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|augend + addend =||sum|
|minuend − subtrahend =||difference|
|multiplicand × multiplier =||product|
|dividend ÷ divisor =||quotient|
|nth root (√)|
|degree √ =||root|
In mathematics, a product is the result of multiplying, or an expression that identifies factors to be multiplied. Thus, for instance, 6 is the product of 2 and 3 (the result of multiplication), and is the product of and (indicating that the two factors should be multiplied together).
The order in which real or complex numbers are multiplied has no bearing on the product; this is known as the commutative law of multiplication. When matrices or members of various other associative algebras are multiplied, the product usually depends on the order of the factors. Matrix multiplication, for example, and multiplication in other algebras is in general non-commutative.
Placing several stones into a rectangular pattern with rows and columns gives
Integers allow positive and negative numbers. The two numbers are multiplied just like natural numbers, except we need an additional rule for the signs:
In words, we have:
Two fractions can be multiplied by multiplying their numerators and denominators:
The rigorous definition of the product of two real numbers is too complicated for this article. But the idea is that one takes a decimal approximation to each real and multiplies the approximations together, and then take better and better approximations.
Two complex numbers can be multiplied by the distributive law and the fact that , as follows:
Complex numbers can be written in polar coordinates:
The geometric meaning is that we multiply the magnitudes and add the angles.
The product of two quaternions can be found in the article on quaternions. However, it is interesting to note that in this case, and are different.
The product operator for the product of a sequence is denoted by the capital Greek letter Pi ∏ (in analogy to the use of the capital Sigma ∑ as summation symbol). The product of a sequence consisting of only one number is just that number itself. The product of no factors at all is known as the empty product, and is equal to 1.
Residue classes in the rings can be added:
Functions to the real numbers can be added or multiplied by adding or multiplying their outputs:
Two functions from the reals to itself can be multiplied in another way, called the convolution.
then the integral
is well defined and is called the convolution.
Under the Fourier transform, convolution becomes multiplication.
The product of two polynomials is given by the following:
By the very definition of a vector space, one can form the product of any scalar with any vector, giving a map .
A scalar product is a bilinear map:
with the following conditions, that for all .
From the scalar product, one can define a norm by letting .
The scalar product also allows one to define an angle between two vectors:
In -dimensional Euclidean space, the standard scalar product (called the dot product) is given by:
The cross product of two vectors in 3-dimensions is a vector perpendicular to the two factors, with length equal to the area of the parallelogram spanned by the two factors.
Given two matrices
their product is given by
In set theory, a Cartesian product is a mathematical operation which returns a set (or product set) from multiple sets. That is, for sets A and B, the Cartesian product A × B is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B.
The empty product has the value of 1 (the identity element of multiplication) just like the empty sum has the value of 0 (the identity element of addition).
Many different kinds of products are studied in mathematics: