Binomial theorem: Difference between revisions

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(delete popular culture section; also see talk page)
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== The first several cases ==
== The first several cases ==


: <math> \begin{align}
<math> \begin{align}
(x + y)^0 &= 1 \\
(x + y)^0 &= 1 \\
(x + y)^1 &= x + y \\
(x + y)^1 &= x + y \\
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There is also '''Newton's binomial theorem''', proved by [[Isaac Newton]], that goes beyond elementary algebra into mathematical analysis, which expands the same sum (''x''&nbsp;+&nbsp;''y'')<sup>''n''</sup> as an infinite series when ''n'' is not an integer or is not positive.
There is also '''Newton's binomial theorem''', proved by [[Isaac Newton]], that goes beyond elementary algebra into mathematical analysis, which expands the same sum (''x''&nbsp;+&nbsp;''y'')<sup>''n''</sup> as an infinite series when ''n'' is not an integer or is not positive.
==In popular culture==
Sir [[W. S. Gilbert]] mentions the binomial theorem at least twice: Once in the Major General's Song in the [[Gilbert and Sullivan]] operetta [[The Pirates of Penzance]] ("About binomial theorem I'm teeming with a lot o' news -- / With many cheerful facts about the square of the hypotenuse"); and again in the poem "My Dream" in his "Bab Ballads," which says of a group of intelligent infants:  "For as their nurses dandle them, / They crow binomial theorem, / With views (it seems absurd to us) / On differential calculus."  <ref>Ian Bradley, ed., ''The Complete Annotated Gilbert and Sullivan'' (Oxford: Oxford University Press, 1996), pp. 218-219.)</ref>
==Notes==
<references/>

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In elementary algebra, the binomial theorem is the identity that states that for any non-negative integer n,

or, equivalently,

where

One way to prove this identity is by mathematical induction.

The first several cases

Newton's binomial theorem

There is also Newton's binomial theorem, proved by Isaac Newton, that goes beyond elementary algebra into mathematical analysis, which expands the same sum (x + y)n as an infinite series when n is not an integer or is not positive.