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Product (mathematics)




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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: [math]\prod[/math]. 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 [math]a_i[/math] we define [math]\prod_{1\leq i\leq n}a_i:=a_1\dotsm a_n[/math]. A rigorous definition is usually given recursively as follows

[math] \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]\prod_{1\leq i\leq n}[/math] is [math]\prod_{i=1}^n[/math].[2][3]

Properties

[math]\prod_{i=1}^n i = 1 \cdot 2 \cdot ... \cdot n = n![/math] ([math]n![/math] is pronounced "[math]n[/math] factorial" or "factorial of [math]n[/math]")
[math]\prod_{i=1}^n x = x^n[/math] (i.e., the usual [math]n[/math]th power operation)
[math]\prod_{i=1}^n n = n^n[/math] (i.e., [math]n[/math] multiplied by itself [math]n[/math] times)
[math]\prod_{i=1}^n c \cdot i = \prod_{i=1}^n c \cdot \prod_{i=1}^n i = c^n \cdot n![/math] (where [math]c[/math] is a constant independent of [math]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]\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, as one generally cannot separate terms inside them before they are raised to an exponent, but with products, this is possible:

[math]\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]\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]

Related pages

References

  1. "Comprehensive List of Algebra Symbols" (in en-US). 2020-03-25. https://mathvault.ca/hub/higher-math/math-symbols/algebra-symbols/. 
  2. "Summation and Product Notation". https://math.illinoisstate.edu/day/courses/old/305/contentsummationnotation.html. 
  3. Weisstein, Eric W.. "Product" (in en). https://mathworld.wolfram.com/Product.html.