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
    4·
    11 months 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.

  • Artisian@lemmy.worldEnglish
    4·
    11 months 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 you pick a door, and you will get what’s behind the door. Before you open it, they open a different door than your choice and reveal 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.

    • Evirisu@kbin.social
      2·
      11 months 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.

    • Beto@lemmy.studio
      2·
      11 months 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·
      11 months 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.

  • FergleFFergleson@infosec.pub
    2·
    11 months 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.”

  • WtfEvenIsExistence3️@reddthat.comEnglish
    2·
    11 months ago

    Collatz conjecture or sometimes known as the 3x+1 problem.

    The question is basically: Does the Collatz sequence eventually reach 1 for all positive integer initial values?

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

    Basically:

    You choose any positive integer, then apply 3x+1 to the number if it’s odd, and divide by 2 if it’s even. The Collatz conjecture says all positive integers eventually becomes a 4 --> 2 --> 1 loop.

    So far, no person or machine has found a positive integer that doesn’t eventually results in the 4 --> 2 --> 1 loop. But we may never be able to prove the conjecture, since there could be a very large number that has a collatz sequence that doesn’t end in the 4-2-1 loop.

  • jonhanson@lemmy.ml
    1·
    11 months ago

    Euler’s identity is pretty amazing:

    e^iπ + 1 = 0

    To quote the Wikipedia page:

    Three of the basic arithmetic operations occur exactly once each: addition, multiplication, and exponentiation. The identity also links five fundamental mathematical constants:[6]

    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.

    The fact that an equation like that exists at the heart of maths - feels almost like it was left there deliberately.

  • Urist@lemmy.ml
    1·
    11 months ago

    Borsuk-Ulam is a great one! In essense it says that flattening a sphere into a disk will always make two antipodal points meet. This holds in arbitrary dimensions and leads to statements such as “there are two points along the equator on opposite sides of the earth with the same temperature”. Similarly one knows that there are two points on the opposite sides (antipodal) of the earth that both have the same temperature and pressure.

  • naura@kbin.social
    1·
    11 months 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.

  • Nfamwap@lemmy.world
    1·
    11 months ago

    11 X 11 = 121

    111 X 111 = 12321

    1111 X 1111 = 1234321

    11111 X 11111 = 123454321

    111111 X 1111111 = 12345654321

  • aggelalex@lemmy.world
    1·
    11 months ago

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

  • mookulator@lemmy.worldEnglish
    1·
    11 months 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.

    • cll7793@lemmy.worldOPEnglish
      1·
      11 months ago

      Thanks for the comment! It is cool and also pretty aesthetically pleasing!

  • timeisart@lemmy.world
    1·
    11 months 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