# Definition:Subtraction

## Contents

## Definition

### Natural Numbers

Let $\N$ be the set of natural numbers.

Let $m, n \in \N$ such that $m \le n$.

Let $p \in \N$ such that $m + n = p$.

Then we define the operation **subtraction** as:

- $n - m = p$

The natural number $p$ is known as the **difference** between $m$ and $n$.

### Integers

The **subtraction** operation in the domain of integers $\Z$ is written "$-$".

As the set of integers is the Inverse Completion of Natural Numbers, it follows that elements of $\Z$ are the isomorphic images of the elements of equivalence classes of $\N \times \N$ where two tuples are equivalent if the difference between the two elements of each tuple is the same.

Thus subtraction can be formally defined on $\Z$ as the operation induced on those equivalence classes as specified in the definition of integers.

That is, the integers being defined as all the difference congruence classes, integer multiplication can be defined directly as the operation induced by natural number multiplication on these congruence classes.

It follows that:

- $\forall a, b, c, d \in \N: \left[\!\left[{\left({a, b}\right)}\right]\!\right]_\boxminus - \left[\!\left[{\left({c, d}\right)}\right]\!\right]_\boxminus = \left[\!\left[{\left({a, b}\right)}\right]\!\right]_\boxminus + \left({- \left[\!\left[{\left({c, d}\right)}\right]\!\right]_\boxminus}\right) = \left[\!\left[{\left({a, b}\right)}\right]\!\right]_\boxminus + \left[\!\left[{\left({d, c}\right)}\right]\!\right]_\boxminus$

Thus integer subtraction is defined between all pairs of integers, such that:

- $\forall x, y \in \Z: x - y = x + \left({-y}\right)$

### Complex Numbers

Let $\left({\C, +, \times}\right)$ be the field of complex numbers.

The operation of **subtraction** is defined on $\C$ as:

- $\forall a, b \in \C: a - b := a + \left({-b}\right)$

where $-b$ is the negative of $b$ in $\C$.

### Extended Real Subtraction

Let $\overline \R$ denote the extended real numbers.

Define **extended real subtraction** or **subtraction on $\overline \R$**, denoted $-_{\overline \R}: \overline \R \times \overline \R \to \overline \R$, by:

- $\forall x, y \in \R: x -_{\overline \R} y := x -_{\R} y$ where $-_\R$ denotes real subtraction
- $\forall x \in \R: x -_{\overline \R} \left({+\infty}\right) = \left({-\infty}\right) -_{\overline \R} x := -\infty$
- $\forall x \in \R: x -_{\overline \R} \left({-\infty}\right) = \left({+\infty}\right) -_{\overline \R} x := +\infty$
- $\left({-\infty}\right) -_{\overline \R} \left({+\infty}\right) := -\infty$
- $\left({+\infty}\right) -_{\overline \R} \left({-\infty}\right) := +\infty$

In particular, the expressions:

- $\left({+\infty}\right) -_{\overline \R} \left({+\infty}\right)$
- $\left({-\infty}\right) -_{\overline \R} \left({-\infty}\right)$

are considered **void** and should be avoided.

### Ring

Let $\left({R, +, \circ}\right)$ be a ring.

The operation of **subtraction** $a - b$ on $R$ is defined as:

- $\forall a, b \in R: a - b := a + \left({-b}\right)$

where $-b$ is the (ring) negative of $b$.

## Also known as

The value $a - b$ (for any of the above definitions) is often called the **difference between $a$ and $b$**.

In this context, whether $a - b$ or $b - a$ is being referred to is often irrelevant, but it pays to be careful.

## Historical Note

The symbol $-$ for subtraction originated in commerce, along with the symbol $+$ for addition, where they were used by German merchants to distinguish underweight and overweight items.

These symbols first appeared in print in $1481$.

## Sources

- 1960: Walter Ledermann:
*Complex Numbers*... (previous) ... (next): $\S 1.1$. Number Systems - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 2$: Example $2.1$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $2$