Степенување: Разлика помеѓу преработките
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Избришана содржина Додадена содржина
с Замена со македонски назив на предлошка, replaced: Cite web → Наведена мрежна страница, cite web → Наведена мрежна страница (11) |
с Замена со македонски назив на предлошка, replaced: cite book → Наведена книга (6) |
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==Експонентот е позитивен реален број==
Доколку основата ''b'' е позитивен реален број, а бидејќи секој ирационален број ''х'' може да се приближува со рационален број ''r'', степенување со експонент ''х'', т.е. ''b''<sup>''х''</sup> може да се дефинира преку [[непрекината функција|непрекинатост]] со <ref name=Denlinger>{{
:<math> b^x = \lim_{r \to x} b^r\quad(r\in\mathbb Q,\,x\in\mathbb R)</math>
<!-- Тука треба нешто за кога b е негативен ирационален број... -->
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The number ''e''<sup>2''πi'' ({{frac|1|''n''}})</sup> is the primitive ''n''th root of unity with the smallest positive [[complex argument]]. (It is sometimes called the '''principal ''n''th root of unity''', although this terminology is not universal and should not be confused with the [[principal value]] of <sup>''n''</sup>√<span style="text-decoration:overline">1</span>, which is 1.<ref>This definition of a principal root of unity can be found in:
*{{
*{{
* "[http://mathworld.wolfram.com/PrincipalRootofUnity.html Principal root of unity]", MathWorld.</ref>)
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*The [[#Combinatorial interpretation|combinatorial interpretation]] of 0<sup>0</sup> is the number of [[empty tuple]]s of elements from the empty set. There is exactly one empty tuple.
*Equivalently, the [[#Exponentiation over sets|set-theoretic interpretation]] of 0<sup>0</sup> is the number of functions from the empty set to the empty set. There is exactly one such function, the [[empty function]].<ref name="Bourbaki">N. Bourbaki, Elements of Mathematics, Theory of Sets, Springer-Verlag, 2004, III.§3.5.</ref>
*The notation <math>\scriptstyle \sum a_nx^n</math> for [[polynomial]]s and [[power series]] rely on defining 0<sup>0</sup> = 1. Identities like <math>\scriptstyle \frac{1}{1-x} = \sum_{n=0}^{\infty} x^n</math> and <math>\scriptstyle e^{x} = \sum_{n=0}^{\infty} \frac{x^n}{n!}</math> and the [[binomial theorem]] <math>\scriptstyle (1 + x)^n = \sum_{k = 0}^n \binom{n}{k} x^k</math> are not valid for {{nowrap|1=''x'' = 0}} unless {{nowrap|1=0<sup>0</sup> = 1}}.<ref>"Some textbooks leave the quantity 0<sup>0</sup> undefined, because the functions ''x''<sup>0</sup> and 0<sup>''x''</sup> have different limiting values when ''x'' decreases to 0. But this is a mistake. We must define {{nowrap|1=''x''<sup>0</sup> = 1}}, for all ''x'', if the binomial theorem is to be valid when {{nowrap|1=''x'' = 0}}, {{nowrap|1=''y'' = 0}}, and/or {{nowrap|1=''x'' = −''y''}}. The binomial theorem is too important to be arbitrarily restricted! By contrast, the function 0<sup>''x''</sup> is quite unimportant".{{
*In [[differential calculus]], the [[power rule]] <math>\scriptstyle \frac{d}{dx} x^n = nx^{n-1}</math> is not valid for {{nowrap|1=''n'' = 1}} at {{nowrap|1=''x'' = 0}} unless {{nowrap|1=0<sup>0</sup> = 1}}.
===In analysis===
On the other hand, when 0<sup>0</sup> arises when trying to determine a [[limit of a function|limit]] of the form <math>\scriptstyle \lim_{x\rarr 0} f(x)^{g(x)}</math>, it must be handled as an [[indeterminate form]].
*Limits involving algebraic operations can often be evaluated by replacing subexpressions by their limits; if the resulting expression does not determine the original limit, the expression is known as an indeterminate form.<ref>{{
::<math> \lim_{t \to 0^+} {t}^{t} = 1, \quad \lim_{t \to 0^+} \left(e^{-\frac{1}{t^2}}\right)^t = 0, \quad \lim_{t \to 0^+} \left(e^{-\frac{1}{t^2}}\right)^{-t} = +\infty, \quad \lim_{t \to 0^+} \left(e^{-\frac{1}{t}}\right)^{at} = e^{-a}</math>.
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====IEEE floating point standard====
The [[IEEE 754-2008]] floating point standard is used in the design of most floating point libraries. It recommends a number of different functions for computing a power:<ref>{{
*<tt>pow</tt> treats 0<sup>0</sup> as 1. This is the oldest defined version. If the power is an exact integer the result is the same as for <tt>pown</tt>, otherwise the result is as for <tt>powr</tt> (except for some exceptional cases).
*<tt>pown</tt> treats 0<sup>0</sup> as 1. The power must be an exact integer. The value is defined for negative bases; e.g., <tt>pown(−3,5)</tt> is −243.
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