Product (mathematics)

In mathematics, a product is a number or a quantity obtained by multiplying two or more numbers together. For example: 4 × 7 = 28 Here, the number 28 is called the product of 4 and 7. The product of 6 and 4 will be 24, because 6 × 4 = 24.

Capital pi

A short way to write the product of many numbers uses the capital Greek letter pi: [math]\displaystyle{ \prod }[/math]. This notation works the same as the Sigma notation. Informally, given a sequence of numbers (or elements of a multiplicative structure with unit) say [math]\displaystyle{ a_i }[/math] we define [math]\displaystyle{ \prod_{1\leq i\leq n}a_i:=a_1\dotsm a_n }[/math]. A rigorous definition is usually given recursively as follows

[math]\displaystyle{ \prod_{1\leq i\leq n}a_i := \begin{cases} 1 & \text{ for } n=0 , \\ \left(\prod_{1\leq i\leq n-1}a_i\right) a_n & \text{ for } n\geq1 . \end{cases} }[/math]

An alternative notation for [math]\displaystyle{ \prod_{1\leq i\leq n} }[/math] is [math]\displaystyle{ \prod_{i=1}^n }[/math].

Properties

[math]\displaystyle{ \prod_{i=1}^n i = 1 \cdot 2 \cdot ... \cdot n = n! }[/math] ([math]\displaystyle{ n! }[/math] is pronounced "[math]\displaystyle{ n }[/math] factorial" or "factorial of [math]\displaystyle{ n }[/math]");

[math]\displaystyle{ \prod_{i=1}^n x = x^n }[/math], i.e., the usual [math]\displaystyle{ n }[/math]th power operation;

[math]\displaystyle{ \prod_{i=1}^n n = n^n }[/math], i.e., we multiply [math]\displaystyle{ n }[/math] by itself [math]\displaystyle{ n }[/math] times;

[math]\displaystyle{ \prod_{i=1}^n c \cdot i = c^n \cdot n! }[/math] where [math]\displaystyle{ c }[/math] is a constant with respect to [math]\displaystyle{ i }[/math].

From the above equation we can see that any number with an exponent can be represented by a product, though it normally is not desirable.

Unlike summation, the sums of two terms cannot be separated into different sums. That is,

[math]\displaystyle{ \prod_{i=1}^4 (3 + 4) \neq \prod_{i=1}^4 3 + \prod_{i=1}^4 4 }[/math],

This can be thought of in terms of polynomials: one generally cannot separate terms inside them before they are raised to an exponent. But the product does,

[math]\displaystyle{ \prod_{i=1}^n a_ib_i=\prod_{i=1}^na_i\prod_{i=1}^nb_i. }[/math]

Relation to Summation

The product of powers with the same base can be written as an exponential of the sum of the powers' exponents:

[math]\displaystyle{ \prod_{i=1}^n a^{c_i} = a^{c_1} \cdot a^{c_2} \dotsm a^{c_n}= a^{c_1 + c_2 + ... + c_n} = a^{(\sum_{i=1}^n c_i)} }[/math]