2025-07-21 Math finds
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@johncarlosbaez@mathstodon.xyz just posted the following:

> List all the numbers
>
> 1, 2, 3, 4, 5, 6, 7, ...
>
> skip every second one:
>
> 1, 3, 5, 7, ...
>
> form the partial sums like this:
>
> 1, 1+3, 1+3+5, 1+3+5+7, ...
>
> Hey, you get the square numbers!
>
> 1, 4, 9, 16, ...
>
> Lots of people know that.

I did not and it blows my mind.

He continues:

> But now list all the numbers
>
> 1, 2, 3, 4, 5, 6, 7, ...
>
> skip every third one:
>
> 1, 2, 4, 5, 7, 8, 10, 11, ...
>
> then form the partial sums:
>
> 1, 1+2, 1+2+4, 1+2+4+5, 1+2+4+5+7 ...
>
> =
>
> 1, 3, 7, 12, 19, ...
>
> then skip every second one:
>
> 1, 7, 19, ....
>
> then form the partial sums again:
>
> 1, 8, 27, ...
>
> Hey, now you get the cubes! You shouldn't trust me based on so
> little evidence, so do some more, or prove it works.
>
> But the cool part is that this pattern goes on forever. If you list
> all the natural numbers starting from 1, skip every nth one, form
> the list of partial sums, skip every (n-1)st one, form the list of
> partial sums, skip every (n-2)nd one, ... blah di blah di blah...
> skip every 2nd one, then form the list of partial sums, you get the
> nth powers!
>
> This is called Moessner's theorem, and I learned about it from
> Michael Fourman. It's in Chapter 7.5 here:

* Jan Rutten, The Method of Coalgebra: Exercises in Coinduction

Moral: anytime you see a pattern in mathematics - one that goes on
infinitely, not a coincidence! - it's probably just the tip of a
bigger iceberg.

Such wonderful stuff.

@plyspomitox@chaos.social commented:

> i was amazed when i learned that 1 / 7 is 0.142857 (repeating) which
> struck me as odd as that contains 14, 0028, 000056, 00000112 and so
> on and so 1/7 is basically the infinite sum of 7×(0.02)ⁿ which is so
> because 1/49 is the infinite sum of 0.02ⁿ ... 0.02 being 1/50

🤯

Sadly, I don't have anything to add. I'm still stuck in my high-school
math and even that is getting worse over the years. But it's small
posts like these that remind me of the cool stuff that's out there.

#Math