Subtraction and division

Integers

Subtraction of natural numbers

We have inverse functions for addition. This is subtraction.

For function \(\oplus\), its inverse is \(\oplus' \), as defined below:

\(a\oplus b=c\)

\(b=c\oplus 'a\)

\(f(a,b)=c\rightarrow f^{-1}(c,b)=a\)

Subtraction

\(a+b=c\rightarrow b=c-a\)

There is no natural number \(b\) that satisfies:

\(3+b=2\)

While addition and multiplication are defined across all natural numbers, subtraction is not.

Properties of subtraction

Subtraction is not commutative:

\(x-y\ne y-x\)

Subtraction is not associative:

\(x-(y-z)\ne (x-y)-z\)

Division

Introduction

We have inverse functions for multiplication. This is division.

These will not necessarily have solutions for natural numbers or integers.

Division of natural numbers

\(a.b=c\rightarrow b=\dfrac{c}{a}\)

Division is not commutative

Division is not commutative:

\(\dfrac{x}{y}\ne \dfrac{y}{x}\)

Division is not associative

\(\dfrac{x}{\dfrac{y}{z}}\ne \dfrac{\dfrac{x}{y}}{z}\)

Division is not left distributive

Division is not left distributive over subtraction:

\(\dfrac{a}{b-c} \ne \dfrac{a}{b} -\dfrac{a}{c}\)

Division is right distributive

Division is right distributive over subtraction:

\(\dfrac{a-b}{c} =\dfrac{a}{b} -\dfrac{b}{c}\)

Division of integers