numbers ℕ ℤ ℚ ℝᐥ
The whole numbers are closed under addition.
Subtraction is the inverse operation of addition.
l + m = n
n - m = l
Define negative integers then integers are closed under subtraction.
Multiplication is repeated addition.
p + p + p = p · 3
s = "p · 0 means 'add p zero times' which is 0.";
s = "p · 1 means 'add p once' which is p.";
s = "Define p·(-q) as 'subtract p q times' then integers are still closed under multiplication.";
Division is the inverse operation of multiplication.
p · q = r , r / q = p
Define rational numbers then they are closed under division.
Exponentiation is repeated multiplication.
p · p · p = p ˄ 3
p˄1 is p.
(p˄m) × (p˄n) = p˄(m+n)
(p˄m) ˄ n = p ˄ (m×n)
(p×q) ˄ n = (p˄n) × (q˄n)
Root is the inverse operation of exponentiation.
p ˄ q = r
r ˅ q = p
Define p ˄ 1/n as p ˅ n ,
then (p˄n) ˄ 1/n = p ˄ (n·(1/n)) = p ˄ 1 = p.
From this time, root is a part of exponent.
The limit of p ˄ x as x approaches 0 from the right is 1.
So define p ˄ 0 as 1.
Define p ˄ (-n) as 1 / (p˄n),
then (p˄n) × (p˄(-n)) = p ˄ (n+(-n)) = p ˄ 0 = 1.
Logarithm is a new inverse operation of exponentiation.
p ˄ q = r , r ⍻ p = q
