# 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 $n = m + p$.

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 tuples 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.

It follows that:

- $\forall a, b, c, d \in \N: \eqclass {\tuple {a, b} } \boxminus - \eqclass {\tuple {c, d} } \boxminus = \eqclass {\tuple {a, b} } \boxminus + \tuple {-\eqclass {\tuple {c, d} } \boxminus} = \eqclass {\tuple {a, b} } \boxminus + \eqclass {\tuple {d, c} } \boxminus$

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

- $\forall x, y \in \Z: x - y = x + \paren {-y}$

### Rational Numbers

Let $\struct {\Q, +, \times}$ be the field of rational numbers.

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

- $\forall a, b \in \Q: a - b := a + \paren {-b}$

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

### Real Numbers

Let $\struct {\R, +, \times}$ be the field of real numbers.

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

- $\forall a, b \in \R: a - b := a + \paren {-b}$

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

### Complex Numbers

Let $\struct {\C, +, \times}$ be the field of complex numbers.

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

- $\forall a, b \in \C: a - b := a + \paren {-b}$

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
- $\forall x \in \R: x -_{\overline \R} \paren {+\infty} = \paren {-\infty} -_{\overline \R} x := -\infty$
- $\forall x \in \R: x -_{\overline \R} \paren {-\infty} = \paren {+\infty} -_{\overline \R} x := +\infty$
- $\paren {-\infty} -_{\overline \R} \paren {+\infty} := -\infty$
- $\paren {+\infty} -_{\overline \R} \paren {-\infty} := +\infty$

In particular, the expressions:

- $\paren {+\infty} -_{\overline \R} \paren {+\infty}$
- $\paren {-\infty} -_{\overline \R} \paren {-\infty}$

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

### Ring

Let $\struct {R, +, \circ}$ be a ring.

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

- $\forall a, b \in R: a - b := a + \paren {-b}$

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.

In certain historical texts, the term **subduction** can sometimes be seen.

## Also see

- Results about
**subtraction**can be found here.

## 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 $2.1$ - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $2$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $2$