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### Zeroth power

For **zero as exponent**, every number *a* ≠ 0 gives

a^{0}= 1

##### Real number

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

−

a^{0}= −1

##### Zero

Only 0^{0} ≝ 1 must be determined *by definition*. You cannot calculate it, because

and a division by zero is not permitted.

##### Imaginary unit

From the definition of the imaginary unit you get

and so it gives

i^{ 0}= 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 *a*^{ 0} = 1.

##### Functions

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

sin

^{0}x= cos^{0}x= 1

##### Logarithms

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

And as ln 1 = 0 you get

##### Infinitely large

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

∞

^{0 }= ?

##### Infinitely small

Infinitely small to the power zero is

∆

x^{0}= 1

because Δ*x* is small, but infinitely small is not zero.

## HistoryThe German mathematician Christoph Rudolff described in 1515 in his book |