Numbers: quantity. Counting: how many. Measuring: how much
concept, word, symbol

Many ways to denote a number. Many forms of a number.
3 = 3.0 = ³/₁ = ⁶/₂ = ⁹/₃ = ... = ⁴²/₁₄ = ... = --3 = √9 = ∛27 = √3√3 = (√3)² = ... = 3.0×10⁰ = 3¹

2 < 4      -4 < -2      the more negative (i.e. the leftmost) is less than
Clearer if "smaller" number is to the left, using <:    x<3   instead of   3>x
Except x>5

2/5 = "two fifths" = 2*1/5 = 2÷5 = .4 = .4(100/100) = 40/100 = 40%
2 is 40% of 5    2 = 40% * 5 = .4 * 5
ratio of 2 to 5.    2:5

¹¹/₃ = 3⅔ = 3.6

-^a/_b = ^-a/_b = ^a/_-b

⅔ x = ^2x/₃ ≠ ²/_3x

^a/_b · ^b/_a = 1      the product of reciprocals equals 1
x · ¹/_x = 1

^a/_b = ^c/_d    ↔    ^b/_a = ^d/_c    flip both

split numerator ✓    (a+b)/c = a/c + b/c
split denominator ✗    a/(b+c) ≠ a/b + a/c

(a/b)/c = a / bc
a/(bc) = ac / b
(a/b)/(c/d) = ad / bc
Fraction facts

Dividing by x is the same as multiplying by its reciprocal, 1/x.
   ÷2 ≡ *½    ÷3 ≡ *⅓    ÷½ ≡ *2    ÷⅔ ≡ *3/2    ÷1½ ≡ ÷3/2 ≡ *⅔

3y = 3·y    juxtaposition means multiplication. = y + y + y

x² = x·x      2x = x+x
x³ = x·x·x    3x = x+x+x
4x³ = 4(x·x·x) = x·x·x + x·x·x + x·x·x +x·x·x

x² = (-x)² ≠ -x²
x³ ≠ (-x)³ = -x³

x⁰ = 1
x¹ = x
1^x = 1
0^x = 0    (except 0⁰ is indeterminate)

3√2   is a (one) number.    ≈4.242640687

(If c=ab then √c =) √(ab) = √a√b      Square root of a product = the product of the square roots of the factors
Simplify a root: √72 = √(36*2) = √36*√2 = 6√2

√10 ≠ 2√5      √10 = √2√5

√a² = |a|
If a>0, √a²=a Ex. a=3, √3²=√9=3
If a<0, √a²=|a| Ex. a=--3, √(-3)²=√9=3=|-3|

x^1/n = ⁿ√x      x^1/2 = √x      x^1/3 = ∛x ...
As n→∞, n^th root (ⁿ√) → 1

x^n/d = ^d√xⁿ = (^d√x)ⁿ
d^th root of the n^th power = n^th power of the d^th root
Raise to the n^th power then take that's d^th root = Take the d^th root then raise that to the n^th power
x^2/3 = ∛(x²) = (∛x)²      9^2/3 = ∛(9²) = (∛9)² ≈4.326749      8^2/3 = ∛(8²) = (∛8)² =4
x^3/2 = √(x³) = (√x)³      9^3/2 = √(9³) = (√9)³ =27      8^3/2 = √(8³) = (√8)³ ≈22.627417

A set is a collection of unique (no duplicates) objects, physical or abstract, called elements or members.
{a, b, c} No order, so same set as {c, a, b}
finite: {9, 3, -2, π}
infinite: {0,1,2,3...}
empty set: {} = ∅
interval: [a,b] all numbers between a and b, inclusive: a≤x≤b
interval: (a,b] all numbers between a and b. includes b but not a: a<x≤b

interval: [b,∞) x≥b