Power (function): Difference between revisions
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:''This page describes the dyadic function. For the monadic function that uses <math>e</math> as a base, see [[Exponential]]. For the iteration operator, see [[Power (operator)]].'' | :''This page describes the dyadic function. For the monadic function that uses <math>e</math> as a base, see [[Exponential]]. For the iteration operator, see [[Power (operator)]].'' | ||
{{Built-in|Power|*}} is a [[dyadic]] [[scalar function]] that computes the [[wikipedia:exponentiation|exponentiation]] function of the two [[argument|arguments]], so that < | {{Built-in|Power|*}} is a [[dyadic]] [[scalar function]] that computes the [[wikipedia:exponentiation|exponentiation]] function of the two [[argument|arguments]], so that <syntaxhighlight lang=apl inline>X*Y</source> is <syntaxhighlight lang=apl inline>X</source> raised to the power <syntaxhighlight lang=apl inline>Y</source>. Power shares the [[glyph]] <syntaxhighlight lang=apl inline>*</source> with the monadic arithmetic function [[Exponential]]. | ||
== Examples == | == Examples == | ||
< | <syntaxhighlight lang=apl> | ||
2*¯1 0 1 2 3 4 5 | 2*¯1 0 1 2 3 4 5 | ||
0.5 1 2 4 8 16 32 | 0.5 1 2 4 8 16 32 | ||
</source> | </source> | ||
A common technique is to choose [[sign]] based on a [[Boolean]] array: | A common technique is to choose [[sign]] based on a [[Boolean]] array: | ||
< | <syntaxhighlight lang=apl> | ||
¯1*1 0 0 1 0 | ¯1*1 0 0 1 0 | ||
¯1 1 1 ¯1 1 | ¯1 1 1 ¯1 1 | ||
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== Properties == | == Properties == | ||
For positive integer < | For positive integer <syntaxhighlight lang=apl inline>Y</source>, <syntaxhighlight lang=apl inline>X*Y</source> equals the [[times|product]] of <syntaxhighlight lang=apl inline>Y</source> copies of <syntaxhighlight lang=apl inline>X</source>. When <syntaxhighlight lang=apl inline>Y</source> is 0, <syntaxhighlight lang=apl inline>X*Y</source> equals 1, possibly except when <syntaxhighlight lang=apl inline>X</source> is also 0 (since [[wikipedia:zero to the power of zero|zero to the power of zero]] is undefined in mathematics). | ||
< | <syntaxhighlight lang=apl> | ||
3*5 | 3*5 | ||
243 | 243 | ||
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[[negate|Negating]] the exponent (right argument) gives the [[reciprocal]] of the return value. | [[negate|Negating]] the exponent (right argument) gives the [[reciprocal]] of the return value. | ||
< | <syntaxhighlight lang=apl> | ||
(2*¯4)=÷2*4 | (2*¯4)=÷2*4 | ||
1 | 1 | ||
</source> | </source> | ||
If the exponent is the [[reciprocal]] of some number n, the result is the n-th [[root]] of the base. For example, a right argument of < | If the exponent is the [[reciprocal]] of some number n, the result is the n-th [[root]] of the base. For example, a right argument of <syntaxhighlight lang=apl inline>÷2</source> gives the [[square root]]. | ||
< | <syntaxhighlight lang=apl> | ||
3*2 | 3*2 | ||
9 | 9 | ||
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Power has two inverses, [[Root]] and [[Logarithm]]: | Power has two inverses, [[Root]] and [[Logarithm]]: | ||
< | <syntaxhighlight lang=apl> | ||
2*3 | 2*3 | ||
8 | 8 | ||
Revision as of 21:32, 10 September 2022
- This page describes the dyadic function. For the monadic function that uses as a base, see Exponential. For the iteration operator, see Power (operator).
*
|
Power (*) is a dyadic scalar function that computes the exponentiation function of the two arguments, so that <syntaxhighlight lang=apl inline>X*Y</source> is <syntaxhighlight lang=apl inline>X</source> raised to the power <syntaxhighlight lang=apl inline>Y</source>. Power shares the glyph <syntaxhighlight lang=apl inline>*</source> with the monadic arithmetic function Exponential.
Examples
<syntaxhighlight lang=apl>
2*¯1 0 1 2 3 4 5
0.5 1 2 4 8 16 32 </source> A common technique is to choose sign based on a Boolean array: <syntaxhighlight lang=apl>
¯1*1 0 0 1 0
¯1 1 1 ¯1 1 </source>
Properties
For positive integer <syntaxhighlight lang=apl inline>Y</source>, <syntaxhighlight lang=apl inline>X*Y</source> equals the product of <syntaxhighlight lang=apl inline>Y</source> copies of <syntaxhighlight lang=apl inline>X</source>. When <syntaxhighlight lang=apl inline>Y</source> is 0, <syntaxhighlight lang=apl inline>X*Y</source> equals 1, possibly except when <syntaxhighlight lang=apl inline>X</source> is also 0 (since zero to the power of zero is undefined in mathematics).
<syntaxhighlight lang=apl>
3*5
243
×/5⍴3
243
1 2 3*0
1 1 1 </source>
Negating the exponent (right argument) gives the reciprocal of the return value.
<syntaxhighlight lang=apl>
(2*¯4)=÷2*4
1 </source>
If the exponent is the reciprocal of some number n, the result is the n-th root of the base. For example, a right argument of <syntaxhighlight lang=apl inline>÷2</source> gives the square root.
<syntaxhighlight lang=apl>
3*2
9
9*÷2
3 </source>
Power has two inverses, Root and Logarithm: <syntaxhighlight lang=apl>
2*3
8
2⍟8
3
3√8
2 </source>
External links
Documentation
- Dyalog
- APLX
- J Dictionary, NuVoc
- BQN