Multiplication, usually denoted by the symbol × or ∙, is a fundamental operation that is defined differently for different mathematical objects.

In arithmetic, multiplication of natural numbers can be defined in terms of repeated addition, for example 3×4 is equal to 3+3+3+3 or 4+4+4. That definition can easily be extended to rational, real and complex numbers.

In abstract algebra, multiplication is an operation that isn't always explicitly defined, but rather is assumed to satisfy some axioms. For example:

• In multiplicative groups, multiplication is assumed to be associative, have an identity element, and that each group element has an inverse.
• In rings, which also have an addition operation, multiplication is assumed to be associative and distributive over addition.
  • In commutative rings, multiplication is assumed to be commutative.
  • In integral domains, multiplication is assumed to satisfy the zero-product rule.
  • In rings with unity, there exists a multiplicative identity.
  • Fields are commutative rings with unity in which every element except 0 (the additive identity) have multiplicative inverses.

Other objects, such as vectors, quaternionsquaternionsquaternionsquaternionsquaternionsquaternionsquaternionsquaternions and matrices have their own definitions of multiplication.

• Multiplication table (natural numbers up to 20)

Nomenclature

The result from multiplying two numbers together is called the product.

Properties

These properties can be useful for doing big multiplies.

• N×M=N÷(1÷M) — multiplication by division
• N×M=exp (ln N+ln M) — multiplication on logarithmic scale
• (A+B)×(C+D)=A×C+A×D+B×C+B×D — subdivided multiplication
• FFT(N)×FFT(M)=convolution(N,M) — FFT multiplication (use the set of digits as the input sets for FFT, then carry after convolution)

Inverse Relationship

Division

can be done with fact families

5 × 5 = 25

25 ÷ 5 = 5

In Real Life

Multiplication is widely used in real life in economics, or in most areas of physics.

Estimation

It is possible to estimate a product by rounding the multiplicands and the multipliers to fewer significant figures, then multiplying the rounded numbers. Alternatively, approximate the numbers on a logarithmic scale, add the values on the scale, then extract the value from the scale.

Applications

If two integers, ${\displaystyle m,n}$ are multiplied together, the product is the sum of ${\displaystyle m}$. ${\displaystyle n}$ times, or equivalently the sum of ${\displaystyle n}$, ${\displaystyle m}$ times. Written out in standard notation this is

${\displaystyle m \cdot n = \underbrace{m + m + \cdots + m}_{n} = \underbrace{n + n + \cdots + n}_{m} = n \cdot m }$

Table of basic multiplications

${\displaystyle *}$	${\displaystyle -\frac{1}{0}}$	${\displaystyle -1}$	${\displaystyle -0}$	${\displaystyle \frac{0}{0}}$	${\displaystyle 0}$	${\displaystyle 1}$	${\displaystyle \frac{1}{0}}$
${\displaystyle -\frac{1}{0}}$	${\displaystyle \frac{1}{0}}$	${\displaystyle \frac{1}{0}}$	${\displaystyle \frac{0}{0}}$	${\displaystyle \frac{0}{0}}$	${\displaystyle \frac{0}{0}}$	${\displaystyle -\frac{1}{0}}$	${\displaystyle -\frac{1}{0}}$
${\displaystyle -1}$	${\displaystyle \frac{1}{0}}$	${\displaystyle 1}$	${\displaystyle 0}$	${\displaystyle \frac{0}{0}}$	${\displaystyle -0}$	${\displaystyle -1}$	${\displaystyle -\frac{1}{0}}$
${\displaystyle -0}$	${\displaystyle \frac{0}{0}}$	${\displaystyle 0}$	${\displaystyle 0}$	${\displaystyle \frac{0}{0}}$	${\displaystyle -0}$	${\displaystyle -0}$	${\displaystyle \frac{0}{0}}$
${\displaystyle \frac{0}{0}}$	${\displaystyle \frac{0}{0}}$	${\displaystyle \frac{0}{0}}$	${\displaystyle \frac{0}{0}}$	${\displaystyle \frac{0}{0}}$	${\displaystyle \frac{0}{0}}$	${\displaystyle \frac{0}{0}}$	${\displaystyle \frac{0}{0}}$
${\displaystyle 0}$	${\displaystyle \frac{0}{0}}$	${\displaystyle -0}$	${\displaystyle -0}$	${\displaystyle \frac{0}{0}}$	${\displaystyle 0}$	${\displaystyle 0}$	${\displaystyle \frac{0}{0}}$
${\displaystyle 1}$	${\displaystyle -\frac{1}{0}}$	${\displaystyle -1}$	${\displaystyle -0}$	${\displaystyle \frac{0}{0}}$	${\displaystyle 0}$	${\displaystyle 1}$	${\displaystyle \frac{1}{0}}$
${\displaystyle \frac{1}{0}}$	${\displaystyle -\frac{1}{0}}$	${\displaystyle -\frac{1}{0}}$	${\displaystyle \frac{0}{0}}$	${\displaystyle \frac{0}{0}}$	${\displaystyle \frac{0}{0}}$	${\displaystyle \frac{1}{0}}$	${\displaystyle \frac{1}{0}}$