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:

  • timeisart@lemmy.world
    46·
    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

    • nUbee@lemmy.world
      10·
      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.

  • Nfamwap@lemmy.world
    41·
    1 year ago

    11 X 11 = 121

    111 X 111 = 12321

    1111 X 1111 = 1234321

    11111 X 11111 = 123454321

    111111 X 1111111 = 12345654321

  • Gogo Sempai@programming.dev
    30·
    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.

  • naura@kbin.social
    28·
    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.

  • backseat@lemmy.world
    27·
    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

    • Thoth19@lemmy.world
      7·
      1 year ago

      Does this really hold for higher values? It seems like a pretty good way of searching for primes esp when combined with other approaches.

      • metiulekm@sh.itjust.worksEnglish
        5·
        1 year ago

        Every prime larger than 3 is either of form 6k+1, or 6k+5; the other four possibilities are either divisible by 2 or by 3 (or by both). Now (6k+1)² − 1 = 6k(6k+2) = 12k(3k+1) and at least one of k and 3k+1 must be even. Also (6k+5)² − 1 = (6k+4)(6k+6) = 12(3k+2)(k+1) and at least one of 3k+2 and k+1 must be even.

    • manishmehra@lemmy.ml
      11·
      1 year ago

      I don’t get it,

      5² = 25 != (2 * 24) + 1

      11² = 121 != (2 * 24) + 1

      Could you please help me understand, thanks!

  • metiulekm@sh.itjust.worksEnglish
    26·
    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: https://math.stackexchange.com/a/18347.

  • parrottail@sh.itjust.worksEnglish
    23·
    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’.

    • Reliant1087@lemmy.world
      51·
      1 year ago

      Incompleteness doesn’t come as a huge surprise when your learn math in an axiomatic way rather than computationally. For me the treacherous part is actually knowing whether something is unprovable because of incompleteness or because no one has found a proof yet.

  • Evirisu@kbin.social
    18·
    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 their students to sum the numbers.

  • jonhanson@lemmy.ml
    18·
    1 year 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
    16·
    1 year 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.

    • Urist@lemmy.ml
      12·
      1 year ago

      Also honorable mentions to the hairy ball theorem for giving us the much needed information that there is always a point on the earth where the wind is not blowing.

  • I Cast Fist@programming.dev
    13·
    1 year ago

    To me, personally, it has to be bezier curves. They’re not one of those things that only real mathematicians can understand, and that’s exactly why I’m fascinated by them. You don’t need to understand the equations happening to make use of them, since they make a lot of sense visually. The cherry on top is their real world usefulness in computer graphics.

  • Cobrachickenwing@lemmy.ca
    163·
    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.