Construction of the Integers

The integers can be constructed from the natural numbers using natural arithmetic operations. An integer is constructed as an ordered pair of natural numbers: (= A (<,> (head A) (tail A))). The first number in the pair (head A) is called the head and represents the positive part of the integer. The second number in the pair (tail A) is called the tail and represents the negative part of the integer.

Two integers A and B are equal if subtracting the negative part from the positive part results in the same number (= (- (head A) (tail A)) (- (head B) (tail B))), but we actually cannot use subtraction yet because subtraction is only defined after the integers have been constructed. But we can use an equivalent equation (= (+ (head A) (tail B)) (+ (head B) (tail A))) /edit/peano_new/arithmetic/integers/equality_min.gh/df-eq to define integer equality.

The additive inverse is found by reversing the order of the negative and positive parts. For example, the integer (2) is represented by (<,> (2) (0)). Its additive inverse, the integer (-n (2)) is represented by (<,> (0) (2)).

Multiple ordered pairs correspond to the same integer. The integer (0) can be represented as (= (0) (<,> (0) (0))) or (= (0) (<,> (1) (1))) or (= (0) (<,> (2) (2))), etc. Each integer represents a class of equivalent ordered pairs.

The difficult part of constructing the integers is to shown that the arithmetic operations are properly defined over these equivalence classes. Integer addition is defined by adding the positive and negative parts (= (+ (<,> a b) (<,> c d)) (<,> (n.+ a c) (n.+ b d))) /edit/peano_new/arithmetic/integers/add_multiply_min.gh/addop. Integer multiplication is defined by the formula (= (* (<,> a b) (<,> c d)) (<,> (n.+ (n.* a c) (n.* b d)) (n.+ (n.* a d) (n.* b c)))) /edit/peano_new/arithmetic/integers/add_multiply_min.gh/mulop. We can prove all the important properties of addition and multiplication with the equivalence classes:

Arithmetic Overview

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