there is a correct answer, but don’t google it (yet). i want to know what you feel in your heart.

okay yeah sure

Sometimes the result is the constructions we met along the way

this fibonacci joke is as bad as the last two you heard combined

Define 0^0 to be the smallest natural number.

I don't think this is possible????

Hello Ryan I am here to help. So the first step is pretty easy: Three cheeseburgers are worth 18, so each one is worth 6. If these are dollars, that's a steal!

From the second equation we get that cheeseburger plus fries-squared is five. Subtracting cheeseburger, which is six, from both sides, we get that fries-squared is negative-one. Math fans will know that there are two solutions to this; either fries are the "imaginary unit" 𝒾 or they are its negative, -𝒾. We'll do the rest of the problem with 𝒾, keeping in mind that at the end we should also take the complex conjugates as solutions.

Finally, we have that cup to the power of fries, minus cup, equals three. Replacing fries with 𝒾, and moving a cup to the other side, we get that cup-to-the-𝒾 is equal to cup-plus-three.

Now, the weird part about this is the cup-to-the-i. The problem with this is that complex exponentiation is technically not a thing. That is to say, there is no one function which is mathematically equal to "input-to-the-power-of-𝒾". In fact, there are infinitely many such functions.

Fortunately, due to reasons that take about six pages to explain (trust me I've done it), there is one particular function that many people have agreed is "the most reasonable one". This is not a mathematical notion, but a human preference. Seeing as this question was presumably written by a human, I am comfortable with using this function.

So, what function is this? Well, given a complex number r∠θ written in polar form (if you don't know what that means don't worry), where -π < θ ≤ π, then (r∠θ)^𝒾 = e^(-θ)∠ln(r).

Applying this to our problem a value r∠θ will be a possible solution for cup if e^(-θ)∠ln(r) = r∠θ + 3. Splitting this into real and imaginary parts, we get two equations: e^(-θ) cos(ln(r)) = r cos(θ) + 3 and e^(-θ) sin(ln(r)) = r sin(θ). We can graph these equations on Desmos:

The possible values of cup are the intersections between the red, green, and purple. There are infinitely many of these which have an angle of around -π/3, and there are two weirdos: One which is a complex number very close to -2.98, and one which is somewhere around -25. The possible values for cup are all of these infinitely many solutions, and also all of their complex conjugates.

They were right, 99% of people can't solve it.

Bizzaro Right Triangles

Okay, we all know that, if you have a right triangle with sides a and b and hypotenuse c, that means a^2 + b^2 = c^2, right?

So you can have 3-4-5 right triangles, and 5-12-13, and 7-24-25, etc.

And technically, a 1-i-0 right triangle follows this pattern, too, which has made the rounds in the math meme community.

But I found something better and weirder. I found a whole family of better and weirder.

I'm gonna skip a bit of the beginning and start here:

That follows from Euler's formula; if you need a walkthrough, I can provide, but I want to start with the above. So then, it follows that:

Which means that an angle of (i*lnx) radians in a right triangle will have an adjacent side of (x^2 + 1) and a hypotenuse of 2x. By Pythagoras, then, the opposite side must be (x^2 - 1)*i.

But please note: if x is odd, then the lengths of all three sides will be even, and thus can be divided by 2.

Which means -- are you holding onto your hats and shoes?!? -- means that the proportions of the right triangle with angle i*ln(3) radians... are 5-4i-3.

And for i*ln(5) radians, they are 13-12i-5.

And so forth, with all the familiar Pythagorean triplets sqrt(2n+1)-n-(n+1) showing up, just with one side imaginary and the hypotenuse and remaining side swapped -- so, (n+1)-n*i-sqrt(2n+1). They still fulfill Pythagoras, every single one.

Which I think is, pardon my directness, fucking terrific. But just as a little bonus, please note that this means the triangle with angle i*ln(2) radians, the proportions are 5-3i-4, which is just delightful IMHO.

@desmos-calculator

i hate looking at the thermometer and seeing some fuckass number like 2

come the fuck on

Switch to freedom units and you'll automatically feel warmer! Just think of how you're going to feel seeing 34 instead of 0!

…girl

math wrapped

you solved 27183 equations you proved 0 of the millennium prize problems you used proof by induction 341 times you used proof by contradiction 177 times you used the pidgeon-hole principle 85 times you cried 70233 times while trying to prove the collatz conjecture you used your calculator for trivial sums that anybody should be able to do 23099 times you spent 443 hours trying to prove the collatz conjecture you had to google trigonometric identities 113 times you forgot to write the +c 56 times you said that dy/dx was a fraction 4 times you ripped 207 pages in half while trying to prove the collatz conjecture

not to be a number nerd on main but 2025 (45^2) will be the only square year most of us ever experience. the last one was 1936 and the next one will be 2116

It's also 1³ + 2³ + 3³ + 4³ + 5³ + 6³ + 7³ + 8³ + 9³.

And 45 is 1+2+3+4+5+6+7+8+9

I love when a textbook has a theorem that is just the most obviously true shit, like girl I sure hope so, if that weren't true I think math would fall apart!

I especially love the conjunction of "this problem is intuitively plain as day" and "it took mathematicians centuries to formally prove this." What do you mean it takes advanced topology to prove every closed curve has an inside and outside