I’ve recently seen a video where someone did an explanation on why 0 to the power of 0 is defined as 1. He went with X to the power of X and made X increasingly smaller decimals. X = 0.9, X = 0.8, and so on. The results for 0.9 to the power of 0.9 and then 0.8 to the power of 0.8 kept getting smaller until about X = 0.4, where it started to increase again, so 0.3 to the power of 0.3 was a bigger number than 0.4 to 0.4. At what number is the exact “turning point” in this kind of series and why is it in such a weird place (as opposed to at 0.5, i.e halfway to 0)?
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> I’ve recently seen a video where someone did an explanation on why 0 to the power of 0 is defined as 1
That video is bad. Why? Because defining(!) 0^0 = 1 has nothing to do with limits. At all. You could just as well take the limit of 0^x for x>0 as x approaches 0, but then you would get 0 instead of 1. There is hardly any limit reason where 0^0 = 1 simplifies anything.
The true reason is way simpler: it is a convention that works extremely well, while any other choice makes things uglier (but not “wrong”). We write down many expressions in ways where it would be very tedious to mention 0 explicitly, such as polynomials or sums.
Also, there is a combinatorial reason that is actually related: m^n is the number of ways to color n items, with one of m colors each, no restrictions. So how many ways are there to color 0 things with 0 colors? Well, 1: do exactly nothing, it is already okay as it is. Nothing to color, after all. Having more colors as options changes nothing, hence m^0 is still 1. But if you have one or more objects but no colors, there is no way to accomplish the task, hence 0^n = 0 for n>0.
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