Why is Math.pow(0, 0) === 1?

Question

We all know that 0⁰ is indeterminate.

But, javascript says that:

Math.pow(0, 0) === 1 // true

Answer 1

The C language definition says (7.12.7.4/2):

A domain error may occur if x is zero and y is zero.

It also says (7.12.1/2):

On a domain error, the function returns an implementation-defined value; if the integer expression math_errhandling & MATH_ERRNO is nonzero, the integer expression errno acquires the value EDOM; if the integer expression math_errhandling & MATH_ERREXCEPT is nonzero, the ‘‘invalid’’ floating-point exception is raised.

By default, the value of math_errhandling is MATH_ERRNO, so check errno for the value EDOM.

Answer 2

According to Wikipedia:

In most settings not involving continuity in the exponent, interpreting 0⁰ as 1 simplifies formulas and eliminates the need for special cases in theorems.

There are several possible ways to treat 0**0 with pros and cons to each (see Wikipedia for an extended discussion).

The IEEE 754-2008 floating point standard recommends three different functions:

• pow treats 0**0 as 1. This is the oldest defined version. If the power is an exact integer the result is the same as for pown, otherwise the result is as for powr (except for some exceptional cases).
• pown treats 0**0 as 1. The power must be an exact integer. The value is defined for negative bases; e.g., pown(−3,5) is −243.
• powr treats 0**0 as NaN (Not-a-Number – undefined). The value is also NaN for cases like powr(−3,2) where the base is less than zero. The value is defined by exp(power'×log(base)).

Answer 3

In C++ The result of pow(0, 0) the result is basically implementation defined behavior since mathematically we have a contradictory situation where N^0 should always be 1 but 0^N should always be 0 for N > 0, so you should have no expectations mathematically as to the result of this either. This Wolfram Alpha forum posts goes into a bit more details.

Although having pow(0,0) result in 1 is useful for many applications as the Rationale for International Standard—Programming Languages—C states in the section covering IEC 60559 floating-point arithmetic support:

Generally, C99 eschews a NaN result where a numerical value is useful. [...] The results of pow(∞,0) and pow(0,0) are both 1, because there are applications that can exploit this definition. For example, if x(p) and y(p) are any analytic functions that become zero at p = a, then pow(x,y), which equals exp(y*log(x)), approaches 1 as p approaches a.

Update C++

As leemes correctly pointed out I originally linked to the reference for the complex version of pow while the non-complex version claims it is domain error the draft C++ standard falls back to the draft C standard and both C99 and C11 in section 7.12.7.4 The pow functions paragraph 2 says (emphasis mine):

[...]A domain error may occur if x is zero and y is zero.[...]

which as far as I can tell means this behavior is unspecified behavior Winding back a bit section 7.12.1 Treatment of error conditions says:

[...]a domain error occurs if an input argument is outside the domain over which the mathematical function is defined.[...] On a domain error, the function returns an implementation-defined value; if the integer expression math_errhandling & MATH_ERRNO is nonzero, the integer expression errno acquires the value EDOM; [...]

So if there was a domain error then this would be implementation defined behavior but in both the latest versions of gcc and clang the value of errno is 0 so it is not a domain error for those compilers.

Update Javascript

For Javascript the ECMAScript® Language Specification in section 15.8 The Math Object under 15.8.2.13 pow (x, y) says amongst other conditions that:

If y is +0, the result is 1, even if x is NaN.