 Expansion of Numbers and Operations by trustle on 2020-06-18 (1) Natural numbers are closed under addition. (2) Subtraction is the inverse operation of addition. l + m = n , n - m = l Define negative integers then integers are closed under subtraction. (3) Multiplication is repeated addition. p + p + p = p × 3 p × 0 means 'add p zero times' which is 0. p × 1 means 'add p once' which is p. Define p × (-n) as 'subtract p n times' then integers are still closed under multiplication. (4) Division is the inverse operation of multiplication. p × q = r , r / q = p Define rational numbers then they are closed under division. (5) Exponentiation is repeated multiplication. p × p × p = p ∧ 3 p ∧ 1 is p. Identities and properties: (p∧m) × (p∧n) = p∧(m+n) (p∧m) ∧ n = p ∧ (m×n) (p×q) ∧ n = (p∧n) × (q∧n) (6) Root is the inverse operation of exponentiation. p ∧ n = q , q ∨ n = p Define p ∧ 1/n as p ∨ n , then ( p ∧ n ) ∧ 1/n = p ∧ ( n × 1/n ) = p ∧ 1 = p. From this, root is no longer the inverse operation 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. (7) Logarithm is a new inverse operation of exponentiation. p ∧ n = q , q ∀ p = n  