52 lines
2.2 KiB
Markdown
52 lines
2.2 KiB
Markdown
---
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title: Factorials
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---
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## Factorials
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<!-- The article goes here, in GitHub-flavored Markdown. Feel free to add YouTube videos, images, and CodePen/JSBin embeds -->
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### Definition of Factorial
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The factorial is to multiply it by every intiger lower then it ending at one. If the initial number is negative the result is infinity.
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A factorial of <span class="texhtml"> n </span>, a non-negative integer, is defined as:
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<span class="texhtml"> n! = 1 * 2 * ... * (n - 1) * n </span>
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A special case arises when <span class="texhtml"> n = 0 </span>. Namely, <span class="texhtml"> 0! = 1 </span>.
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### Convenience of Factorials
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The definition above provides you with convenience in certain computations. For example, factorials inside fractions can often be simplified as follows:
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Example 1: <span class="texhtml"> 7! / 5! = (1 * 2 * 3 * 4 * 5 * 6 * 7) / (1 * 2 * 3 * 4 * 5) = 6 * 7 = 42 </span>
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Example 2: <span class="texhtml"> (n + 1)! / n! = (1 * 2 * ... * n * (n + 1)) / (1 * 2 * ... * n) = n + 1 </span>
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### Alternative Definition
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Alternatively, factorials can be defined as follows:
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<span class="texhtml"> 0! = 1 </span>
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<span class="texhtml"> n! = n * (n - 1)! </span> if <span class="texhtml"> n > 0 </span>
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This recursive definition means the exact same as the traditional definition. Applying this to the second example above, we get:
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<span class="texhtml"> (n + 1)! / n! = (n + 1) * n! / n! = n + 1 </span>
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### Aside: Extension to Non-Integers
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Note that factorial as defined above applies only to non-negative integers. Actually, there is a generalization of factorials that extends to non-integers as well, which is the Gamma function. In particular, for any natural number <span class="texhtml"> n </span>, you have <span class="texhtml"> n! = Gamma(n + 1) = n * Gamma(n) </span>.
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For more, see <a href='https://en.wikipedia.org/wiki/Factorial#Extension_of_factorial_to_non-integer_values_of_argument' target='_blank' rel='nofollow'>Extension of factorial to non-integer values of argument</a>.
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<span class="texhtml">
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One tricky example that many may not know if 0! = 1. For futher proof see the link under More Information.
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#### More Information:
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<a href='http://www.purplemath.com/modules/factorial.htm' target='_blank' rel='nofollow'>Factorials</a>
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