Definition:Multiplication

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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 on the notation for multiplication:

  • $n \times m$, which is usually used when numbers are under consideration, for example: $3 \times 5 = 15$, but can be used in the context of algebra where extra clarity is needed
  • $n m$, which is most common in algebra, but not with numbers unless (for obvious reasons) the numbers are placed in parenthesis, for example: $\paren 3 \paren 4 = 12$
  • $n \cdot m$ or $n . m$, which have their uses in algebra, but have the danger of being confused with the decimal point
  • $n * m$, specifically in the field of computer science, but occasionally encroaching into mathematics.


Also see


Commutativity of Multiplication

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


Associativity of Multiplication

On all the above number sets, we have that 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