof:

Proof of theorem 8.1

.


9. Fractions

Instead of writing , you can write the  under the , like this: . Here’s the formal definition:

Definition 9.1 (fraction):  

The  is pronounced “  over ” or “  divided by .” Dividing  by  is called division.

The symbol  is called a fraction. For example,  is a fraction. To write a fraction, you write a horizontal line; the number above that line is called the fraction’s top (or numerator); the number below that line is called fraction’s bottom (or denominator). For example, in the fraction , the 2 is called the top (or numerator); the 3 is called the bottom (or denominator). In the fraction , the top (numerator) is ; the bottom (denominator) is .

Here are theorems about fractions:

Theorem 9.1 (top one):                                   

Theorem 9.2 (bottom one):                    

Theorem 9.3 (top zero):                            

Theorem 9.4 (both zero):                              

Theorem 9.5 (both same):                         (if )

Theorem 9.6 (remove bottom):             (if )

Theorem 9.7 (top negative):                   

Theorem 9.8 (multiply by fraction): 

Theorem 9.9 (add fractions):                 

Here are the proofs:

Proof of theorem 9.1

.

Proof of theorem 9.2

.

Proof of theorem 9.3

.

Proof of theorem 9.4

In theorem 9.3, make  be 0.

Proof of theorem 9.5

.

Proof of theorem 9.6

.

Proof of theorem 9.7

.

Proof of theorem 9.8

.

Proof of theorem 9.9

.

10. Solving equations

Axiom 8.1 leads to these theorems about “if” and “iff”:

Theorem 10.1 (multiply by both sides):

iff  (assuming )

Theorem 10.2 (divide into both sides):

iff  (assuming )

Theorem 10.3 (factor removed or zero):

iff (  or )

Theorem 10.4 (product zero):

 iff (  or )

Theorem 10.5 (roots from factors):

 iff (  or )

Theorem 10.6 (product not zero):

 iff (  and )

Theorem 10.7 (reciprocal test):

if  then

Theorem 10.8 (division test):

 iff  (assuming )

Theorem 10.9 (solve linear equations):

 iff  (assuming )

Here are the proofs:

Proof of theorem 10.1

If , obviously .

Conversely,

if , we have “ ,” which simplifies to “ .”

Proof of theorem 10.2

If , obviously .

Conversely, if , we have “ ,” which simplifies to “ .”

Proof of theorem 10.3

If , obviously .

If ,  (because  and  are both 0).

Conversely, if , theorem 10.1 says (  or ).

Proof of theorem 10.4

If  or  then .

Conversely,

if  and , we have “ ” which simplifies to “ .”

Proof of theorem 10.5

 

iff (  or )

iff (  or ).

Proof of theorem 10.6

 iff not (  or ) iff (  and ).

Proof of theorem 10.7

If

then

then .

Proof of theorem 10.8

 

iff

iff .

Proof of theorem 10.9

 

iff

iff .

Theorem 10.8 lets you calculate answers to division problems. For example, since , you have .

 

11. Zero’s reciprocal

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 .” We’ll use a smarter approach: we’ll define the reciprocal of 0 to be 0, by adding this axiom:

Axiom 11.1 (zero reciprocal):

That axiom saves us from having to say “assuming ” so often, though we’ll still have to say “assuming ” occasionally.

That axiom leads to these theorems:

Theorem 11.1 (bottom zero):                                             

Theorem 11.2 (zero means reciprocal is zero):   iff

Theorem 11.3 (reciprocal not zero):                                iff

Theorem 11.4 (double reciprocal):                                 

Theorem 11.5 (reciprocal of negative):                        

Theorem 11.6 (reciprocal of product):                         

Theorem 11.7 (reciprocal of quotient):                        

Theorem 11.8 (reciprocate both sides):                        iff

Here are the proofs:

Proof of theorem 11.1

.

Proof of theorem 11.2

If , axiom 11.1 says .

Conversely, if  and  were not 0,

we’d have “ ,” contradicting theorem 7.3.

Proof of theorem 11.3

This theorem just restates theorem 11.2.

Proof of theorem 11.4

If  then .

If  then  so theorem 10.7 says .

Proof of theorem 11.5

If  then both sides of that equation are 0.

If  then ( , so theorem 10.7 says .

Proof of theorem 11.6

If  or  is 0 then  and  are both 0.

If neither  nor  is 0 then ,

so theorem 10.7 says .

Proof of theorem 11.7

.

Proof of theorem 11.8

If , obviously .

Conversely, if , we have “ ,” which simplifies to “ .”


 

12. Change denominator

The theorems about reciprocals lead to these theorems about changing the denominator:

Theorem 12.1 (bottom negative):           

Theorem 12.2 (both negative):                  

Theorem 12.3 (multiply fractions):       

Theorem 12.4 (multiply both):                   (if  or )

Theorem 12.5 (add any fractions):         

                                                                                               (if  and )

Theorem 12.6 (divide by fraction):        

Theorem 12.7 (flip both sides):                   iff

Theorem 12.8 (cross-multiply):                     iff

                                                                                               (assuming  and )

Theorem 12.9 (divide across):                    iff

                                                                                               (assuming  and )

Here are the proofs:

Proof of theorem 12.1

.

Proof of theorem 12.2

.

Proof of theorem 12.3

.

Proof of theorem 12.4

If  then .

If  and  then .

Proof of theorem 12.5

.

Proof of theorem 12.6

.

Proof of theorem 12.7

 iff  iff .

Proof of theorem 12.8

 iff  iff  iff .

Proof of theorem 12.9

 iff  iff  iff .


13. Percent

The symbol % is pronounced “percent” and means .

Definition 13.1 (percent):     

For example, 3% means , which is . For another example,  means , which can also be written as 7%.

 

14. Positive reciprocals

Here’s another axiom about reciprocals:

Axiom 14.1 (reciprocal positive):   if  is positive, so is

That axiom leads to this theorem:

Theorem 14.1 (fraction positive):  if  and  are positive, so is

Here’s the proof:

Proof of theorem 14.1

Suppose  and  are positive.

Since  is positive, axiom 14.1 says  is positive.

Since  and  are positive,  is positive, so  is positive.


 

Order

Let’s explore order and its consequences.

 

15. Exceeds

You’ve been using the equal sign, which is this symbol:

=

Now we’ll examine a different sign, which looks like this:

> 

The symbol > is pronounced “exceeds” or
is bigger than” or “is more than” or “is greater than.” For example, the statement  is pronounced “5 exceeds 3” or “5 is bigger than 3” or “5 is more than 3” or “5 is greater than 3.”

The symbol > is called the greater-than sign.

Here’s an informal explanation:

On a vertical number line, “ ” means  is higher than .

On a horizontal number line, “ ” means  is somewhere to the right of .

But now let’s get formal!

One way to get formal would be to create axioms about the symbol >. But here’s a better way: define the symbol > in terms of what we know already, so we can use the axioms and theorems we developed already, without have to invent more axioms. Here’s the trick.…

To compute whether , just compute whether  is positive.

Definition 15.1 (exceeds):                           iff  is positive

Definition 15.2 (double exceeds):     iff  and


Those definitions lead to these theorems:

Theorem 15.1 (exceeds zero):                           iff  is positive

Theorem 15.2 (add to exceeds):                       iff

Theorem 15.3 (subtract from exceeds):     iff

Theorem 15.4 (exceeds is transitive):        if  then

Theorem 15.5 (sum the exceeds):                  if  and

                                                                                                    then

Theorem 15.6 (exceeds itself):                        ” is false

Theorem 15.7 (exceeds can’t reverse): if  then “ ” is false

Theorem 15.8 (multiply exceeds):           iff

                                                                                                    (assuming  is positive)

Here are the proofs:

Proof of theorem 15.1

 

iff  is positive

iff  is positive.

Proof of theorem 15.2

 

iff  is positive

iff  is positive

iff  is positive

iff .

Proof of theorem 15.3

 

iff

iff .

Proof of theorem 15.4

If

then  and

then  and  are positive

then  is positive

then  is positive

then .

Proof of theorem 15.5

If  and

then  and  are positive

then  is positive

then  is postivie

then .

Proof of theorem 15.6

If “ ” were true

we’d have  is positive

then 0 is positive, but that contradicts axiom 7.2.

Proof of theorem 15.7

If “ ” were true

we'd have  and

then , but that contradicts theorem 15.6.

Proof of theorem 15.8

If

then  is positive

then  is positive (since  is positive)

then  is positive

then .

Conversely, if ,

then  is positive

then  is positive

then  is positive (since  is positive)

then  is positive

then .

16. Undercuts

The opposite of > is <, which is pronounced “undercuts” or “is smaller than” or “is less than.” For example, the statement  is pronounced “3 undercuts 5” or “3 is smaller than 5” or “3 is less than 5.” The symbol < is called the less-than sign.

Definition 16.1 (undercuts):                          iff

Definition 16.2 (double undercuts):         iff  and

Each theorem about “exceeds” leads to a theorem about “undercuts”:

Theorem 16.1 (zero undercuts):                          iff  is positive

Theorem 16.2 (add to undercuts):                      iff

Theorem 16.3 (subtract from undercuts):    iff

Theorem 16.4 (undercuts is transitive):       if  then

Theorem 16.5 (sum the undercuts):                 if  and

                                                                                                         then

Theorem 16.6 (undercuts itself):                       ” is false

Theorem 16.7 (undercuts can’t reverse):     if  then “ ” is false

Theorem 16.8 (multiply undercuts):               iff

                                                                                                    (assuming  is positive)

The proofs are easy.

 

17. Flip the sign

These theorems relate < to >:

Theorem 17.1 (flip if negate):                             iff

Theorem 17.2 (flip if reciprocate):                if  then

Here are the proofs:

Proof of theorem 17.1

iff  is positive

iff  is positive

iff  is positive

iff

iff .

Proof of theorem 17.2

If

then  (since  is positive)

then

then

then  (since  and )

then

then .


18. Or equals

Here are more definitions:

Definition 18.1 (grequal):       iff  or

Definition 18.2 (lequal):           iff  or

Each theorem about “<” leads to a theorem about “ ”:

Theorem 18.1 (zero lequal):                                    iff  is 0 or positive

Theorem 18.2 (add to lequal):                                iff

Theorem 18.3 (subtract from lequal):              iff

Theorem 18.4 (lequal is transitive):                 if  then

Theorem 18.5 (sum the lequals):                        if  and

                                                                                                         then

Theorem 18.6 (lequal itself):                                

Theorem 18.7 (lequal can’t reverse):               if  then “ ” is false

Theorem 18.8 (flip lequal if negate):              iff

Theorem 18.9 (flip lequal if reciprocate):      if  then

The proofs are easy.


 

Exponents

Let’s explore exponents and their consequences.

 

19. Exponents

We’ve discussed addition (written as ), subtraction ( ), multiplication ( ), and division ( ). Now we’ll discuss an operation called exponentiation ( ).

In the operation , the  is called the exponent (or power); the  is called the base; the  is called “the base  raised to the  power” or just “  to the .”

 is called “  to the 2” or “  squared.”

 is called “  to the 3” or “  cubed.”

 is called “  to the 4” or “  to the 4th.”

 is called “  to the 5” or “  to the 5 th.”

Compute powers before computing anything else:

Grouping 5.1 (powers before adding):           

Grouping 5.2 (powers before negating):            

Grouping 5.3 (powers before multiplying):     

Here are the basic axioms about powers:

Axiom 19.1 (power of 1):      

Axiom 19.2 (add powers):     if (  or )

Those axioms lead to these theorems:

Theorem 19.1 (square):                            

Theorem 19.2 (next power):                    if (  or )

Theorem 19.3 (previous power):         if (  or )

Theorem 19.4 (power of zero):              if

Theorem 19.5 (cube):                                      

Theorem 19.6 (square of negative):  

Here are the proofs:

Proof of theorem 19.1

.

Proof of theorem 19.2

.

Proof of theorem 19.3

.

Proof of theorem 19.4

.

Proof of theorem 19.5

.

Proof of theorem 19.6

.


20. Multiply binomials

Here are advanced theorems about multiplication:

Theorem 20.1 (FOIL):                                           

                                                                                                   

Theorem 20.2 (square a sum):                        

Theorem 20.3 (square a difference):      

Theorem 20.4 (difference of squares): 

Theorem 20.5 (equal squares):                        iff

Theorem 20.6 (factor by guessing):             ,

                                                                                                    if  and

Theorem 20.7 (factor all by guessing):  ,

                                                                                                    if  and

                                                                                                    and

Here are the proofs:

Proof of theorem 20.1

 

 

.

Proof of theorem 20.2

 

 

 

.

Proof of theorem 20.3

 

 

 

.

Proof of theorem 20.4

 

 

.

Proof of theorem 20.5

 

iff

iff

iff  or

iff  or

iff .

Proof of theorem 20.6

 

 

 

.

Proof of theorem 20.7

 

 

 

 

 

.

Theorem 20.1 is called the FOIL theorem (or FOIL method) because it says that to compute , add together these products:

the First term in each factor:   

the Outer terms:                              

the Inner terms:                               

the Last term in each factor:     

 

21. Cubes

Here are advanced theorems about :

Theorem 21.1 (difference of cubes): 

Theorem 21.2 (sum of cubes):                   

Theorem 21.3(cube a sum):                              

                                                              

Here are the proofs:

Proof of theorem 21.1

 

 

.

Proof of theorem 21.2

 

 

.

Proof of theorem 21.3

 

 

 

 

.


22. Zero power

This axiom is more advanced:

Axiom 22.1 (zero power):         

That axiom leads to these theorems:

Theorem 22.1 (zero to the zero):                       

Theorem 22.2 (power gives zero):                     iff (  and )

Theorem 22.3 (negative power):                       

Theorem 22.4 (negative power fraction):    

Theorem 22.5 (-1 power):                                      

Theorem 22.6 (-1 power fraction):                  

Theorem 22.7 (subtract powers):                      if (  or )

Here are the proofs:

Proof of theorem 22.1

Apply axiom 22.1 to the case when  is 0.

Proof of theorem 22.2

If  then .

If  then  (because if  were 0 we’d have , contradicting theorem 7.3).

Conversely, if (  and ) then .

Proof of theorem 22.3

If , then  and  are both 1.

If  and , then  and  are both 0.

If , then we have