# Definition:Multiplication

## Definition

**Multiplication** is the basic operation $\times$ everyone is familiar with.

For example:

- $3 \times 4 = 12$
- $13 \cdotp 2 \times 7 \cdotp 7 = 101 \cdotp 64$

### Natural Numbers

Let $+$ denote addition.

The binary operation $\times$ is recursively defined on $\N$ as follows:

- $\forall m, n \in \N: \begin{cases} m \times 0 & = 0 \\ m \times \left({n + 1}\right) & = m \times n + m \end{cases}$

This operation is called **multiplication**.

Equivalently, **multiplication** can be defined as:

- $\forall m, n \in \N: m \times n := +^n m$

where $+^n m$ denotes the $n$th power of $m$ under $+$.

### Integers

The multiplication operation in the domain of integers $\Z$ is written $\times$.

Let us define $\eqclass {\tuple {a, b} } \boxtimes$ as in the formal definition of integers.

That is, $\eqclass {\tuple {a, b} } \boxtimes$ is an equivalence class of ordered pairs of natural numbers under the congruence relation $\boxtimes$.

$\boxtimes$ is the congruence relation defined on $\N \times \N$ by $\tuple {x_1, y_1} \boxtimes \tuple {x_2, y_2} \iff x_1 + y_2 = x_2 + y_1$.

In order to streamline the notation, we will use $\eqclass {a, b} {}$ to mean $\eqclass {\tuple {a, b} } \boxtimes$, as suggested.

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 multiplication 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:

- $\forall a, b, c, d \in \N: \eqclass {a, b} {} \times \eqclass {c, d} {} = \eqclass {a \times c + b \times d, a \times d + b \times c} {}$ or, more compactly, as $\eqclass {a c + b d, a d + b c} {}$.

This can also be defined as:

- $n \times m = +^n m = \underbrace {m + m + \cdots + m}_{\text{$n$ copies of $m$} }$

and the validity of this is proved in Index Laws for Monoids.

### Modulo Multiplication

Let $m \in \Z$ be an integer.

Let $\Z_m$ be the set of integers modulo $m$:

- $\Z_m = \set {\eqclass 0 m, \eqclass 1 m, \ldots, \eqclass {m - 1} m}$

where $\eqclass x m$ is the residue class of $x$ modulo $m$.

The operation of **multiplication modulo $m$** is defined on $\Z_m$ as:

- $\eqclass a m \times_m \eqclass b m = \eqclass {a b} m$

### Rational Numbers

The **multiplication operation** in the domain of rational numbers $\Q$ is written $\times$.

Let $a = \dfrac p q, b = \dfrac r s$ where $p, q \in \Z, r, s \in \Z \setminus \set 0$.

Then $a \times b$ is defined as $\dfrac p q \times \dfrac r s = \dfrac {p \times r} {q \times s}$.

This definition follows from the definition of and proof of existence of the quotient field of any integral domain, of which the set of integers is one.

### Real Numbers

The multiplication operation in the domain of real numbers $\R$ is written $\times$.

From the definition, the real numbers are the set of all equivalence classes $\eqclass {\sequence {x_n} } {}$ of Cauchy sequences of rational numbers.

Let $x = \eqclass {\sequence {x_n} } {}, y = \eqclass {\sequence {y_n} } {}$, where $\eqclass {\sequence {x_n} } {}$ and $\eqclass {\sequence {y_n} } {}$ are such equivalence classes.

Then $x \times y$ is defined as:

- $\eqclass {\sequence {x_n} } {} \times \eqclass {\sequence {y_n} } {} = \eqclass {\sequence {x_n \times y_n} } {}$

### Complex Numbers

The **multiplication operation** in the domain of complex numbers $\C$ is written $\times$.

Let $z = a + i b, w = c + i d$ where $a, b, c, d \in \R$.

Then $z \times w$ is defined as:

- $\paren {a + i b} \times \paren {c + i d} = \paren {a c - b d} + i \paren {a d + b c}$

## Terminology

### Multiplicand

Let $a \times b$ denote the operation of multiplication on two objects.

The object $b$ is known as the **multiplicand of $a$**.

That is, it is the object which is to be multiplied by the multiplier.

### Multiplier

Let $a \times b$ denote the operation of multiplication on two objects.

The object $a$ is known as the **multiplier of $b$**.

That is, it is the object which is to multiply the multiplicand.

### Product

Let $a \times b$ denote the operation of multiplication on two objects $a$ and $b$.

Then the result $a \times b$ is referred to as the **product** of $a$ and $b$.

## Notation

There are several variants of the notation for **multiplication**:

- $n \times m$

This is usually used when numbers are under consideration, for example: $3 \times 5 = 15$.

However, it can be used in the context of algebra where extra clarity is needed.

- $n m$

This is most common in algebra, but not with numbers, as it is difficult to make it obvious where one number ends and the next number begins.

- $\paren n \paren m$

This form can be used for either symbols denoting variables or numbers, for example: $\paren 3 \paren 4 = 12$.

- $n \cdot m$ or $n . m$

These have their uses in algebra, but the dot has the danger of being confused with the decimal point when used for numbers.

- $n * m$

This notation specifically evolved in the field of computer science, but can occasionally be seen encroaching into mathematics.

Its use is not recommended, as it can be confused with other operations that use the same or similar notation, for example convolution.

## Also see

### Commutativity of Multiplication

On all the above number sets, we have that multiplication is commutative:

- Natural Number Multiplication is Commutative
- Integer Multiplication is Commutative
- Modulo Multiplication is Commutative
- Rational Multiplication is Commutative
- Real Multiplication is Commutative
- Complex Multiplication is Commutative

### Associativity of Multiplication

On all the above number sets, we have that multiplication is associative:

- Natural Number Multiplication is Associative
- Integer Multiplication is Associative
- Modulo Multiplication is Associative
- Rational Multiplication is Associative
- Real Multiplication is Associative
- Complex Multiplication is Associative

## Historical Note

The symbol $\times$ for multiplication was invented by William Oughtred.

However, he was criticized by Leibniz, as it was in his view too similar to the letter $x$.

Leibniz himself preferred the symbol $\cdot$.

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