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. As another example, the product of 6 and 4 is 24, because 6 times 4 is 24. The product of two positive numbers is positive, just as the product of two negative numbers is positive as well (e.g., -6 × -4 = 24).

Pi product notation

A short way to write the product of many numbers is to use the capital Greek letter pi: ${\displaystyle \prod }$. This notation (or way of writing) is in some ways similar to the Sigma notation of summation.^[1]

Informally, given a sequence of numbers (or elements of a multiplicative structure with unit) say ${\displaystyle a_{i}}$ we define ${\displaystyle \prod _{1\leq i\leq n}a_{i}:=a_{1}\dotsm a_{n}}$. A rigorous definition is usually given recursively as follows

${\displaystyle \prod _{1\leq i\leq n}a_{i}:={\begin{cases}1&{\text{ for }}n=0,\\\left(\prod _{1\leq i\leq n-1}a_{i}\right)a_{n}&{\text{ for }}n\geq 1.\end{cases}}}$

An alternative notation for ${\displaystyle \prod _{1\leq i\leq n}}$ is ${\displaystyle \prod _{i=1}^{n}}$.^[2][3]

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:

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

Related pages

• Cartesian product
• Cross product
• Dot product

References

1. "Comprehensive List of Algebra Symbols". Math Vault. 2020-03-25. Retrieved 2020-08-16.
2. "Summation and Product Notation". math.illinoisstate.edu. Archived from the original on 2023-08-29. Retrieved 2020-08-16.
3. Weisstein, Eric W. "Product". mathworld.wolfram.com. Retrieved 2020-08-16.