Shit like this makes me realise why people become mathematicians. You just play around with numbers and find funny facts about them.
this post was submitted on 27 Apr 2025
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So, years ago in college in Linear Algebra our professor said to us to study about idempotent matrices. So I checked out that wiki page and saw the example for 2x2 matrix, that are composed by the numbers 3, -6, 1 and -2. And I was like wait a second, 3×-2=-6 there's no way they are not relationship there, so I started trying other numbers, and found and proved (using induction) that any n, -n(n-1), 1, -(n-1) is an idempotent matrix. At the test there were no questions about that, and I was short of 0.5 poits to pass the class without having to present a final exam and I told my professor that I spent a lot of time learning that and that even discovered something and proved he pass me the chart and asked me to proved it, after that he gave the missing points. Was really good.
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I myself once learned 380 digits of π, when I was a crazy high-school kid. My never-attained ambition was to reach the spot, 762 digits out in the decimal expansion, where it goes "999999", so that I could recite it out loud, come to those six 9s, and then impishly say, "and so on!"
—Douglas Hofstadter
That would be an amazing party trick.
Actually come to think of it, even more amazing in the age of smart phones, when it's possible to easily verify to numbers you're reciting.
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Then you try to figure out why they do be like that
I mean, mathematics are an invention. A useful one, sure, but the whole thing is just made up by people playing around with numbers and going “what if we had a new, different kind of numbers…”
This reminds me of a friend who said "I like to think of dividing by zero as giving zero instead of infinity, because it means you can keep doing math on it" and I just thought that was so pure.
there are number systems that work differently
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gonna need this in every base
I'll start with base 2:
1/1 = 1
Base 3:
21 / 12 = 1.1012101210121012
I'm gonna need a formal proof for this.
We should be friends
that's base 8 tho
Every base is base 10
Who are You, Who are so Wise in the Ways of Science?
I just noticed what the numbers are. It really is easy to memorize. So convenient.
Unfortunately, it requires remembering 8, so it kinda defeats the purpose.
987654312÷123456789
Change the 21 at the end of the first number to 12 and its perfect. It was only ever 9 away.
WOOOAH🤯
Witch! Begone foul demon, and take your dark sorcery with you!
The funniest part is that some people will never understand the absolute crusade that some mathematicians might fight over this one day
I wonder if there’s a related infinite sequence which converges on 8?
This sequence approximates an integer to arbitrary precision, not 8 specifically though, and never perfectly.
I tried it out using other bases, and the rule seems to be that doing this in base n results in n-2 with remainder n-1. So it doesn't ever actually converge, but the remainder becomes small very fast.
never perfectly
eyes you in binary
The sequence in base 2 is only 1/1.
Wonder how close base-16 gets.
FEDCBA987654321 / 123456789ABCDEF
Off by '1.82959E–16' !
Hmmmm.....
Edit: you can kinda think of it being 0, plus the 1/1 that would have ended up as a remainder in larger bases. In base 2, it just ends up being a full 1.
(n * 8 + 1) / n
9876543210987654321 / 1234567890123456789 = 8,0000000729000
strange coincidence
See my other comment, it's no coincide– there's a pattern. I would love to see an actual proof for it though, I don't know enough to say why it behaves that way.
It contains the number 8 though. So how is that useful
Well, simple. Jest substitute that 8 with the above approximation.
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