Difference between revisions of "Power"
From Encyclopedia of Mathematics
(Importing text file) |
(See also Exponential function and Exponential function, real) |
||
| (One intermediate revision by one other user not shown) | |||
| Line 1: | Line 1: | ||
| − | + | {{MSC|97Fxx}} | |
| − | + | In its primary meaning (an integer and positive power), it is the product of a number of equal factors and is written in the form $a^n = a\cdots a$ ($n$ times), where $a$ is the base, $n$ the exponent and $a^n$ the power. The basic properties of powers are: | |
| − | Further generalizations of the idea of a power include: zero powers: | + | $$a^n\cdot a^m = a^{n+m},\quad (ab)^n = a^n b^n,\quad \frac{a^n}{a^m} = a^{n-m},\quad (a^n)^m = a^{nm}.$$ |
| + | Further generalizations of the idea of a power include: zero powers: $a^0 = 1 $, | ||
| + | negative powers: $a^{-n} = 1/a^n$ (when $a\ne 0$); fractional powers: $a^{n/m} = (a^{1/m})^n$, where $a^{1/m} = \sqrt[m]{a}$ (for $a>0$); and a power with an irrational exponent: $\def\a{\alpha}a^\a = \lim_{r_n\to\a} a^{r_n}$, where $r_n$ is an arbitrary sequence of rational numbers tending to $\a$. Powers with a complex base (see | ||
| + | [[De Moivre formula|de Moivre formula]]) and powers with a complex base and complex exponent (by definition: $z^u = e^{u\mathrm{Ln} z}$) are also studied. | ||
| − | + | See also [[Exponential function]] and [[Exponential function, real]]. | |
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
Latest revision as of 21:50, 31 December 2015
2020 Mathematics Subject Classification: Primary: 97Fxx [MSN][ZBL]
In its primary meaning (an integer and positive power), it is the product of a number of equal factors and is written in the form $a^n = a\cdots a$ ($n$ times), where $a$ is the base, $n$ the exponent and $a^n$ the power. The basic properties of powers are:
$$a^n\cdot a^m = a^{n+m},\quad (ab)^n = a^n b^n,\quad \frac{a^n}{a^m} = a^{n-m},\quad (a^n)^m = a^{nm}.$$ Further generalizations of the idea of a power include: zero powers: $a^0 = 1 $, negative powers: $a^{-n} = 1/a^n$ (when $a\ne 0$); fractional powers: $a^{n/m} = (a^{1/m})^n$, where $a^{1/m} = \sqrt[m]{a}$ (for $a>0$); and a power with an irrational exponent: $\def\a{\alpha}a^\a = \lim_{r_n\to\a} a^{r_n}$, where $r_n$ is an arbitrary sequence of rational numbers tending to $\a$. Powers with a complex base (see de Moivre formula) and powers with a complex base and complex exponent (by definition: $z^u = e^{u\mathrm{Ln} z}$) are also studied.
See also Exponential function and Exponential function, real.
How to Cite This Entry:
Power. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Power&oldid=15779
Power. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Power&oldid=15779
This article was adapted from an original article by BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article