# Definition:Subtraction

## 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.

## Abstract Algebra

In the context of abstract algebra, the concept of subtraction is defined as follows:

### Ring Subtraction

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 $b$.

### Field Subtraction

Let $\struct {F, +, \times}$ be a field.

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

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

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

## Linear Algebra

### Vector Subtraction

Let $\struct {F, +_F, \times_F}$ be a field.

Let $\struct {G, +_G}$ be an abelian group.

Let $V := \struct {G, +_G, \circ}_R$ be the corresponding vector space over $F$.

Let $\mathbf x$ and $\mathbf y$ be vectors of $V$.

Then the operation of (vector) subtraction on $\mathbf x$ and $\mathbf y$ is defined as:

$\mathbf x - \mathbf y := \mathbf x + \paren {-\mathbf y}$

where $-\mathbf y$ is the negative of $\mathbf y$.

The $+$ on the right hand side is vector addition.

### Arrow Representation

Let $\mathbf u$ and $\mathbf v$ be vector quantities of the same physical property.

Let $\mathbf u$ and $\mathbf v$ be represented by arrows embedded in the plane such that:

$\mathbf u$ is represented by $\vec {AB}$
$\mathbf v$ is represented by $\vec {AC}$

that is, so that the initial point of $\mathbf v$ is identified with the initial point of $\mathbf u$. Then their (vector) difference $\mathbf u - \mathbf v$ is represented by the arrow $\vec {CB}$.

## Terminology

The symbol $-$ is known as the minus sign.

Hence:

$5 - 3$

$5$ minus $3$

### Minuend

Let $a - b$ denote the operation of subtraction on two objects.

The object $a$ is known as the minuend of $a - b$.

### Subtrahend

Let $a - b$ denote the operation of subtraction on two objects.

The object $b$ is known as the subtrahend of $a - b$.

### Difference

Let $a - b$ denote the operation of subtraction on two objects $a$ and $b$.

Then the result $a - b$ is referred to as the difference of $a$ and $b$.

## Also known as

The result $a - b$ of a subtraction operation 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 some historical texts, the term subduction can sometimes be seen.

## Examples

### Example: $x + 3 = 5$

The equation:

$x + 3 = 5$

has the solution:

$x = 2$

## 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 $1481$.

However, Regiomontanus was the first to use it in its current shape, in an unpublished manuscript from $1456$.