# 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:

- $5 - 3$

is usually read:

*$5$***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 $1456$.

## Sources

- 1951: B. Hague:
*An Introduction to Vector Analysis*(5th ed.) ... (previous) ... (next): Chapter $\text I$: Definitions. Elements of Vector Algebra: $1$. Scalar and Vector Quantities - 1960: Walter Ledermann:
*Complex Numbers*... (previous) ... (next): $\S 1.1$. Number Systems - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 2$: Compositions: Example $2.1$ - 1967: Michael Spivak:
*Calculus*... (previous) ... (next): Part $\text I$: Prologue: Chapter $1$: Basic Properties of Numbers - 1973: C.R.J. Clapham:
*Introduction to Mathematical Analysis*... (previous) ... (next): Chapter $1$: Axioms for the Real Numbers: $2$. Fields - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $2$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $2$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**difference**:**1.** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**difference**:**1.** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**difference** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**subtraction**