# What mathematical truth is the most astonishing?

There are many things in mathematics that are counter intuitive despite being true. What amazes you most in the realm of mathematics?

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0.999…=9/9=1 It’s a complete mindfuck for me.

@Dutchess_III I said that 0.999…=9/9.

One proof is found here. In my Calculus class last semester, I was able to prove it by showing the sum of a series limited to infinity.

Another simple one is

⅓ =.33333333333333

⅔ =.66666666666666

3/3 =.9999999999999=1

That the sum of all natural numbers is -1/12.

@ragingloli I’ve watched a few videos explaining this proof and I still don’t really get it. I feel like they make a few jumps that don’t really make much sense, but it’s more than likely not making sense due to my lack of higher math understanding.

What I found surprising is that there are sets greater than infinity, eg the set of real numbers. The idea is very counter intuitive to me and yet Cantor’s proof is simple and convincing.

1=0

x = y.

Then x^2 = xy.

Subtract the same thing from both sides:

x^{2 – y}2 = xy – y^2.

Dividing by (x-y), obtain

x + y = y.

Since x = y, we see that

2 y = y.

Thus 2 = 1, since we started with y nonzero.

Subtracting 1 from both sides,

**1 = 0**.

@ragingloli I think that value was only the sum of all integers in the realm of string theory. I’m not familiar with string theory.

How could they know what the sum of ALL natural numbers is, since they could never get to the end of them?

@Dutchess_III The proof goes something like this. Like I said above, I don’t really follow and think they made a few jumps that don’t really make sense. IE the way they decide to add these series together. If it really were just the sum of all natural numbers it would be 1+2+3+4+5…... =∞

The sum of all natural numbers = infinity.

Look at i=1∑n [i]. The nth term is n. Now if we lim(n->∞), the series is obviously divergent, which by definition means that the sum is not a finite number.

The “proof” in @El_Cadejo‘s link is apparent nonsense. The famously diverging sum S=1–1+1—..., which alternates 0 or 1, cannot be arbitrarily assigned an “average” value of ½. That’s hand-waving at its worst! There is no *sum of natural numbers*—infinity is not a number. Not that this isn’t good tongue-in-cheek fun raised by @ragingloli, but let’s not go out on mathematical limbs that won’t support us…

I know infinity is not a number, but there is no end to numbers so that’s infinity. My head is starting to hurt, like it always does when I contemplate infinity.

Talking more about infinite decimals, the other day I was looking at mathy stuff and came across Midy’s Theorem. I couldn’t find any biographical info on Midy whatsoever. His theorem adresses how if you have an integer over a prime integer, adding one half of the digits in a repeating decimal with the other half will result in all 9s.

For example: 1/7=0.142857 142857 142857… and so on.

If you add the first half of those digits (1,4, and 2) with the second half of digits (8, 5, 7), you get 999. 1+8, 4+5, and 2+7. The corresponding numbers will be compliments, sort of like how genes work.

Another example: 1/17=0.0588235294117647…. 05882352+94117647=99999999

The proof for this looks very rigorous, and I don’t understand it.

Also, for an even period of repetition, if you group the digits into couplets and then add them, you get a multiple of 9. It seems like it’s a corollary for Midy’s Theorem.

Ex: For 1/7=.142857, 14+28+57=11*9. For 1/17=0.0588235294117647, 05+88+23+52+94+11+76+47=396=44*9.

Taking my ball and going home now.

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