Zeroth power

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For zero as exponent, every number a ≠ 0 gives

a⁰ = 1

You can calculate it with

and also with

(−a)⁰ = 1

That looks quite strange. We better check it, and find

Take care however, as

−a⁰ = −1

Only 0⁰ ≝ 1 must be determined by definition. You cannot calculate it, because

and a division by zero is not permitted.

and so it gives

i⁰ = 1

The imaginary unit i itself has no real value. Because every number is also a complex number it is correct that for every number is a⁰ = 1.

For functions, like the sine, cosine, etc. and also for the logarithm applies (f (a))⁰ = 1. Here you sometimes use a special notation, as shown

sin⁰x = cos⁰x = 1

From the definition of the logarithm follows that you can write every number as a power, also

1 = e^ln 1

And as ln 1 = 0 you get

1 = e^ln 1 = e⁰ = 1

Infinity is not a number (it has no fixed value), and so applies

∞⁰= ?

Infinitely small to the power zero is

∆x⁰ = 1

The German mathematician Christoph Rudolff described in 1515 in his book Die Coss, that x⁰ = 1.