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Zero to the power of zero

Calculations usually give a correct result if you take 00 ≝ 1, but you must pay attention, and sometimes you have to check the calculation carefully.

 


Explanation

You can multiply every number with 0. The result always remains the same, because

0 × 7 = 0
0 × 6958 = 0

You can also multiply 0 with itself



For every power n ≠ 0 applies 0n = 0. Why don't we know this for it not for n = 0 ? Let us have a look at it. We know that


what you can also write as


The exponent may become very small, the result remains zero. Therefore we now take Δx and write very cheeky

because infinitely small is not zero. Is this correct or not? Well, this shows that there is a difference between infinitely small and zero. And therefore 00 ≝ 1 must be determined by definition, because you cannot calculate it.

 


Additional information

The American computer scientist Donald Knuth contends strongly: 00 has to be 1.

 


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