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Proof that 0 is infinity (lemmy.nz)

submitted 3 weeks ago* (last edited 3 weeks ago) by xiao@sh.itjust.works to c/Math_history@sh.itjust.works

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1 = 1

1 = (-1) + 2

1 = (-2) + 3

1 = (-3) + 4

1 = (-4) + 5 ... And so on...

By adding term by term to all these equations, we get: 1 + 1 + 1 + 1 + 1 + 1 + ... = 1 + (-1) + 2 + (-2) + 3 + (-3) + 4 + (-4) + 5 + ...

In the expression on the right, all the terms cancel out in pairs, giving: 1 + 1 + 1 + 1 + 1 + 1 + ... = 0

The expression on the left, consisting of an infinite sum of terms equal to 1, tends to infinity. Thus, 0 is equal to infinity.

And yet 0 is not equal to infinity. So where is the mistake?

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[–] niemcycle@lemmy.ca 5 points 3 weeks ago

Not a mathematician but here's my guess:

spoiler

The question assumes you can just cancel out all the values on the right side. However, by summing the first three terms we see: 1 + 1 + 1 = 1 + (-1) + 2 + (-2) + 3 3 = 3

This continues as the numbers get bigger and bigger, so no matter how large the sum of the ones on the LHS is, there will always be it's equivalent on the RHS to equal it, after the previous values are cancelled out.

Not sure how to prove rigorously, infinities always confuse me a bit when it comes to what is and isn't allowed.