What are the most mindblowing fact in mathematics?

Submitted ⁨⁨1⁩ ⁨year⁩ ago⁩ by ⁨cll7793@lemmy.world⁩ to ⁨[deleted]⁩

What concepts or facts do you know from math that is mindblowing, awesome, or simply fascinating?

Here are some I would like to share:

Gödel’s incompleteness theorems: There are some problems in math so difficult that it can never be solved no matter how much time you put into it.
Halting problem: It is impossible to write a program that can figure out whether or not any input program loops forever or finishes running. (Undecidablity)

The Busy Beaver function

Now this is the mindblowing one. What is the largest non-infinite number you know? Graham’s Number? TREE(3)? TREE(TREE(3))? This one will beat it easily.

The Busy Beaver function produces the fastest growing number that is theoretically possible. These numbers are so large we don’t even know if you can compute the function to get the value even with an infinitely powerful PC.
In fact, just the mere act of being able to compute the value would mean solving the hardest problems in mathematics.
Σ(1) = 1
Σ(4) = 13
Σ(6) > 10^10^10^10^10^10^10^10^10^10^10^10^10^10^10
Σ(17) > Graham’s Number
Σ(27) If you can compute this function the Goldbach’s conjecture is false.
Σ(744) If you can compute this function the Riemann hypothesis is false.

Sources:

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Comments

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_pheel_@lemmy.world ⁨1⁩ ⁨year⁩ ago
There are more ways to arrange a deck of 52 cards than there are atoms on Earth.

52 Factorial

I feel this one is quite well known, but it’s still pretty cool.

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- Nibodhika@lemmy.world ⁨1⁩ ⁨year⁩ ago
  An extension of that is that every time you shuffle a deck of cards there’s a high probability that that particular arrangement has never been seen in the history of mankind.
  
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  - billiam0202@lemmy.world ⁨1⁩ ⁨year⁩ ago
    With the caveat that it’s not the first shuffle of a new deck. Since card decks come out of the factory in the same order, the probability that the first shuffle will result in an order that has been seen before is a little higher than on a deck that has already been shuffled.
    
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  - Elderos@lemmings.world ⁨1⁩ ⁨year⁩ ago
    assuming a perfect mechanical shuffle, I think the odds are near zero. humans don’t shuffle perfectly though!
    
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Artisian@lemmy.world ⁨1⁩ ⁨year⁩ ago
For the uninitiated, the monty Hall problem is a good one.

Start with 3 closed doors, and an announcer who knows what’s behind each. The announcer says that behind 2 of the doors is a goat, and behind the third door is ~~a car~~ student debt relief, but doesn’t tell you which door leads to which. They then let’s you pick a door, you get what’s behind the door. Before you open it, they open a different door than your choice and reveals a goat. Then the announcer says you are allowed to change your choice.

So should you switch?

The answer turns out to be yes. 2/3rds of the time you are better off switching. But even famous mathematicians didn’t believe it at first.

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- Evirisu@kbin.social ⁨1⁩ ⁨year⁩ ago
  I know the problem is easier to visualize if you increase the number of doors. Let's say you start with 1000 doors, you choose one and the announcer opens 998 other doors with goats. In this way is evident you should switch because unless you were incredibly lucky to pick up the initial door with the prize between 1000, the other door will have it.
  
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  - Artisian@lemmy.world ⁨1⁩ ⁨year⁩ ago
    I now recall there was a numberphile with exactly that visualisation! It’s a clever visual
    
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  - dandroid@dandroid.app ⁨1⁩ ⁨year⁩ ago
    This is so mind blowing to me, because I get what you’re saying logically, but my gut still tells me it’s a 50/50 chance.
    
    But I think the reason it is true is because the other person didn’t choose the other 998 doors randomly. So if you chose any of the other 998 doors, it would still be between the door you chose and the winner, other than the 1/1000 chance that you chose right at the beginning.
    
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  - Kissaki@feddit.de ⁨1⁩ ⁨year⁩ ago
    I don’t find this more intuitive. It’s still one or the other door.
    
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  - clumsyninza@lemmy.world ⁨1⁩ ⁨year⁩ ago
    How do we even come up with such amazing problems right ? It’s fascinating.
    
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  - Sharkwellington@lemmy.one ⁨1⁩ ⁨year⁩ ago
    This is fantastic, thank you.
    
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- Sharkwellington@lemmy.one ⁨1⁩ ⁨year⁩ ago
  
  But even famous mathematicians didn’t believe it at first.
  
  They emphatically did not believe it at first. Marilyn vos Savant was flooded with about 10,000 letters after publishing the famous 1990 article, and had to write two followup articles to clarify the logic involved.
  
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  - Artisian@lemmy.world ⁨1⁩ ⁨year⁩ ago
    Oh that’s cool - I had heard one or two examples only. Is there some popular writeup of the story from Savant’s view?
    
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- Ethalis@jlai.lu ⁨1⁩ ⁨year⁩ ago
  I know it to be true, I’ve heard it dozens of times, but my dumb brain still refuses to accept the solution everytime. It’s kind of crazy really
  
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  - Elderos@lemmings.world ⁨1⁩ ⁨year⁩ ago
    To me, it makes sense because there was initially 2 chances out of 3 for the prize to be in the doors you did not pick. Revealing a door does not reset the odds of the whole problem, it is still more likely that the prize is in one of the door you did not pick, and a door was removed from that pool. Imo, the key element here is that your own door cannot be revealed early, so it is never “tested”, and this ultimately make the other door more “vouched” for, statistically, and since you know that the door was more likely to be in the other set to begin with, well, might as well switch!
    
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  - Num10ck@lemmy.world ⁨1⁩ ⁨year⁩ ago
    like on paper the odds on your original door was 1/3 and the option door is 1/2, but in reality with the original information both doors were 1/3 and now with the new information both doors are 1/2.
    
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  - emokidforever@lemmy.world ⁨1⁩ ⁨year⁩ ago
    This explanation really helped me make sense of it: Monty Hall Problem (best explanation) - Numberphile
    
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- jscari@lemmy.world ⁨1⁩ ⁨year⁩ ago
  It took me a while to wrap my head around this, but here’s how I finally got it:
  
  There are three doors and one prize, so the odds of the prize being behind any particular door are 1/3. So let’s say you choose door #1. There’s a 1/3 chance that the prize is behind door #1 and, therefore, a 2/3 chance that the prize is behind either door #2 OR door #3.
  
  Now here’s the catch. Monty opens door #2 and reveals that it does not contain the prize. The odds are the same as before – a 1/3 chance that the prize is behind door #1, and a 2/3 chance that the prize is behind either door #2 or door #3 – but now you know definitively that the prize isn’t behind door #2, so you can rule it out. Therefore, there’s a 1/3 chance that the prize is behind door #1, and a 2/3 chance that the prize is behind door #3. So you’ll be twice as likely to win the prize if you switch your choice from door #1 to door #3.
  
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- clumsyninza@lemmy.world ⁨1⁩ ⁨year⁩ ago
  It’s a good one.
  
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mookulator@lemmy.world ⁨1⁩ ⁨year⁩ ago
The four-color theorem is pretty cool.

You can take any map of anything and color it in using only four colors so that no adjacent “countries” are the same color. Often it can be done with three!

Maybe not the most mind blowing but it’s neat.

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- cll7793@lemmy.world ⁨1⁩ ⁨year⁩ ago
  Thanks for the comment! It is cool also pretty aesthetically pleasing!
  
  Image
  
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  - Reliant1087@lemmy.world ⁨1⁩ ⁨year⁩ ago
    Your map made me think how interesting US would be if there were 4 major political parties. Maybe no one will win the presidential election 🤔
    
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- Blyfh@lemmy.world ⁨1⁩ ⁨year⁩ ago
  What about a hypothetical country that is shaped like a donut, and the hole is filled with four small countries? One of the countries must have the color of one of its neighbors, no?
  
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  - Afrazzle@sh.itjust.works ⁨1⁩ ⁨year⁩ ago
    I think the four small countries inside would each only have 2 neighbours. So you could take 2 that are diagonal and make them the same colour.
    
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  - BitSound@lemmy.world ⁨1⁩ ⁨year⁩ ago
    In that image, you could color yellow into purple since it’s not touching purple. Then, you could color the red inner piece to yellow, and have no red in the inner pieces.
    
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- sanguinepar@lemmy.world ⁨1⁩ ⁨year⁩ ago
  I read an interesting book about that once, will need to see if I can find the name of it.
  
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  - L3s@lemmy.world [bot] ⁨1⁩ ⁨year⁩ ago
    Read the author as Robin Williams
    
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- Artisian@lemmy.world ⁨1⁩ ⁨year⁩ ago
  Note you’ll need the regions to be connected (or allow yourself to color things differently if they are the same ‘country’ but disconnected). I forget if this causes problems for any world map.
  
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  - rycee@lemmy.world ⁨1⁩ ⁨year⁩ ago
    I suspect that the Belgium-Netherlands border defies any mathematical description.
    
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- dandroid@dandroid.app ⁨1⁩ ⁨year⁩ ago
  If you had a 3 dimensional map, would you need more colors to achieve the same results?
  
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- eran_morad@lemmy.world ⁨1⁩ ⁨year⁩ ago
  this whole thread is the shit.
  
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- clumsyninza@lemmy.world ⁨1⁩ ⁨year⁩ ago
  Isn’t the proof of this theorem like millions of pages long or something (proof done by a computer ) ? I mean how can you even be sure that it is correct ? There might be some error somewhere.
  
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BourneHavoc@lemmy.world ⁨1⁩ ⁨year⁩ ago
I came here to find some cool, mind-blowing facts about math and have instead confirmed that I’m not smart enough to have my mind blown. I am familiar with some of the words used by others in this thread, but not enough of them to understand, lol.

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- Maruki_Hurakami@lemm.ee ⁨1⁩ ⁨year⁩ ago
  Same here! Great post but I’m out! lol
  
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  - Ziro427@lemmy.world ⁨1⁩ ⁨year⁩ ago
    Nonsense! I can blow both your minds without a single proof or mathematical symbol, observe!
    
    There are different sizes of infinity.
    
    Think of integers, or whole numbers; 1, 2, 3, 4, 5 and so on. How many are there? Infinite, you can always add one to your previous number.
    
    Now take odd numbers; 1, 3, 5, 7, and so on. How many are there? Again, infinite because you just add 2 to the previous odd number and get a new odd number.
    
    Both of these are infinite, but the set of numbers containing odd numbers is by definition smaller than the set of numbers containing all integers, because it doesn’t have the even numbers.
    
    But they are both still infinite.
    
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  - cll7793@lemmy.world ⁨1⁩ ⁨year⁩ ago
    There was a response I left in the main comment thread but I’m not sure if you will get the notification. I wanted to post it again so you see it
    
    Response below Please feel free to ask any questions! Math is a wonderful field full of beauty but unfortunately almost all education systems fail to show this and instead makes it seem like raw robotic calculations instead of creativity.
    
    Math is best learned visually and with context to more abstract terms. 3Blue1Brown is the best resource in my opinion for this!
    
    Here’s a mindblowing fact for you along with a video from 3Blue1Brown. Imagine you are sliding a 1,000,000 kg box and slamming it into a 1 kg box on an ice surface with no friction. The 1 kg box hits a wall and bounces back to hit the 1,000,000 kg box again.
    
    The number of bounces that appear is related to Pi. Crazy right? Why would pi appear here? If you want to learn more here’s a video from the best math teacher in the world.
    
    www.youtube.com/watch?v=HEfHFsfGXjs
    
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- cll7793@lemmy.world ⁨1⁩ ⁨year⁩ ago
  Please feel free to ask any questions! Math is a wonderful field full of beauty but unfortunately almost all education systems fail to show this and instead makes it seem like raw robotic calculations instead of creativity.
  
  Math is best learned visually and with context to more abstract terms. 3Blue1Brown is the best resource in my opinion for this!
  
  Here’s a mindblowing fact for you along with a video from 3Blue1Brown. Imagine you are sliding a 1,000,000 kg box and slamming it into a 1 kg box on an ice surface with no friction. The 1 kg box hits a wall and bounces back to hit the 1,000,000 kg box again.
  
  The number of bounces that appear is related to Pi. Crazy right? Why would pi appear here? If you want to learn more here’s a video from the best math teacher in the world.
  
  www.youtube.com/watch?v=HEfHFsfGXjs
  
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  - BourneHavoc@lemmy.world ⁨1⁩ ⁨year⁩ ago
    Thanks! I appreciate the response. I’ve seen some videos on 3blue1brown and I’ve really enjoyed them. I think if I were to go back and fill in all the blank spots in my math experience/education I would enjoy math quite a bit.
    
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  - JustZ@lemmy.world ⁨1⁩ ⁨year⁩ ago
    I don’t know why it appears here or why I feel this way, but pictures the box bouncing off the wall and back feels intuitively round to me.
    
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betheydocrime@lemmy.world ⁨1⁩ ⁨year⁩ ago
For me, personally, it’s the divisible-by-three check. You know, the little shortcut you can do where you add up the individual digits of a number and if the resulting sum is divisible by three, then so is the original number.

That, to me, is black magic fuckery. Much like everything else in this thread I have no idea how it works, but unlike everything else is actually a handy trick that I use semifrequently

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- Zippy@lemmy.world ⁨1⁩ ⁨year⁩ ago
  All numbers are dividible by 3. Just saying.
  
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BodePlotHole@lemmy.world ⁨1⁩ ⁨year⁩ ago
The utility of Laplace transforms in regards to differential systems.

In engineering school you learn to analyze passive DC circuits early on using not much more than ohms law and Thevenin’s Theoram. This shit can be taught to elementary schoolers.

Then a little while later, you learn how to do non-finear differential equations to help work complex systems, whether it’s electrical, mechanical, thermal, hydrolic, etc. This shit is no walk in the park.

Then Laplace transforms/identities come along and let you turn non-linear problems in time-based space, into much simpler problems in frequency-based space. Shit blows your mind.

THEN a mafacka comes along and teaches you that these tools can be used to turn complex differential system problems (electrical, mechanical, thermal, hydrolic, etc) into simple DC circuits you can analyze/solve in frequency-based space, then convert back into time-based space for the answers.

I know this is super applied calculus shit, but I always love that sweet spot where all the high-concept math finally hits the pavement.

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- djmarcone@lemmy.world ⁨1⁩ ⁨year⁩ ago
  And then they tell you that the fundamental equations for thermal, fluid, electrical and mechanical are all basically the same when you are looking at the whole Laplace thing. It’s all the same…
  
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- Spedwell@lemmy.world ⁨1⁩ ⁨year⁩ ago
  ABSOLUTELY. I just recently capped off the Diff Eq, Signals, and Controls courses for my undergrad, and truly by the end you feel like a fucking wizard. It’s crazy how much problem-solving/system modeling power there is in such a (relatively) simple, easy to apply, and beautifully elegant mathematical tool.
  
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calexil@lemmy.world ⁨1⁩ ⁨year⁩ ago
e^(pi i) = -1

like, what?

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- SandLight@lemmy.world ⁨1⁩ ⁨year⁩ ago
  3Blue 1Brown actually explains that one in a way that makes it seem less coincidental and black magic. Totally worth a watch
  
  www.youtube.com/watch?v=v0YEaeIClKY
  
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timeisart@lemmy.world ⁨1⁩ ⁨year⁩ ago
Multiply 9 times any number and it always “reduces” back down to 9 (add up the individual numbers in the result)

For example: 9 x 872 = 7848, so you take 7848 and split it into 7 + 8 + 4 + 8 = 27, then do it again 2 + 7 = 9 and we’re back to 9

It can be a huge number and it still works:

9 x 987345734 = 8886111606

8+8+8+6+1+1+1+6+0+6 = 45

4+5 = 9

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- nUbee@lemmy.world ⁨1⁩ ⁨year⁩ ago
  I suspect this holds true to any base x numbering where you take the highest valued digit and multiply it by any number. Try it with base 2 (1), 4 (3), 16 (F) or whatever.
  
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Nfamwap@lemmy.world ⁨1⁩ ⁨year⁩ ago
11 X 11 = 121

111 X 111 = 12321

1111 X 1111 = 1234321

11111 X 11111 = 123454321

111111 X 1111111 = 12345654321

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- Mr_Dr_Oink@lemmy.world ⁨1⁩ ⁨year⁩ ago
  You could include 1 x 1 = 1
  
  But thats so cool. Maths is crazy.
  
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- RandallFlagg@lemm.ee ⁨1⁩ ⁨year⁩ ago
  holt shit
  
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- tom42@lemmy.world ⁨1⁩ ⁨year⁩ ago
  Amazing in deed!
  
  Just a small typo in the very last factor 1111111.
  
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aggelalex@lemmy.world ⁨1⁩ ⁨year⁩ ago
The Fourier series. Musicians may not know about it, but everything music related, even harmony, boils down to this.

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WtfEvenIsExistence@reddthat.com ⁨1⁩ ⁨year⁩ ago
en.wikipedia.org/wiki/Collatz_conjecture

Here’s a Veritasium Video about it: youtu.be/094y1Z2wpJg

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- Caboose12000@lemmy.world ⁨1⁩ ⁨year⁩ ago
  maybe this will make more sense when I watch the veritasium video, but I don’t have time to do that until the weekend. How is 3x+1 unprovable? won’t all odd numbers multiplied by 3 still be odd? and won’t adding 1 to an odd number always make it even? and aren’t all even numbers by definition divisible by 2? I’m struggling to see how there could be u certainty in this
  
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  - WtfEvenIsExistence@reddthat.com ⁨1⁩ ⁨year⁩ ago
    The number 26 reaches as high as 40 before falling back to 4-2-1 loop. The very next number, 27, goes up to 9232 before it stops going higher. The uncertainty is that what if there is a special number that doesn’t just stop at a peak, but goes on forever. I’m not really good at explaining things, so you’re gonna have to watch the video.
    
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  - mipadaitu@lemmy.world ⁨1⁩ ⁨year⁩ ago
    The unproven part is that it eventually will reach 1, not that it’s not possible to do the computation. Someone may find a number loop that doesn’t eventually reach 1.
    
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Valmond@lemmy.mindoki.com ⁨1⁩ ⁨year⁩ ago
Quickly a game of chess becomes a never ever played game of chess before.

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- beto@lemmy.studio ⁨1⁩ ⁨year⁩ ago
  Related: every time you shuffle a deck of cards you get a sequence that has never happened before. The chance of getting a sequence that has occurred is stupidly small.
  
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  - my_hat_stinks@lemmy.world ⁨1⁩ ⁨year⁩ ago
    Most of the time, but only as long as the shuffle is actually random. A perfect riffle shuffle on a brand new deck will get you the same result every time, and 8 perfect riffles on a row get you back to where you started.
    
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- dQw4w9WgXcQ@lemm.ee ⁨1⁩ ⁨year⁩ ago
  I’m guessing this is more pronounced at lower levels. At high level chess, I often hear commentators comparing the moves to their database of games, and it often takes 20-30 moves before they declare that they have now reached a position which has never been reached in a professional game. The high level players have been grinding openings and their counters and the counters to the counters so deeply that a lot of the initial moves can be pretty common.
  
  Also, high levels means that games are narrowing more towards the “perfect” moves, meaning that repetition from existing games are more likely.
  
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  - Valmond@lemmy.mindoki.com ⁨1⁩ ⁨year⁩ ago
    Oh yes I agree totally!
    
    At those (insane) levels “everyone” knows the “best” move after the choice of opening and for a long time.
    
    But for the millions of games played every day, I guess it’s a bit less :-)
    
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Jordan117@lemmy.world ⁨1⁩ ⁨year⁩ ago
Euler’s identity, which elegantly unites some of the most fundamental constants in a single equation:

e^(iπ)+1=0

Euler’s identity is often cited as an example of deep mathematical beauty. Three of the basic arithmetic operations occur exactly once each: addition, multiplication, and exponentiation. The identity also links five fundamental mathematical constants:

The number 0, the additive identity.

The number 1, the multiplicative identity.

The number π (π = 3.1415…), the fundamental circle constant.

The number e (e = 2.718…), also known as Euler’s number, which occurs widely in mathematical analysis.

The number i, the imaginary unit of the complex numbers.

Furthermore, the equation is given in the form of an expression set equal to zero, which is common practice in several areas of mathematics.

Stanford University mathematics professor Keith Devlin has said, “like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler’s equation reaches down into the very depths of existence”. And Paul Nahin, a professor emeritus at the University of New Hampshire, who has written a book dedicated to Euler’s formula and its applications in Fourier analysis, describes Euler’s identity as being “of exquisite beauty”.

Mathematics writer Constance Reid has opined that Euler’s identity is “the most famous formula in all mathematics”. And Benjamin Peirce, a 19th-century American philosopher, mathematician, and professor at Harvard University, after proving Euler’s identity during a lecture, stated that the identity “is absolutely paradoxical; we cannot understand it, and we don’t know what it means, but we have proved it, and therefore we know it must be the truth”.
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- dandroid@dandroid.app ⁨1⁩ ⁨year⁩ ago
  This is the one that made me say out loud, “math is fucking weird”
  
  I started trying to read the explanations, and it just got more and more complicated. I minored in math. But the stuff I learned seems trivial by comparison. I have a friend who is about a year away from getting his PhD in math. I don’t even understand what he’s saying when he talks about math.
  
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- aja@lemm.ee ⁨1⁩ ⁨year⁩ ago
  Time for a deep dive. Wish me luck, lads.
  
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  - sadbehr@lemmy.nz ⁨1⁩ ⁨year⁩ ago
    Let us know what you find in the depths.
    
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FergleFFergleson@infosec.pub ⁨1⁩ ⁨year⁩ ago
The one I bumped into recently: the Coastline Paradox

“The coastline paradox is the counterintuitive observation that the coastline of a landmass does not have a well-defined length. This results from the fractal curve–like properties of coastlines; i.e., the fact that a coastline typically has a fractal dimension.”

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naura@kbin.social ⁨1⁩ ⁨year⁩ ago
Seeing mathematics visually.

I am a huge fan of 3blue1brown and his videos are just amazing. My favorite is linear algebra. It was like an out of body experience. All of a sudden the world made so much more sense.

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- cumcum69@lemmy.world ⁨1⁩ ⁨year⁩ ago
  His video about understanding multiple dimensions was what finally made it click for me
  
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joel_feila@lemmy.world ⁨1⁩ ⁨year⁩ ago
e^(I*pi) = -1

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- elscallr@lemmy.world ⁨1⁩ ⁨year⁩ ago
  My favorite form, just slightly different,
  
  e^(i*π)+1=0
  
  Euler’s identity directly relates all the really cool mathematical constants into one elegant formula: e, i, π, 1, and 0
  
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  - joel_feila@lemmy.world ⁨1⁩ ⁨year⁩ ago
    I think that is the cooler form but I first saw Euler’s ID as =-1
    
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PsychoNewt@lemmy.world ⁨1⁩ ⁨year⁩ ago
This is my silly contribution: 70% of 30 is equal to 30% of 70. This applies to other numbers and can be really helpful when doing percentages in your head. 15% of 77 is equal to 77% of 15.

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- 0nXYZ@lemmy.world ⁨1⁩ ⁨year⁩ ago
  I’ve seen this one used in the news when they want to make one side of a statistic stand out.
  
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Etterra@lemmy.world ⁨1⁩ ⁨year⁩ ago
That you can have 5 apples, divide them zero times, and somehow end up with math shitting itself inside-out at you even though the apples are still just sitting there.

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metiulekm@sh.itjust.works ⁨1⁩ ⁨year⁩ ago
Imagine a soccer ball. The most traditional design consists of white hexagons and black pentagons. If you count them, you will find that there are 12 pentagons and 20 hexagons.

Now imagine you tried to cover the entire Earth in the same way, using similar size hexagons and pentagons (hopefully the rules are intuitive). How many pentagons would be there? Intuitively, you would think that the number of both shapes would be similar, just like on the soccer ball. So, there would be a lot of hexagons and a lot of pentagons. But actually, along with many hexagons, you would still have exactly 12 pentagons, not one less, not one more. This comes from the Euler’s formula, and there is a nice sketch of the proof here: math.stackexchange.com/a/18347.

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backseat@lemmy.world ⁨1⁩ ⁨year⁩ ago
The square of any prime number >3 is one greater than an exact multiple of 24.

For example, 7² = 49= (2 * 24) + 1

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parrottail@sh.itjust.works ⁨1⁩ ⁨year⁩ ago
Godel’s incompleteness theorem is actually even more subtle and mind-blowing than how you describe it. It states that in any mathematical system, there are truths in that system that cannot be proven using just the mathematical rules of that system. It requires adding additional rules to that system to prove those truths. And when you do that, there are new things that are true that cannot be proven using the expanded rules of that mathematical system.

"It’s true, we just can’t prove it’.

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randon31415@lemmy.world ⁨1⁩ ⁨year⁩ ago
That e^pi-pi = 20

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Evirisu@kbin.social ⁨1⁩ ⁨year⁩ ago
A simple one: Let's say you want to sum the numbers from 1 to 100. You could make the sum normally (1+2+3...) or you can rearrange the numbers in pairs: 1+100, 2+99, 3+98.... until 50+51 (50 pairs). So you will have 50 pairs and all of them sum 101 -> 101*50= 5050. There's a story who says that this method was discovered by Gauss when he was still a child in elementary school and their teacher asked them to sum the numbers.

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brainandforce@kbin.social ⁨1⁩ ⁨year⁩ ago
This is a common one, but the cardinality of infinite sets. Some infinities are larger than others.

The natural numbers are countably infinite, and any set that has a one-to-one mapping to the natural numbers is also countably infinite. So that means the set of all even natural numbers is the same size as the natural numbers, because we can map 0 > 0, 1 > 2, 2 > 4, 3 > 6, etc.

But that suggests we can also map a set that seems larger than the natural numbers to the natural numbers, such as the integers: 0 → 0, 1 → 1, 2 → –1, 3 → 2, 4 → –2, etc. In fact, we can even map pairs of integers to natural numbers, and because rational numbers can be represented in terms of pairs of numbers, their cardinality is that of the natural numbers. Even though the cardinality of the rationals is identical to that of the integers, the rationals are still dense, which means that between any two rational numbers we can find another one. The integers do not have this property.

But if we try to do this with real numbers, even a limited subset such as the real numbers between 0 and 1, it is impossible to perform this mapping. If you attempted to enumerate all of the real numbers between 0 and 1 as infinitely long decimals, you could always construct a number that was not present in the original enumeration by going through each element in order and appending a digit that did not match a decimal digit in the referenced element. This is Cantor's diagonal argument, which implies that the cardinality of the real numbers is strictly greater than that of the rationals.

The best part of this is that it is possible to construct a set that has the same cardinality as the real numbers but is not dense, such as the Cantor set.

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ArchmageAzor@lemmy.world ⁨1⁩ ⁨year⁩ ago
I find the logistic map to be fascinating. The logistic map is a simple mathematical equation that surprisingly appears everywhere in nature and social systems. It is a beautiful representation of how complex behavior can emerge from a straightforward rule. Imagine a population of creatures with limited resources that reproduce and compete for those resources. The logistic map describes how the population size changes over time as a function of its current size, and it reveals fascinating patterns. When the population is small, it grows rapidly due to ample resources. However, as it approaches a critical point, the growth slows, and competition intensifies, leading to an eventual stable population. This concept echoes in various real-world scenarios, from describing the spread of epidemics to predicting traffic jams and even modeling economic behaviors. It’s used by computers to generate random numbers, because a computer can’t actually generate truly random numbers. Veritasium did a good video on it: www.youtube.com/watch?v=ovJcsL7vyrk

I find it fascinating how it permeates nature in so many places. It’s a universal constant, but one we can’t easily observe.

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Cobrachickenwing@lemmy.ca ⁨1⁩ ⁨year⁩ ago
How Gauss was able to solve 1+2+3…+99+100 in the span of minutes. It really shows you can solve math problems by thinking in different ways and approaches.

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Squids@sopuli.xyz ⁨1⁩ ⁨year⁩ ago
Here’s a fun one - you know the concept of regular polyhedra/platonic solids right? 3d shapes where every edge, angle, and face is the same? How many of them are there?

Did you guess 48?

There’s way more regular solids out there than the bog standard set of DnD dice! Some of them are easy to understand, like the Kepler-poisont solids which basically use a pentagramme in various orientations for the face shape (hey the rules don’t say the edges can’t intersect!) To uh…This thing. And more! This video is a fun breakdown (both mathematically and mentally) of all of them.

Unfortunately they only add like 4 new potential dice to your collection and all of them are very painful.

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ribakau@lemmy.world ⁨1⁩ ⁨year⁩ ago
Fermat’s Last Theorem

x^n + y^n = z^n has no solutions where n > 2 and x, y and z are all natural numbers. It’s hard to believe that, knowing that it has an infinite number of solutions where n = 2.

Pierre de Format, after whom this theorem was named, famously claimed to have had a proof by leaving the following remark in some book that he owned: “I have a proof of this theorem, but there is not enough space in this margin”. It took mathematicians several hundred years to actually find the proof.

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BitSound@lemmy.world ⁨1⁩ ⁨year⁩ ago
Not so much a fact, but I’ve always liked the prime spirals: en.wikipedia.org/wiki/Ulam_spiral

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Rhynoplaz@lemmy.world ⁨1⁩ ⁨year⁩ ago
I heard that Pythagoras killed a man on a fishing trip because he solved a problem first.

That’s a pretty wild math tale!

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