# Product (mathematics)

In mathematics, a product is a number or a quantity obtained by multiplying two or more numbers together. For example: 5 × 4 = 20. Here, the number 20 is called the product of 5 and 4. The product of 6 and 4 will be 24, because 6 × 4 = 24.

## Capital pi

A short notation for long multiplication expressions is the product notation. It uses the capital Greek letter pi: $\prod$. This works the same as the Sigma notation.

### Properties

$\prod_{i=1}^n i = 1 \cdot 2 \cdot ... \cdot n = n!$ (n! means n factorial)
$\prod_{i=1}^n n = n^n$ because we multiply n by itself n times.
$\prod_{i=1}^n c \cdot i = c^n \cdot n!$ where c is a constant.
$\prod_{i=1}^n x = x^n$

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,

$\prod_{i=1}^4 (3 + 4) \neq \prod_{i=1}^4 3 + \prod_{i=1}^4 4$.

This can be thought of in terms of polynomials: you normally cannot separate terms inside them before they are raised to an exponent!

### Relation to Summation

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

$\prod_{i=1}^n a^{c_i} = a^{c_1} \cdot a^{c_2} \cdot ... \cdot a^{c_n}= a^{c_1 + c_2 + ... + c_n} = a^{(\sum_{i=1}^n c_i)}$