Expansion of Numbers and Operations by EngliSea on 2021-10-29 ᐥ+addition (−subtraction) ↺ ·multiplication (/division) ↺ ˄exponentiation (˅root ⍻logarithm)ᐥ 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