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

The symbol “=” also leads to two definitions:

Definition 1.1 (double
equality):

Definition 1.2
(inequality):

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

Theorem 1.1 (symmetry):

Theorem 1.2 (transitive): if

Theorem 1.3 (dichotomy):

Here are the proofs:

Proof of theorem 1.1

If

Proof of theorem 1.2

Suppose

Proof of theorem 1.3

Either

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

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)

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

Problem Answer (and reason)

If

If

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)

3. Negatives

The symbol “

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):

Theorem 3.3 (negative
test):

Theorem 3.4 (double
negative):

Theorem 3.5 (distribute
negative):

Theorem 3.6 (negate both
sides):

Here are the proofs:

Proof of theorem 3.1

Proof of theorem 3.2

If

Proof of theorem 3.3

iff

iff

Proof of theorem 3.4

so

so

Proof of theorem 3.5

(

so

Proof of theorem 3.6

If

Exercises

Simplify:

Problem How to begin Final answer

Evaluate:

Problem Answer (and reason)

If

If

If

If

If

Prove:

To prove Answer (proof)

0 (by axiom 3.1)

iff

iff

iff

iff

iff

Solve (rewrite so the equation’s left side is

Problem Solution

by adding

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

by adding

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

by adding 1 to both sides, so you get:

Then simplify both sides, so you get:

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

Then simplify the left side, so you get:

by adding 1 to both sides, so you get:

Then simplify both sides, so you get:

To
make

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

Then simplify the left side, so you get:

by
adding

Then simplify the left side, so you get:

by adding 1 to both sides, so you get:

Then simplify both sides, so you get:

Then rewrite the equation so

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

Question Solution

What number plus 7 is 0?

The
answer is

The negative of what
number is

The answer is 2

The
answer is

What number, when added to

The answer is 4

4. Subtraction

The symbol “

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):

Theorem 4.7 (solve simple
equation):

Theorem 4.8 (difference is
solution):

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

Solve (rewrite so

Problem Solution

by adding

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

by adding

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

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

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

Then simplify both sides, so you get:

by adding

Then simplify both sides, so you get:

To make the

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

The
answer is

(The
answer is not

The
answer is

5. Multiplication

The multiplication symbol “

Mathematicians are usually too lazy to write the raised dot.
Instead of writing “

But you must write the raised dot in “

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

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

Axiom 7.4 (product
positive): if

Those axioms lead to these theorems:

Theorem 7.1 (two positive): 2 is positive

Theorem 7.2 (positive not
zero): if

Theorem 7.3 (one not
zero):

Theorem 7.4 (negative not
positive): if

Here are the proofs:

Proof of theorem 7.1

Since 1 is positive, axiom
7.3 says

Proof of theorem 7.2

Suppose

Proof of theorem 7.3

Since 1 is positive, theorem
7.2 says

Proof of theorem 7.4

If

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

the reciprocal of 4
or just reciprocal 4
or bar over 4 or

bar 4 or over 4 or a 4^{th}. More
generally,

the reciprocal of

bar

Here’s another way to pronounce reciprocals:

Warning: I use the symbol

Reciprocals have this axiom:

Axiom 8.1 (reciprocal):

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

Definition 9.1 (fraction):

The

The symbol

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):

Theorem 9.6 (remove
bottom):

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

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):

Theorem 10.2 (divide into both sides):

Theorem 10.3 (factor removed or zero):

Theorem 10.4 (product zero):

Theorem 10.5 (roots from factors):

Theorem 10.6 (product not zero):

Theorem 10.7 (reciprocal test):

if

Theorem 10.8 (division test):

Theorem 10.9 (solve linear equations):

Here are the proofs:

Proof of theorem 10.1

If

Conversely,

if

Proof of theorem 10.2

If

Conversely, if

Proof of theorem 10.3

If

If

Conversely, if

Proof of theorem 10.4

If

Conversely,

if

Proof of theorem 10.5

iff (

iff (

Proof of theorem 10.6

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

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

Axiom 11.1 (zero
reciprocal):

That axiom saves us from having to say “assuming

That axiom leads to these theorems:

Theorem 11.1 (bottom
zero):

Theorem 11.2 (zero means
reciprocal is zero):

Theorem 11.3 (reciprocal not
zero):

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):

Here are the proofs:

Proof of theorem 11.1

Proof of theorem 11.2

If

Conversely, if

we’d have “

Proof of theorem 11.3

This theorem just restates theorem 11.2.

Proof of theorem 11.4

If

If

Proof of theorem 11.5

If

If

Proof of theorem 11.6

If

If neither

so theorem 10.7 says

Proof of theorem 11.7

Proof of theorem 11.8

If

Conversely, if

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):

Theorem 12.5 (add any
fractions):

(if

Theorem 12.6 (divide by fraction):

Theorem 12.7 (flip both
sides):

Theorem 12.8 (cross-multiply):

(assuming

Theorem 12.9 (divide
across):

(assuming

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

If

Proof of theorem 12.5

Proof of theorem 12.6

Proof of theorem 12.7

Proof of theorem 12.8

Proof of theorem 12.9

13. Percent

The symbol % is pronounced “percent” and means

Definition 13.1 (percent):

For example, 3% means

14. Positive reciprocals

Here’s another axiom about reciprocals:

Axiom 14.1 (reciprocal
positive): if

That axiom leads to this theorem:

Theorem 14.1 (fraction
positive): if

Here’s the proof:

Proof of theorem 14.1

Suppose

Since

Since

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

The symbol > is called the greater-than sign.

Here’s an informal explanation:

On a vertical number line,
“

On a horizontal number
line, “

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

Definition 15.1 (exceeds):

Definition 15.2 (double
exceeds):

Those definitions lead to these theorems:

Theorem 15.1 (exceeds
zero):

Theorem 15.2 (add to
exceeds):

Theorem 15.3 (subtract
from exceeds):

Theorem 15.4 (exceeds is
transitive): if

Theorem 15.5 (sum the
exceeds): if

then

Theorem 15.6 (exceeds
itself): “

Theorem 15.7 (exceeds
can’t reverse): if

Theorem 15.8 (multiply exceeds):

(assuming

Here are the proofs:

Proof of theorem 15.1

iff

iff

Proof of theorem 15.2

iff

iff

iff

iff

Proof of theorem 15.3

iff

iff

Proof of theorem 15.4

If

then

then

then

then

then

Proof of theorem 15.5

If

then

then

then

then

Proof of theorem 15.6

If “

we’d have

then 0 is positive, but that contradicts axiom 7.2.

Proof of theorem 15.7

If “

we'd have

then

Proof of theorem 15.8

If

then

then

then

then

Conversely, if

then

then

then

then

then

16. Undercuts

The opposite of > is <, which is pronounced “undercuts” or “is smaller than” or “is less than.” For
example, the statement

Definition 16.1
(undercuts):

Definition 16.2 (double
undercuts):

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

Theorem 16.1 (zero
undercuts):

Theorem 16.2 (add to
undercuts):

Theorem 16.3 (subtract
from undercuts):

Theorem 16.4 (undercuts is
transitive): if

Theorem 16.5 (sum the
undercuts): if

then

Theorem 16.6 (undercuts
itself): “

Theorem 16.7 (undercuts
can’t reverse): if

Theorem 16.8 (multiply
undercuts):

(assuming

The proofs are easy.

17. Flip the sign

These theorems relate < to >:

Theorem 17.1 (flip if
negate):

Theorem 17.2 (flip if
reciprocate): if

Here are the proofs:

Proof of theorem 17.1

iff

iff

iff

iff

iff

Proof of theorem 17.2

If

then

then

then

then

then

then

18. Or equals

Here are more definitions:

Definition 18.1 (grequal):

Definition 18.2 (lequal):

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

Theorem 18.1 (zero
lequal):

Theorem 18.2 (add to
lequal):

Theorem 18.3 (subtract
from lequal):

Theorem 18.4 (lequal is
transitive): if

Theorem 18.5 (sum the
lequals): if

then

Theorem 18.6 (lequal
itself):

Theorem 18.7 (lequal can’t
reverse): if

Theorem 18.8 (flip lequal
if negate):

Theorem 18.9 (flip lequal
if reciprocate): if

The proofs are easy.

Exponents

Let’s explore exponents and their consequences.

19. Exponents

We’ve discussed addition (written as

In the operation

^{th}.”

^{ 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):

Those axioms lead to these theorems:

Theorem 19.1 (square):

Theorem 19.2 (next power):

Theorem 19.3 (previous
power):

Theorem 19.4 (power of
zero):

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):

Theorem 20.6 (factor by
guessing):

if

Theorem 20.7 (factor all
by guessing):

if

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

iff

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

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):

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):

Here are the proofs:

Proof of theorem 22.1

Apply axiom 22.1 to the
case when

Proof of theorem 22.2

If

If

Conversely, if (

Proof of theorem 22.3

If

If

If