A basic math guide listing rules & properties for algebra & general math.

math-guide.txt

# Tables #

Addition table

 +    1   2   3   4   5   6   7   8   9  10
    ---------------------------------------
 1 |  2   3   4   5   6   7   8   9  10  11
 2 |  3   4   5   6   7   8   9  10  11  12
 3 |  4   5   6   7   8   9  10  11  12  13
 4 |  5   6   7   8   9  10  11  12  13  14
 5 |  6   7   8   9  10  11  12  13  14  15
 6 |  7   8   9  10  11  12  13  14  15  16
 7 |  8   9  10  11  12  13  14  15  16  17
 8 |  9  10  11  12  13  14  15  16  17  18
 9 | 10  11  12  13  14  15  16  17  18  19
10 | 11  12  13  14  15  16  17  18  19  20

Multiplication table

 *    1   2   3   4   5   6   7   8   9  10
    ---------------------------------------
 1 |  1   2   3   4   5   6   7   8   9  10
 2 |  2   4   6   8  10  12  14  16  18  20
 3 |  3   6   9  12  15  18  21  24  27  30
 4 |  4   8  12  16  20  24  28  32  36  40
 5 |  5  10  15  20  25  30  35  40  45  50
 6 |  6  12  18  24  30  36  42  48  54  60
 7 |  7  14  21  28  35  42  49  56  63  70
 8 |  8  16  24  32  40  48  56  64  72  80
 9 |  9  18  27  36  45  54  63  72  81  90
10 | 10  20  30  40  50  60  70  80  90 100



# Order of Operations #

The order of operations (BODMAS) is a set of rules for us to agree on multi-operation expressions.

RULES:
1. Do things in parentheses first.
2. Do exponents & roots next.
3. Divide & multiply, left to right.
4. Add & subtract, left to right.

EXAMPLES:
5 + 3 * 7 --> 5 + 21 --> 26
3 * (5² - 1) --> 3 * (25 - 1) --> 3 * 24 --> 72
5 - 2 + 10 + 3 --> 3 + 10 + 3 --> 16



# Fractions #

Fractions are ratios of 2 integers. They can be used as a fallback for decimals (0.3333333333… = ¹/₃).
Terminating decimals, like 2.491, don't HAVE to be converted into fractions, but can be: 2⁴⁹¹/₁₀₀₀

RULES:
Doing division or exponents using fractions requires parentheses. 1/⁴/₇ --> 1/(4/7)
Mixed fractions must always have parentheses in expressions. 5²/₉ --> (5 + 2/9)

EXAMPLES:
1/4 = 0.25
(2 + 1/3) * 10 = 23¹/₃
3/5 - 2 = -1.4



# Percentages #

A percent means "out of 100". You can use them as equivalent fractions or relative change.
0% is empty. 100% is whole. In general, x% = 0.01x.

RULES:
x% of y = 0.01xy
x as a % of y = x/y * 100
x + y% = x * (1 + 0.01y)

EXAMPLES:
60% of 35 = 21
80 as a % of 125 = 64
30 + 20% = 36



# Negative Numbers #

Negative numbers are just numbers less than 0. They're included in the set of integers.
{…-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6…}

RULES:
Negative numbers in expression form must be in parentheses. -5 --> (-5)
Also true for fractions. -⁵/₆ --> (-5/6)
Also true for mixed fractions. -1³/₇ --> (-1 - 3/7)

EXAMPLES:
5 - 8 = -3
(-3) * (-5) = 15
7.41/(-2) = -3.705



# Exponents #

Exponents are defined as repeated multiplication. (6³ = 6 * 6 * 6 = 216)
Exponents are only allowed to be integers. Bases can still be any number.

RULES:
aⁿ * bⁿ = (ab)ⁿ
aⁿ/bⁿ = (a/b)ⁿ
aⁿ * aᵐ = aⁿ⁺ᵐ
aⁿ/aᵐ = aⁿ⁻ᵐ
(aⁿ)ᵐ = aⁿᵐ

EXAMPLES:
6² = 36
2¹⁶ = 65536
2⁶ - 2 = 62



# Pi #

Pi is an irrational constant defined as a circle's circumference divided by its diameter.
The decimal expansion is infinite & aperiodic: 3.1415926535…

EXAMPLES:
2π = 6.2831853071… (1 revolution)
180/π = 57.2957795130… (1 radian to degrees)
π² - π² = 0



# Arithmetic #

Addition & subtraction are inverse operations of each other, as well as multiplication & division.
Addition & multiplication have the commutative & associative properties (you can switch the
order or calculate differently & get the same answer), but subtraction & division do NOT have those properties.

EXAMPLES:
(2 + 9) + 3 = 2 + (9 + 3)
6 * 2 = 2 * 6
7 + 14 = 21, 21 - 14 = 7



# Roots #

Roots are defined as the inverse of exponents. (³√(8) = 2)
The radical sign √() without a root is a square root by default.
Roots have to be integers, but bases can be any number.

EXAMPLES:
√(2) = 1.4142135623…
√(121) = 11
⁵√(59049) = 9



# Distributive Property #

The distributive property says that we can expand a list of additions or subtractions, all multiplied by a factor.
It also partially works with division. You can have the number on the right, but NOT on the left.

EXAMPLES:
3 * (4 + 6) = 3 * 4 + 3 * 6
(7 - 3)/2 = 7/2 - 3/2
(7 + 2 + 6 - 1) * 8 = 7 * 8 + 2 * 8 + 6 * 8 - 1 * 8



# Distance #

To calculate 2D Euclidean distance, we use the formula: d = √(a² + b²)
Also works for 3 dimensions: d = √(a² + b² + c²), dimensions = length of sum
Useful for things like the Pythagorean theorem, or absolute value: |a| = √(a²)
The absolute value is the distance from 0, & the answer is never negative.

EXAMPLES:
√(3² + 4²) = 5
√(9² + 1²) = 9.0553851381…
√(1² + 1² + 1²) = 1.7320508075…



# Scientific Notation #

Scientific notation is a way to express very big or very small numbers in a compact form, using powers of 10.

EXAMPLES:
5.4 * 10⁸ = 540000000
2 * 10⁻¹⁰ = 0.0000000002
3.51 * 10² = 351



# Real Numbers #

Real numbers are basically all the numbers on a continuous line. They can be positive, negative, or zero.
They can be integers, terminating decimals, fractions, mixed fractions, or infinite decimals.
A real number can have any number of digits on either side of the decimal point.

0     a     -a     a.b     -a.b     ⁱ/ⱼ     -ⁱ/ⱼ     aⁱ/ⱼ     -aⁱ/ⱼ     a.b…     -a.b…

EXAMPLES:
241
-4.523
7²/₆



# Prime Factorization #

Prime numbers are integers that only have 2 factors.
The first few prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43…
Every positive integer has its own unique prime factors. 42 = 2 * 3 * 7

EXAMPLES:
54 = 2 * 3 * 3 * 3
91 = 7 * 13
1000 = 2 * 2 * 2 * 5 * 5 * 5