No Stupid Questions

11 X 11 = 121

111 X 111 = 12321

1111 X 1111 = 1234321

11111 X 11111 = 123454321

111111 X 1111111 = 12345654321

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.

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'.

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.

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.

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.

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.

There are more infinite real numbers between 0 and 1 than whole numbers.

https://en.wikipedia.org/wiki/Countable_set

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.

The Monty hall problem makes me irrationally angry.

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.

The Julia and Mandelbrot sets always get me. That such a complex structure could arise from such simple rules. Here's a brilliant explanation I found years back: https://www.karlsims.com/julia.html

Integrals. I can have an area function, integrate it, and then have a volume.

And if you look at it from the Rieman sum angle, you are pretty much adding up an infinite amount of tiny volumes (the area * width of slice) to get the full volume.

Quickly a game of chess becomes a never ever played game of chess before.

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."

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.

One thing that definitely feels like "magic" is Monstrous Moonshine (https://en.wikipedia.org/wiki/Monstrous_moonshine) and stuff related to the j-invariant e.g. the fact that exp(pi*sqrt(163)) is so close to an integer (https://en.wikipedia.org/wiki/Heegner_number#Almost_integers_and_Ramanujan.27s_constant). I hardly understand it at all but it seems mind-blowing to me, almost in a suspicious way.

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.

Told ya. That's why i don't like math: the syntax is awkward.

e^(pi i) = -1

like, what?

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

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

Maybe a bit advanced for this crowd, but there is a correspondence between logic and type theory (like in programming languages). Roughly we have

Proposition ≈ Type

Proof of a prop ≈ member of a Type

Implication ≈ function type

and ≈ Cartesian product

or ≈ disjoint union

true ≈ type with one element

false ≈ empty type

Once you understand it, its actually really simple and "obvious", but the fact that this exists is really really surprising imo.

https://en.m.wikipedia.org/wiki/Curry%E2%80%93Howard_correspondence

You can also add topology into the mix:

https://en.m.wikipedia.org/wiki/Homotopy_type_theory

As someone who took maths in university for two years, this has successfully given me PTSD, well done Lemmy.

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.