Schanuel's conjecture Puns

tl;dr I wanted to try something different. This is an introduction to Schanuel's conjecture, explaining its significance and hinting at related topics. There are some exercises, and theorems are stated, but not proved. Prerequisites: first-year university math, linear algebra, and preferably introductory abstract algebra. Those with Galois Theory under their belt should find everything simple and intuitive.

I. Transcendental Numbers

A complex number is called algebraic if it is the solution to some polynomial equation with rational coefficients. For example, √2 is an algebraic number, since it's a complex number and it solves the polynomial equation x^2 - 2 = 0. The number 2×3^(1/3) is also algebraic: it is a solution of the polynomial equation x^3 - 24 = 0.

> Exercise: Prove that every rational number is algebraic.

> Exercise: Explain why the golden ratio is algebraic.

At this point, you might wonder: are there any numbers that are not algebraic?! Euler wondered about the same thing in the 1700s. He called non-algebraic numbers transcendental, and suspected that e and π were both transcendental numbers. It was a good fifty years after Euler's death that Liouville managed to prove the existence of transcendental numbers for the first time. Liouville used a technique called Diophantine Approximation to prove that the number L=∑*k∈ℕ* 10^(-k!) is, in a certain technical sense, easier to approximate using rationals than any irrational algebraic number, and so L cannot be algebraic.

Liouville's proof required a fairly deep understanding of diophantine approximation of algebraic numbers. These days, we have a much quicker way of constructing transcendental numbers, using Cantor's diagonal argument. Since the algebraic numbers form a countable set, we can make a list of all the real algebraic numbers, and diagonalization will produce a real number absent from our list. That number cannot be algebraic, so it will have to be transcendental.

> Exercise: Prove that the real algebraic numbers are countable.

Thanks to Cantor, first year undergraduates can answer questions that baffled the great Euler! Thus advances mathematics.

However, it's very hard to prove the transcendence of numbers that were not specifically constructed for the purpose of proving them transcendental. Euler's suspicions about e and π being transcendental were confirmed only in the late 19th century. The transcendence of the following numbers is still unknown: π+e, π−e

I don't want to step on anybody's toes here, but the amount of non-dad jokes here in this subreddit really annoys me. First of all, dad jokes CAN be NSFW, it clearly says so in the sub rules. Secondly, it doesn't automatically make it a dad joke if it's from a conversation between you and your child. Most importantly, the jokes that your CHILDREN tell YOU are not dad jokes. The point of a dad joke is that it's so cheesy only a dad who's trying to be funny would make such a joke. That's it. They are stupid plays on words, lame puns and so on. There has to be a clever pun or wordplay for it to be considered a dad joke.

Again, to all the fellow dads, I apologise if I'm sounding too harsh. But I just needed to get it off my chest.

Do your worst!

I'm surprised it hasn't decade.

For context I'm a Refuse Driver (Garbage man) & today I was on food waste. After I'd tipped I was checking the wagon for any defects when I spotted a lone pea balanced on the lifts.

I said "hey look, an escaPEA"

No one near me but it didn't half make me laugh for a good hour or so!

Edit: I can't believe how much this has blown up. Thank you everyone I've had a blast reading through the replies 😂

It really does, I swear!

They’re on standbi

Buenosdillas

Pilot on me!!

Dad jokes are supposed to be jokes you can tell a kid and they will understand it and find it funny.

This sub is mostly just NSFW puns now.

If it needs a NSFW tag it's not a dad joke. There should just be a NSFW puns subreddit for that.

Edit* I'm not replying any longer and turning off notifications but to all those that say "no one cares", there sure are a lot of you arguing about it. Maybe I'm wrong but you people don't need to be rude about it. If you really don't care, don't comment.

When I got home, they were still there.

What did 0 say to 8 ?

" Nice Belt "

So What did 3 say to 8 ?

" Hey, you two stop making out "

Nothing, he was gladiator.

I won't be doing that today!

[Removed]

You take away their little brooms

This morning, my 4 year old daughter.

Daughter: I'm hungry

Me: nerves building, smile widening

Me: Hi hungry, I'm dad.

She had no idea what was going on but I finally did it.

Thank you all for listening.

There hasn't been a post all year!

Why

It’s pronounced “Noel.”

After all his first name is No-vac

What, then, is Chinese rap?

Edit:

Notable mentions from the comments:

• Spanish/Swedish/Swiss/Serbian hits

• French/Finnish art

• Country/Canadian rap

• Chinese/Country/Canadian rock

• Turkish/Tunisian/Taiwanese rap

There hasn't been a single post this year!

(Happy 2022 from New Zealand)

Nothing, it just waved

Him: I can explain everything!

(It's his best joke yet I think)

Bob

So that I could frequently say, "I am going to walk 5 miles now."

Edit: My most popular post on Reddit! 😀 Thank you for the awards.

Just to clarify, 12345678

Me grabbing a soda from my (what I thought was) half full 12pk...

Notices there's only 2;

Me: "Awe man... This is a damn bird box!" Her: "What the hell does that mean?!" Me: (Pulls both cans out & shows them to her) "It's only got Toucans."

I'm not ashamed to admit the look on her face was glorious.

I was just sitting there doing nothing.

“BOOM”?!