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Zero is both the biggest advancement in math and its biggest problem. Think about it, what is 1^0? It’s 1. What is 1^1? Also 1. You can’t have two different equations with only one different digit and still get the same answer, because that means the logic isn’t fully rational.
I’ve heard it explained as the ‘1’ you get from a zero exponent is the result of no multiplication being done. If you multiply any number times 1, you get that number, so not multiplying any number means you get one.
Problem is, that ‘1’ is being inserted to make the equation seem to make sense, but the logic is backward compared to every other exponent. In non-zero exponents, there’s no extra variables, just the numbers being multiplied. We didn’t need to multiply anything times one, so we didn’t.
If we work backward to show where this magical ‘1’ comes from, we can do aᶜ/aᵈ we get aᶜ⁻ᵈ. We can then say that, for example, a²/a²=a⁰, and we know that a²/a²=1. There it is. But this gives us a new problem: a⁰ is, by the very nature of its nomenclature, stating that no operation is being done. If no operation is done in math, we should be left with 0 because, well, nothing should be there. Instead, we’re left with 1 because that’s the baseline of non-zero multiplication.
Our number system moved to using zero as its smallest value, and the smallest value in multiplication is now also zero… Except when it isn’t. One is still the baseline when we’re not multiplying by zero. This results in exponents showing an operation in a no-operation equation. It’s like zero was tacked onto multiplication and the rest never got updated.
A math genius needs to come along and discover what we’re all doing wrong. It sure as hell won’t be me, I just like finding such problems and pointing them out.