What concepts or facts do you know from math that is mind blowing, 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 mind blowing 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) > 101010101010101010101010101010 (10s are stacked on each other)
  • Σ(17) > Graham’s Number
  • Σ(27) If you can compute this function the Goldbach conjecture is false.
  • Σ(744) If you can compute this function the Riemann hypothesis is false.

Sources:

  • Gogo Sempai@programming.dev
    2·
    1 year ago

    Goldbach’s Conjecture: Every even natural number > 2 is a sum of 2 prime numbers. Eg: 8=5+3, 20=13+7.

    https://en.m.wikipedia.org/wiki/Goldbach’s_conjecture

    Such a simple construct right? Notice the word “conjecture”. The above has been verified till 4x10^18 numbers BUT no one has been able to prove it mathematically till date! It’s one of the best known unsolved problems in mathematics.

    • Beto@lemmy.studio
      2·
      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.

    • dQw4w9WgXcQ@lemm.ee
      1·
      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.

  • aggelalex@lemmy.world
    1·
    1 year ago

    The Fourier series. Musicians may not know about it, but everything music related, even harmony, boils down to this.

  • FergleFFergleson@infosec.pub
    1·
    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.”

  • timeisart@lemmy.world
    1·
    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

    Also here’s a cool video about some more mind blowing math facts

  • mookulator@lemmy.worldEnglish
    1·
    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.

  • Nfamwap@lemmy.world
    1·
    1 year ago

    11 X 11 = 121

    111 X 111 = 12321

    1111 X 1111 = 1234321

    11111 X 11111 = 123454321

    111111 X 1111111 = 12345654321

  • betheydocrime@lemmy.world
    0·
    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 in this thread it’s actually a handy trick that I use semifrequently

    • JonC@programming.devEnglish
      1·
      1 year ago

      That one’s actually really easy to prove numerically.

      Not going to type out a full proof here, but here’s an example.

      Let’s look at a two digit number for simplicity. You can write any two digit number as 10*a+b, where a and b are the first and second digits respectively.

      E.g. 72 is 10 * 7 + 2. And 10 is just 9+1, so in this case it becomes 72=(9 * 7)+7+2

      We know 9 * 7 is divisible by 3 as it’s just 3 * 3 * 7. Then if the number we add on (7 and 2) also sum to a multiple of 3, then we know the entire number is a multiple of 3.

      You can then extend that to larger numbers as 100 is 99+1 and 99 is divisible by 3, and so on.