Product (mathematics)

result of multiplying

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 notationEdit

A short way to write the product of many numbers is to use the capital Greek letter pi:  . 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   we define  . A rigorous definition is usually given recursively as follows

 

An alternative notation for   is  .[2][3]

PropertiesEdit

  (  is pronounced "  factorial" or "factorial of  ")
  (i.e., the usual  th power operation)
  (i.e.,   multiplied by itself   times)
  (where   is a constant independent of  )

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,

 ,

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:

 

Relation to SummationEdit

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

 

Related pagesEdit

ReferencesEdit

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