Product (mathematics)

Properties

${\displaystyle \prod _{i=1}^{n}i=1\cdot 2\cdot ...\cdot n=n!}$ (${\displaystyle n!}$ is pronounced "${\displaystyle n}$ factorial" or "factorial of ${\displaystyle n}$")

${\displaystyle \prod _{i=1}^{n}x=x^{n}}$ (i.e., the usual ${\displaystyle n}$th power operation)

${\displaystyle \prod _{i=1}^{n}n=n^{n}}$ (i.e., ${\displaystyle n}$ multiplied by itself ${\displaystyle n}$ times)

${\displaystyle \prod _{i=1}^{n}c\cdot i=\prod _{i=1}^{n}c\cdot \prod _{i=1}^{n}i=c^{n}\cdot n!}$ (where ${\displaystyle c}$ is a constant independent of ${\displaystyle i}$)

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,

${\displaystyle \prod _{i=1}^{4}(3+4)\neq \prod _{i=1}^{4}3+\prod _{i=1}^{4}4}$,

This can be thought of in terms of polynomials, as one generally cannot separate terms inside them before they are raised to an exponent, but with products, this is possible:

${\displaystyle \prod _{i=1}^{n}a_{i}b_{i}=\prod _{i=1}^{n}a_{i}\prod _{i=1}^{n}b_{i}.}$

Relation to Summation

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

${\displaystyle \prod _{i=1}^{n}a^{c_{i}}=a^{c_{1}}\cdot a^{c_{2}}\dotsm a^{c_{n}}=a^{c_{1}+c_{2}+...+c_{n}}=a^{(\sum _{i=1}^{n}c_{i})}}$