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### Zeroth power

For zero as exponent, every number a ≠ 0 gives

a0 = 1

##### Real number

You can calculate it with and also with That looks quite strange. We better check it, and find Take care however, as ##### Zero

Only 00 ≝ 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 ((a)) 0 = 1. Here you sometimes use a special notation, as shown

sin0x = cos0x = 1

##### Logarithms

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

1 = eln 1

And as ln 1 = 0 you get

1 = eln 1 = e0 = 1

##### Infinitely large

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

= ?

##### Infinitely small

Infinitely small to the power zero is

x0 = 1

because Δx is small, but infinitely small is not zero.

##### History

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