# 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:

$$\dfrac{a-b}{c} =\dfrac{a}{b} -\dfrac{b}{c}$$