Numbers: quantity. Counting: how many. Measuring: how much
Quantitative information, processing, reasoning.
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

Real number line: infinite in both directions and infinite in every interval. (Infinite in length and in depth.)

[Demo: Desmos: turn off Y axis, Grid. Zoom in to 0]

Number as distance from 0.
|a| = distance of a from 0
|a-b| = |b-a| distance between a and b

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

Addition: move to the right.
Subtraction: move to the left.

3/5 = "three fifths" = 3*1/5 = 3÷5 = .6 = .6(100/100) = 60/100 = 60%
3 is 60% of 5    3 = 60% * 5 = .6 * 5
ratio of 3 to 5.    3:5
"three out of five"

¹¹/₃ = 3⅔ = 3.6
improper fraction, mixed number, decimal number

Rarely seen: 6/6, 21/21, etc
because = 1

the bigger the denominator, the smaller the fraction.
Ex. 1/2, 1/3, 1/4, 1/5 ...
Ex. 5/2, 5/3, 5/4, 5/5, 5/6 ...

-^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/(b/c) = ac / b
(a/b)/(c/d) = ad / bc
(a-b)/(b-a) = -1
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.    3y = 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)

30	31	32	33	34		3-1	3-2	3-3
1	3	9	27	81		1/3	1/32 = 1/9	1/33 = 1/27

FOIL is double distribution: (a+b)(c+d) = a(c+d)+b(c+d)    [ and = c(a+b)+d(a+b) ]

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
√6 = √2√3

5-√2    can't be simplified.

√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|

√9 = 3    "principle" square root, i.e. positive
-√9 = -3    the negative root

x = √9   →   x=3
x = -√9   →   x=-3
Both: x = ±√9   →   Convenience notation for x=3 and/or x=-3

√2 ≈ 1.414     2√2 ≈ 2.828     √2 /2 ≈ 0.707
√3 ≈ 1.732 (GW's birthday)     √3 /2 ≈ 0.866
√5 ≈ 2.236

Number > 1 → √ is smaller than it:
   a>1, √a < a    Ex. a=4, √4=2 < 4
Number < 1 → √ is bigger than it:
   a<1, √a > a    Ex. a=¼, √¼=½ > ¼

Nth root: ⁿ√x = y ↔ yⁿ=x
Ex. √x = y ↔ y²=x
Ex. ∛x = y ↔ y³=x
Ex. ⁴√x = y ↔ y⁴=x
As n→∞, n^th root (ⁿ√) → 1

-Nth root: ^-n√x = y ↔ y^-n=x
Ex. ^-4√x = y ↔ y^-4=x
^-n√x = 1 / ⁿ√x [ = x^-1/n = 1 / x^1/n ]

x^1/n = ⁿ√x      x^1/2 = √x      x^1/3 = ∛x ...

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

Prime numbers

Memorable ones:
1111111111111111111      (19 1's) ≈ 1.1 quintillion
12345678910987654321
1 2 3...2445 2446 2445...3 2 1    (17,347 digits)
2G-1 = 2³¹-1 = 2,147,483,647    Largest 32-bit 2's-complement integer.
100...06660...001    (130 0's each side) Belphagor's prime. palindrome.
googol+267 = 10¹⁰⁰+267