Definition:Multiplication

Modulo Multiplication
The multiplication operation on $\Z_m$, the set of integers modulo $m$, is defined by the rule:


 * $\left[\!\left[{a}\right]\!\right]_m \times_m \left[\!\left[{b}\right]\!\right]_m = \left[\!\left[{a b}\right]\!\right]_m$

Although the operation of multiplication modulo $m$ is denoted by the symbol $\times_m$, if there is no danger of confusion, the conventional multiplication symbols $\times, \cdot$ etc. are often used instead.

More usually, though, the notation $a b \left({\bmod\, m}\right)$ is used instead of $\left[\!\left[{a}\right]\!\right]_m \times_m \left[\!\left[{b}\right]\!\right]_m$.

It means the same thing and, although obscuring the true meaning behind modulo arithmetic, is more streamlined and less unwieldy.

See modulo multiplication.

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 $p, q \in \Z, r, s \in \Z \setminus \left\{{0}\right\}$.

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 $\left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right]$ of Cauchy sequences of rational numbers.

Let $x = \left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right], y = \left[\!\left[{\left \langle {y_n} \right \rangle}\right]\!\right]$, where $\left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right]$ and $\left[\!\left[{\left \langle {y_n} \right \rangle}\right]\!\right]$ are such equivalence classes.

Then $x \times y$ is defined as $\left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right] \times \left[\!\left[{\left \langle {y_n} \right \rangle}\right]\!\right] = \left[\!\left[{\left \langle {x_n \times y_n} \right \rangle}\right]\!\right]$.

The operation of multiplication on the real numbers is well-defined.

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 $\left({a + i b}\right) \times \left({c + i d}\right) = \left({ac - bd}\right) + i \left({ad + bc}\right)$.

This follows by the facts that:
 * the Real Numbers form a Field and thus multiplication distributes over addition;
 * the entity $i$ is such that $i^2 = -1$.

Notation
There are several variants on the notation for multiplication:


 * $n \times m$, which is usually used only when numbers are under consideration, e.g. $3 \times 5 = 15$;
 * $nm$, which is most common in algebra, but not with numbers unless parentheses are put round the numbers, e.g. $\left({3}\right)\left({4}\right) = 12$, for obvious reasons;
 * $n \cdot m$ or $n . m$, which have their uses in algebra, but has the danger of being confused with the decimal point.

Also see

 * Quaternion Multiplication

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

Complex Multiplication

 * : $6.4$