➊ Addition
The natural numbers are closed under addition.
➋ Subtraction is the inverse operation of addition.
l + m = n
n - m = l
Define 0 and negative integers then integers are closed under subtraction.
➌ Multiplication is repeated addition.
p + p + p = p·3
p·0 means 'add p zero times' which is 0.
p·1 means 'add p one time' which is p.
Define p·(-q) as 'subtract p q times' then integers are still closed under multiplication.
➍ Division is the inverse operation of multiplication.
p · q = r
r / q = p
Define rational numbers then they are closed under division.
➎ Exponentiation is repeated multiplication.
p · p · p = p ˄ 3
p ˄ 1 means 'multiply p one time' which means 'add p one time' that is p.
(p˄m)·(p˄n) = p˄(m+n)
(p˄m)˄n = p˄(m·n)
(p·q)˄n = (p˄n)·(q˄n)
➏ 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 exponentiation.
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.
➐ Logarithm is a new inverse operation of exponentiation.
p ˄ q = r
r ⍻ p = q
