(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 |