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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



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.



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



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.



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

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