Zero to the zero power: Difference between revisions

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Some programming languages are not exactly consistent (with other programming languages) when raising zero to the zeroth power: 0⁰

Task

Show the results of raising zero to the zeroth power.

If your computer language objects to 0**0 or 0^0 at compile time, you may also try something like:

 x = 0
 y = 0
 z = x**y
 say  'z='  z

Show the result here.
And of course use any symbols or notation that is supported in your computer language for exponentiation.

See also

The Wiki entry: Zero to the power of zero.
The Wiki entry: History of differing points of view.
The MathWorld™ entry: exponent laws.
- Also, in the above MathWorld™ entry, see formula (9): $x^{0}=1$ .
The OEIS entry: The special case of zero to the zeroth power

8th

Output:

1

AutoHotkey

<lang AutoHotkey>MsgBox % 0 ** 0</lang>

Output:

Ada

<lang Ada>with Ada.Text_IO, Ada.Integer_Text_IO, Ada.Long_Integer_Text_IO,

 Ada.Long_Long_Integer_Text_IO, Ada.Float_Text_IO, Ada.Long_Float_Text_IO,
 Ada.Long_Long_Float_Text_IO;

use Ada.Text_IO, Ada.Integer_Text_IO, Ada.Long_Integer_Text_IO,

 Ada.Long_Long_Integer_Text_IO, Ada.Float_Text_IO, Ada.Long_Float_Text_IO,
 Ada.Long_Long_Float_Text_IO;

procedure Test5 is

  I    : Integer           := 0;
  LI   : Long_Integer      := 0;
  LLI  : Long_Long_Integer := 0;
  F    : Float             := 0.0;
  LF   : Long_Float        := 0.0;
  LLF  : Long_Long_Float   := 0.0;
  Zero : Natural           := 0;

begin

  Put ("Integer           0^0 = "); 
  Put (I ** Zero, 2);   New_Line;
  Put ("Long Integer      0^0 = ");
  Put (LI ** Zero, 2);  New_Line;
  Put ("Long Long Integer 0^0 = ");
  Put (LLI ** Zero, 2); New_Line;
  Put ("Float           0.0^0 = ");           
  Put (F ** Zero);   New_Line;
  Put ("Long Float      0.0^0 = ");      
  Put (LF ** Zero);  New_Line;
  Put ("Long Long Float 0.0^0 = "); 
  Put (LLF ** Zero); New_Line;

end Test5; </lang>

Output:

Integer           0^0 =  1
Long Integer      0^0 =  1
Long Long Integer 0^0 =  1
Float           0.0^0 =  1.00000E+00
Long Float      0.0^0 =  1.00000000000000E+00
Long Long Float 0.0^0 =  1.00000000000000000E+00

ALGOL 68

Works with: ALGOL 68G version Any - tested with release 2.6.win32

<lang algol68>print( ( 0 ^ 0, newline ) ) </lang>

Output:

+1

Applesoft BASIC

]? 0^0
1

Bc

0 ^ 0
1

Befunge-93 doesn't have explicit support for exponentiation, but there are a couple of fingerprint extensions for Befunge-98 which add that functionality. The example below makes use of the FPDP fingerprint (double precision floating point).

Note that the result is potentially dependent on the underlying language of the interpreter, but all those tested so far have returned 1. Interpreters that don't support Befunge-98, or don't support this fingerprint, should just terminate (possibly with a warning).

Output:

1.000000

Bracmat

Output:

Burlesque

BBC BASIC

<lang bbcbasic> PRINT 0^0</lang>

Output:

C

Works with: C99

This example uses the standard pow function in the math library. 0^0 is given as 1. <lang c>#include <stdio.h>

include <math.h>
include <complex.h>

int main() { printf("0 ^ 0 = %f\n", pow(0,0));

       double complex c = cpow(0,0);

printf("0+0i ^ 0+0i = %f+%fi\n", creal(c), cimag(c)); return 0; }</lang>

Output:

0 ^ 0 = 1.000000
0+0i ^ 0+0i = nan+nani

C++

<lang cpp>#include <iostream>

include <cmath>
include <complex>

int main() {

 std::cout << "0 ^ 0 = " << std::pow(0,0) << std::endl;
 std::cout << "0+0i ^ 0+0i = " <<
   std::pow(std::complex<double>(0),std::complex<double>(0)) << std::endl;
 return 0;

}</lang>

Output:

0 ^ 0 = 1
0+0i ^ 0+0i = (nan,nan)

C#

<lang csharp>using System;

namespace ZeroToTheZeroeth {

   class Program
   {
       static void Main(string[] args)
       {
           double k = Math.Pow(0, 0);
           Console.Write("0^0 is {0}", k);           
       }
   }

}</lang>

Output:

0^0 is 1

Clojure

user=> (use 'clojure.math.numeric-tower)
user=> (expt 0 0)
1

; alternative java-interop route:
user=> (Math/pow 0 0)
1.0

COBOL

<lang cobol>identification division. program-id. zero-power-zero-program. data division. working-storage section. 77 n pic 9. procedure division.

   compute n = 0**0.
   display n upon console.
   stop run.</lang>

Output:

Common Lisp

> (expt 0 0)
1

D

<lang d>void main() {

   import std.stdio, std.math, std.bigint, std.complex;

   writeln("Int:     ", 0 ^^ 0);
   writeln("Ulong:   ", 0UL ^^ 0UL);
   writeln("Float:   ", 0.0f ^^ 0.0f);
   writeln("Double:  ", 0.0 ^^ 0.0);
   writeln("Real:    ", 0.0L ^^ 0.0L);
   writeln("pow:     ", pow(0, 0));
   writeln("BigInt:  ", 0.BigInt ^^ 0);
   writeln("Complex: ", complex(0.0, 0.0) ^^ 0);

}</lang>

Output:

Int:     1
Ulong:   1
Float:   1
Double:  1
Real:    1
pow:     1
BigInt:  1
Complex: 1+0i

Dc

0 0^p

Output:

EchoLisp

trying the 16 combinations all return the integer 1

(lib 'bigint) (define zeroes '(integer: 0 inexact=float: 0.000 complex: 0+0i bignum: #0)) (for* ((z1 zeroes) (z2 zeroes)) (write (expt z1 z2)))

   →  1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

</lang>

Eiffel

<lang Eiffel>print (0^0)</lang>

Output:

Elixir

Elixir uses Erlang's :math for power operations and can handle zero to the zero power. <lang Elixir>

math.pow(0,0)

</lang>

Output:

1.0

ERRE

<lang ERRE> ..... PRINT(0^0) ..... </lang>

Output:

F#

In the REPL:

> let z = 0.**0.;;

val z : float = 1.0

Factor

<lang factor>USING: math.functions.private ; ! ^complex 0 0 ^ C{ 0 0 } C{ 0 0 } ^complex</lang>

Output:

--- Data stack:
NAN: 8000000000000
C{ NAN: 8000000000000 NAN: 8000000000000 }

Forth

Output:

1.

Of course in an embedded program we would be tempted to "pre-calculate" the answer :-)

Output:

0 ^0 . 1 ok

Fortran

<lang Fortran> program zero double precision :: i, j double complex :: z1, z2 i = 0.0D0 j = 0.0D0 z1 = (0.0D0,0.0D0) z2 = (0.0D0,0.0D0) write(*,*) 'When integers are used, we have 0^0 = ', 0**0 write(*,*) 'When double precision numbers are used, we have 0.0^0.0 = ', i**j write(*,*) 'When complex numbers are used, we have (0.0+0.0i)^(0.0+0.0i) = ', z1**z2 end program </lang>

Output:

 When integers are used, we have 0^0 =            1
 When double precision numbers are used, we have 0.0^0.0 =    1.0000000000000000     
 When complex numbers are used, we have (0.0+0.0i)^(0.0+0.0i) =  (             NaN,             NaN)

FreeBASIC

<lang freebasic>' FB 1.05.0 Win64

Print "0 ^ 0 ="; 0 ^ 0 Sleep</lang>

Output:

0 ^ 0 = 1

Go

Go does not have an exponentiation operator but has functions in the standard library for three types, float64, complex128, and big.Int. As of Go 1.3, all are documented to return 1. <lang go>package main

import (

   "fmt"
   "math"
   "math/big"
   "math/cmplx"

)

func main() {

   fmt.Println("float64:    ", math.Pow(0, 0))
   var b big.Int
   fmt.Println("big integer:", b.Exp(&b, &b, nil))
   fmt.Println("complex:    ", cmplx.Pow(0, 0))

}</lang>

Output:

float64:     1
big integer: 1
complex:     (1+0i)

FutureBasic

<lang futurebasic> include "ConsoleWindow"

print 0^0 </lang> Output:

Groovy

Translation of: Java

Test: <lang groovy>println 0**0</lang>

Output:

Haskell

<lang haskell>import Data.Complex

main = do

 print $ 0 ^ 0
 print $ 0.0 ^ 0
 print $ 0 ^^ 0
 print $ 0 ** 0
 print $ (0 :+ 0) ^ 0
 print $ (0 :+ 0) ** (0 :+ 0)</lang>

Output:

1
1.0
1.0
1.0
1.0 :+ 0.0
NaN :+ NaN

Icon and Unicon

"Works" in both languages: <lang unicon>procedure main()

   write(0^0)

end</lang>

Output:

->z2z

Run-time error 204
File z2z.icn; Line 2
real overflow, underflow, or division by zero
Traceback:
   main()
   {0 ^ 0} from line 2 in z2z.icn
->

J

Java

<lang java>System.out.println(Math.pow(0, 0));</lang>

Output:

1.0

JavaScript

Works with: Node.js

In interactive mode: <lang javascript>> Math.pow(0, 0); 1</lang>

jq

jq version 1.4 does not have a builtin "power" function. If it were to be defined using the exp and log builtins as 'log * y | exp', then 0 | power(0) would yield null, and therefore a definition that makes a special case of 0^0 should be considered, e.g. along the following lines: <lang jq>def power(y): y as $y | if $y == 0 then 1 elif . == 0 then 0 else log * $y | exp end;</lang>

This definition will however be unsatisfactory for many purposes because it does not maintain precision for integer values of the input (.) and y.

Julia

Try all combinations of complex, float, rational, integer and boolean. <lang Julia> zs = Any[zero(Complex),

        zero(FloatingPoint),
        zero(Rational),
        zero(Integer),
        zero(Bool)]

for i in zs, j in zs

   println(i, "^", j, " = ", i^j, " (", typeof(i^j), ")")

end </lang> Note that if zs is not annotated as being of type Any all of the zeros will be promoted to complex when zs is constructed.

Output:

0 + 0im^0 + 0im = 1.0 + 0.0im (Complex{Float64})
0 + 0im^0.0 = 1.0 + 0.0im (Complex{Float64})
0 + 0im^0//1 = 1.0 + 0.0im (Complex{Float64})
0 + 0im^0 = 1 + 0im (Complex{Int64})
0 + 0im^false = 1 + 0im (Complex{Int64})
0.0^0 + 0im = 1.0 + 0.0im (Complex{Float64})
0.0^0.0 = 1.0 (Float64)
0.0^0//1 = 1.0 (Float64)
0.0^0 = 1.0 (Float64)
0.0^false = 1.0 (Float64)
0//1^0 + 0im = 1.0 + 0.0im (Complex{Float64})
0//1^0.0 = 1.0 (Float64)
0//1^0//1 = 1.0 (Float64)
0//1^0 = 1//1 (Rational{Int64})
0//1^false = 1//1 (Rational{Int64})
0^0 + 0im = 1.0 + 0.0im (Complex{Float64})
0^0.0 = 1.0 (Float64)
0^0//1 = 1.0 (Float64)
0^0 = 1 (Int64)
0^false = 1 (Int64)
false^0 + 0im = 1.0 + 0.0im (Complex{Float64})
false^0.0 = 1.0 (Float64)
false^0//1 = 1.0 (Float64)
false^0 = true (Bool)
false^false = true (Bool)

Lua

No need to try different data types or with / without decimal points as all numbers in Lua are stored in double-precision floating-point format. <lang Lua>print(0^0)</lang>

Output:

Maple

Output:

However, for consistency with IEEE-754 numerics, we also have a NaN result for the equivalent floating-point exponentiation: <lang Maple>0^0.0</lang>

Output:

Float(undefined)

Mathematica

Output:

Indeterminate

MATLAB / Octave

<lang Matlab>0^0 complex(0,0)^0</lang>

Output:

1
1

Mercury

<lang Mercury>:- module zero_to_the_zero_power.

- interface.

- import_module io.

- pred main(io::di, io::uo) is det.

- implementation.

- import_module float, int, integer, list, string.

main(!IO) :-

  io.format("    int.pow(0, 0) = %d\n", [i(pow(0, 0))], !IO),
  io.format("integer.pow(zero, zero) = %s\n",
       [s(to_string(pow(zero, zero)))], !IO),
  io.format("  float.pow(0.0, 0) = %.1f\n", [f(pow(0.0, 0))], !IO).

- end_module zero_to_the_zero_power.</lang>

Output:

    int.pow(0, 0) = 1
integer.pow(zero, zero) = 1
  float.pow(0.0, 0) = 1.0

МК-61/52

The result is error message.

NetRexx

Output:

0**0=1

NewLISP

Output:

Nim

<lang nim>import math

echo pow(0, 0)</lang>

Output:

1.0

OCaml

In the interpreter:

# 0.0 ** 0.0;;
- : float = 1.
# Complex.pow Complex.zero Complex.zero;;
- : Complex.t = {Complex.re = nan; Complex.im = nan}
# #load "nums.cma";;
# open Num;;
# Int 0 **/ Int 0;;                 
- : Num.num = Int 1

Oforth

<lang Oforth>0 0 pow println</lang>

Output:

ooRexx

<lang oorexx>/**********************************************************************

21.04.2014 Walter Pachl
- - - - /

Say 'rxCalcpower(0,0) ->' rxCalcpower(0,0) Say '0**0 ->' 0**0

requires rxmath library</lang>

Output:

rxCalcpower(0,0)  -> 1
0**0              -> 1

PARI/GP

Output:

%1 = 1

Pascal

Works with: Free Pascal

Library: math

<lang Pascal>program ZToZ; uses

 math;

begin

 write('0.0 ^ 0 :',IntPower(0.0,0):4:2);
 writeln('   0.0 ^ 0.0 :',Power(0.0,0.0):4:2);

end.</lang>

output

0.0 ^ 0 :1.00   0.0 ^ 0.0 :1.00

Perl

<lang perl>print 0 ** 0, "\n";

use Math::Complex;

print cplx(0,0) ** cplx(0,0), "\n";</lang>

Output:

1
1

Perl 6

Works with: Rakudo version 2016.11

<lang perl 6>say ' type n n**n exp(n,n)'; say '-------- -------- -------- --------';

for 0, 0.0, FatRat.new(0), 0e0, 0+0i {

   printf "%8s  %8s  %8s  %8s\n", .^name, $_, $_**$_, exp($_,$_);

}</lang>

Output:

    type         n      n**n  exp(n,n)
--------  --------  --------  --------
     Int         0         1         1
     Rat         0         1         1
  FatRat         0         1         1
     Num         0         1         1
 Complex      0+0i      1+0i      1+0i

Phix

Fair enough, I have no strong opinions on this matter, so I have just removed the test/error that was present in previous versions. Should you for any reason want to change it back, just edit builtins/VM/pPower.e, search for the two mods dated 3/11/15 (32 and 64 bit, both are two lines, test eax/rax; jz :e102cr0tple0), save and rebuild (run "p -c p"), which should take less than 10 seconds. <lang Phix>?power(0,0)</lang>

Output:

PHP

Output:

1
1

PicoLisp

Output:

1

PL/I

<lang pli> zhz: Proc Options(Main);

Dcl a dec float(10) Init(1);
Dcl b dec float(10) Init(0);
Put skip list('1**0=',a**b);
Put skip list('0**1=',b**a);
Put skip list('0**0=',b**b);
End;</lang>

Output:

1**0=                    1.000000000E+0000
0**1=                    0.000000000E+0000
0**0=
IBM0682I  ONCODE=1553  X in EXPONENT(X) was invalid.
   At offset +0000025B in procedure with entry ZHZ

PowerShell

PureBasic

<lang PureBasic> If OpenConsole()

 PrintN("Zero to the zero power is " + Pow(0,0))
 PrintN("")
 PrintN("Press any key to close the console")
 Repeat: Delay(10) : Until Inkey() <> ""
 CloseConsole()

EndIf </lang>

Output:

Zero to the zero power is 1

Python

<lang python>from decimal import Decimal from fractions import Fraction for n in (Decimal(0), Fraction(0, 1), complex(0), float(0), int(0)): try: n1 = n**n except: n1 = '<Raised exception>' try: n2 = pow(n, n) except: n2 = '<Raised exception>' print('%8s: ** -> %r; pow -> %r' % (n.__class__.__name__, n1, n2))</lang>

Output:

 Decimal: ** -> '<Raised exception>'; pow -> '<Raised exception>'
Fraction: ** -> Fraction(1, 1); pow -> Fraction(1, 1)
 complex: ** -> (1+0j); pow -> (1+0j)
   float: ** -> 1.0; pow -> 1.0
     int: ** -> 1; pow -> 1

R

<lang rsplus>print(0^0)</lang>

Output:

Racket

<lang racket>#lang racket

as many zeros as I can think of...

(define zeros (list

              0  ; unspecified number type
              0. ; hinted as float
              #e0 ; explicitly exact
              #i0 ; explicitly inexact
              0+0i ; exact complex
              0.+0.i ; float inexact
              ))

(for*((z zeros) (p zeros))

 (printf "(~a)^(~a) = ~s~%" z p
 (with-handlers [(exn:fail:contract:divide-by-zero? exn-message)]
   (expt z p))))</lang>

Output:

(0)^(0) = 1
(0)^(0.0) = 1.0
(0)^(0) = 1
(0)^(0.0) = 1.0
(0)^(0) = 1
(0)^(0.0+0.0i) = "expt: undefined for 0 and 0.0+0.0i"
(0.0)^(0) = 1
(0.0)^(0.0) = 1.0
(0.0)^(0) = 1
(0.0)^(0.0) = 1.0
(0.0)^(0) = 1
(0.0)^(0.0+0.0i) = +nan.0+nan.0i
(0)^(0) = 1
(0)^(0.0) = 1.0
(0)^(0) = 1
(0)^(0.0) = 1.0
(0)^(0) = 1
(0)^(0.0+0.0i) = "expt: undefined for 0 and 0.0+0.0i"
(0.0)^(0) = 1
(0.0)^(0.0) = 1.0
(0.0)^(0) = 1
(0.0)^(0.0) = 1.0
(0.0)^(0) = 1
(0.0)^(0.0+0.0i) = +nan.0+nan.0i
(0)^(0) = 1
(0)^(0.0) = 1.0
(0)^(0) = 1
(0)^(0.0) = 1.0
(0)^(0) = 1
(0)^(0.0+0.0i) = "expt: undefined for 0 and 0.0+0.0i"
(0.0+0.0i)^(0) = 1
(0.0+0.0i)^(0.0) = 1.0+0.0i
(0.0+0.0i)^(0) = 1
(0.0+0.0i)^(0.0) = 1.0+0.0i
(0.0+0.0i)^(0) = 1
(0.0+0.0i)^(0.0+0.0i) = +nan.0+nan.0i

REXX

<lang rexx>/*REXX program shows the results of raising zero to the zeroth power.*/ say '0 ** 0 (zero to the zeroth power) ───► ' 0**0</lang>
using PC/REXX
using Personal REXX
using REGINA
using ooRexx

Output:

0 ** 0  (zero to the zeroth power) ───►  1

using R4

Output:

Error 26 : Invalid whole number (SYNTAX)
Information: 0 ** 0 is undefined
Error occurred in statement# 2
Statement source: say '0 ** 0  (zero to the zeroth power) ───► ' 0**0
Statement context: C:\ZERO_TO0.REX, procedure: ZERO_TO0

using ROO

Output:

Error 26 : Invalid whole number (SYNTAX)
Information: 0 ** 0 is undefined
Error occurred in statement# 2
Statement source: say '0 ** 0  (zero to the zeroth power) ───► ' 0**0
Statement context: C:\ZERO_TO0.REX, procedure: ZERO_TO0

Ring

Ruby

<lang ruby>require 'bigdecimal'

[0, 0.0, Complex(0), Rational(0), BigDecimal.new("0")].each do |n|

 printf "%10s: ** -> %s\n" % [n.class, n**n]

end</lang>

Output:

    Fixnum: ** -> 1
     Float: ** -> 1.0
   Complex: ** -> 1+0i
  Rational: ** -> 1/1
BigDecimal: ** -> 0.1E1

Rust

<lang rust>fn main() {

   println!("{}",0u32.pow(0));

}</lang>

Output:

S-lang

<lang S-lang>print(0^0);</lang>

Output:

1.0

Scala

Library: Scala

<lang Scala> assert(math.pow(0, 0) == 1, "Scala blunder, should go back to school !")</lang>

Scheme

<lang scheme>(display (expt 0 0)) (newline) (display (expt 0.0 0.0)) (newline) (display (expt 0+0i 0+0i)) (newline)</lang>

Output:

1
1.0
1.0

Seed7

<lang seed7>$ include "seed7_05.s7i";

 include "float.s7i";
 include "complex.s7i";

const proc: main is func

 begin
   writeln("0      ** 0   = " <& 0 ** 0);
   writeln("0.0    ** 0   = " <& 0.0 ** 0);
   writeln("0.0    ** 0.0 = " <& 0.0 ** 0.0);
   writeln("0.0+0i ** 0   = " <& complex(0.0) ** 0);
 end func;

</lang>

Output:

0      ** 0   = 1
0.0    ** 0   = 1.0
0.0    ** 0.0 = 1.0
0.0+0i ** 0   = 1.0+0.0i

Sidef

<lang ruby>[0, Complex(0, 0)].each {|n|

   say n**n;
   say n.pow(n);
   say pow(n, n);
   say Math.pow(n, n);

}</lang>

Output:

Taking the 0'th root of a number and raising it back to the zero power, we also get a 1:

Smalltalk

<lang smalltalk> 0 raisedTo: 0 0.0 raisedTo: 0.0 </lang>

Output:

1
1.0

smart BASIC

-- Scott A. Rossell, 12-31-16 <lang qbasic>PRINT 0^0</lang>

Output:

SQL

<lang SQL> SQL> select power(0,0) from dual; </lang>

Output:

POWER(0,0)
----------
         1

Standard ML

In the interpreter:

- Math.pow (0.0, 0.0);
val it = 1.0 : real

Swift

<lang swift>import Darwin print(pow(0.0,0.0))</lang>

Output:

1.0

Tcl

Interactively… <lang tcl>% expr 0**0 1 % expr 0.0**0.0 1.0</lang>

uBasic/4tH

<lang>Print 0^0</lang>

Output:

1

0 OK, 0:9

Ursa

Cygnus/X Ursa is written in Java, and as a result returns 1.0 when raising 0 to the 0. <lang ursa>> out (pow 0 0) endl console 1.0</lang>

VBScript

<lang vb>WScript.Echo 0 ^ 0</lang>

Output:

XLISP

1 [2] </lang>

zkl

<lang zkl>(0.0).pow(0) //--> 1.0 var BN=Import("zklBigNum"); // big ints BN(0).pow(0) //--> 1</lang>

@@ Line 534: / Line 534: @@
 =={{header|MATLAB}} / {{header|Octave}}==
-<lang Matlab>0^0</lang>
+<lang Matlab>0^0
+complex(0,0)^0</lang>
 {{out}}
-<pre>1</pre>
+<pre>1
+</pre>
 =={{header|Mercury}}==