Multiplication ( $\times$ )

Multiplication combines two factors into their product of numbers. For a natural number

n

n x

is the sum of

n

copies of

x

; with no copies the result is zero. This repeated-addition interpretation directly explains integer grouping, while multiplication of arbitrary real numbers also describes scaling by noninteger factors.

A rectangle of side lengths

3

and

6

has area

18

. Split the second side into lengths

2

and

4

. The two areas are

6

and

12

, giving the visible identity

3 (2 + 4) = 3 \cdot 2 + 3 \cdot 4 = 18.

(1)

Rotating the same rectangle interchanges the sides without changing its area, explaining

3 \cdot 6 = 6 \cdot 3

. These pictures illustrate distributivity and commutativity for ordinary positive numbers; matrix multiplication and other operations require their own rules.

Figure 1.
Partitioning a rectangle explains distributivity; partitioning a unit square gives an exact product of fractions
. **Details**.

For a fractional example, take

3/2

of a quantity and then

2/3

of that result: the combined scale is

(2/3) (3/2) = 1

. The two fractions are multiplicative inverses. In the unit square on the right, the overlap of a strip occupying

3/4

of the width with one occupying

2/3

of the height contains six of twelve equal cells. Its area is

(3/4) (2/3) = 1/2

. The product stays exact even though the decimal expansion of

2/3

repeats. ^[ref]

Binary multiplication ( $X Y = \sum_{i} y_{i} 2^{i} X$ )

 0  0

For nonnegative integers with binary representation

Y = \sum_{i = 0}^{w - 1} y_{i} 2^{i}

, multiply by

X

by summing the shifted terms

y_{i} 2^{i} X

. Each term is either zero or a left bit shift of

X

. An unsigned

w

-bit input pair needs at most

2 w

product bits, because

(2^{w} - 1)^{2} < 2^{2 w}

. This expansion follows from distributivity and motivates a binary multiplier.

Multiplication table

 1  0

A multiplication table lists the products of the factors labeling its rows and columns. Each cell is determined by one multiplication; the same arrangement can serve as a spreadsheet tabular model.

Product of numbers ( $\prod_{i = 1}^{n} a_{i}$ )

 0  0

The product of numbers

a_{1}, \dots, a_{n}

is the result of their iterated multiplication. The empty product is

1

, the multiplicative identity. For positive factors,

lo g (\prod_{i} a_{i}) = \sum_{i} lo g a_{i}

, turning a multiplicative conservation law into an additive derivative identity.

Exponentiation ( $a^{b}$ )

 2  0

For a nonnegative integer exponent, exponentiation iterates multiplication:

a^{0} = 1

and

a^{b + 1} = a a^{b}

. This convention defines

0^{0} = 1

in primitive recursive arithmetic. Extensions to other exponents may use different domains and conventions.

Fifth power ( $a^{5}$ )

 0  0

A value of the form

a^{5}

, the product of five copies of

a

Fourth power ( $a^{4}$ )

 1  0

A value of the form

a^{4}

, the product of four copies of

a

nth root ( $k n$ )

 2  0

Integer nth root ( $⌊ n^{1/ k} ⌋$ )

 0  0

Integer root bound

 0  0

Tetration ( $^{h} a$ )

 1  0

Repeated exponentiation with right association:

^{1} a = a

and

^{h + 1} a = a^{^{h} a}

. For example,

^{3} a = a^{a^{a}}

, rather than

(a^{a})^{a}

. Computing its remainder modulo an integer often uses a chain of smaller exponent moduli; applying Euler's theorem requires checking its coprimality hypothesis at every level.

Addition chain

 1  0

An addition chain for a positive integer

k

1 = a_{0} < a_{1} < \dots < a_{r} = k

, where each

a_{t}

is the sum of two earlier entries, which may coincide. It computes

x^{k}

r

multiplications by reusing previously computed powers. For example,

1, 2, 3, 6, 12, 15

computes

x^{15}

in five multiplications.

Star chain

 0  0

An addition chain restricted to updates

a_{t + 1} = a_{t} + a_{i}

with

0 \leq i \leq t

. Each new power uses the most recently computed power as one factor. This restriction reduces the search space, but a shortest star chain need not be a shortest unrestricted addition chain for every target.

Cube of a number ( $x^{3}$ )

 0  0

The cube of a number

x

x^{3} = x \cdot x \cdot x

, using the exponent three. For example,

2^{3} = 8

and

(- 2)^{3} = - 8

Integer cube ( $n^{3}$ )

 0  0

A value

n^{3}

for an integer

n

Exponent ( $b in a^{b}$ )

 0  0

An exponent specifies the power in exponentiation: in

a^{b}

a

is the base and

b

is the exponent. For a positive integer exponent,

2^{3} = 2 \cdot 2 \cdot 2 = 8

. For a positive real number base, a real exponent is defined by

a^{b} = exp (b lo g a)

using the exponential function and natural logarithm. The Lp norm uses its exponent

p

to determine how strongly large absolute values affect the result.

Square of a number ( $a^{2}$ )

 0  0

The square of a number

a

is the product

a \cdot a

. For a real number, it is nonnegative. The sum of squares of real coordinates is the squared Euclidean norm of the corresponding vector.

Hyperoperation ( $a [n] b$ )

 1  0

The hyperoperation hierarchy successively iterates basic arithmetic operations. In the usual indexing, successor is followed by addition, multiplication, exponentiation and their higher iterates. A sequence beginning with addition shifts this index by one. Each fixed rank is a primitive recursive function; this does not imply that the evaluator taking the rank as an additional variable is primitive recursive.

Empty product ( $\prod_{i \in \emptyset} a_{i} = 1$ )

 1  0

A product over the empty set has value one, the identity for multiplication. This makes splitting a product into successive pieces valid even when one piece contains no factors. In particular, the factorial convention is