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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: 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: [math]\prod[/math]. This works the same as the Sigma notation.

Properties

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

[math]\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)} [/math]hu:Szorzás