# Big math/science results within easy reach

OP

#### jsilvela

##### Senior Member
Forum Donor
Oh that is some dangerous stuff you got over there. That can end friendships! In fact it can even be a anti example to the thread, in that, it is very easy to contruct the problem and very hard to grasp why it works the way it does for a human brain that has evolved to survive in equatorial grasslands.
Speaking of anti-examples for this thread, my favorite obvious result that is hard to prove is the Jordan Curve Theorem.
(BTW I have not yet seen the proof.)

And, like the "Einstein tile" in page 1, this also is an example of false advertisement. The theorem has nothing to do with basketball.

#### linear diver

##### Member

Here is one from 'Baby' Rudin that shows that there are gaps in the rational number system.

#### pseudoid

##### Major Contributor
Forum Donor
I don't know if it is exactly counter-intuitive, but using the example of a deck of playing cards and the factorial function can be fun. The uniqueness of a random shuffle, for example, and the likelihood that any two decks of cards have ever been shuffled the same way, or ever will be. 68 digit numbers can be hard to get your head around.
You are forcing me to go OT...
We enjoy watching the "Fool Us" show with Penn&[not]Teller.
They got FU'd recently by an algorithmic-machine having to guess two random cards that were pulled from an unshuffled deck.
Both cards were guessed correctly. Them two supposed to be 'masters' of previous but similar card-theory exploits.
I hate that show, I have never figured a single one of their contestants' magic acts.

OP

#### jsilvela

##### Senior Member
Forum Donor

Here is one from 'Baby' Rudin that shows that there are gaps in the rational number system.

OK, now that I've read in detail: I think the gap there is the irrationality of square-root of 2.
I.e. here Rudin is constructing the Dedekind cut for square-root(2), right?
But you could construct similar sets C and D, leaving, say, 1 in the middle, even if it's not necessary to define 1 via Dedekind cuts.

I've only read Rudin in sections. I admire it, but have never bought it.

#### holla

##### Member

Damn you, now I am thinking about the Ackermann function again and I won't sleep for another few nights.

#### holla

##### Member
Speaking of anti-examples for this thread, my favorite obvious result that is hard to prove is the Jordan Curve Theorem.

Not so hard to prove if you are a Redditor:

#### tomtoo

##### Major Contributor
I don't know if it is exactly counter-intuitive, but using the example of a deck of playing cards and the factorial function can be fun. The uniqueness of a random shuffle, for example, and the likelihood that any two decks of cards have ever been shuffled the same way, or ever will be. 68 digit numbers can be hard to get your head around.

Just add a digit then you are at 69. Thats much more easy to remember. And hey, who cares about that 1 digit? We learned at least experienced mechanical engineers not.

OP

#### jsilvela

##### Senior Member
Forum Donor
Not so hard to prove if you are a Redditor:
I love this!
They even wrote it in LaTeX using the Computer Modern font for academic looks. 100 points, would recommend.

EDIT: in true ASR spirit I want to point out that they got the curly quotes around "theorem" wrong. If you're not going to distinguish left curly from right curly double quotes, that's just shoddy. Tssk

#### BDWoody

##### Chief Cat Herder
Moderator
Forum Donor
Just add a digit then you are at 69. Thats much more easy to remember. And hey, who cares about that 1 digit? We learned at least experienced mechanical engineers not.

I suppose getting my head around 69 might be more intuitive at that.

What's an order of magnitude among friends?

#### tomtoo

##### Major Contributor
I suppose getting my head around 69 might be more intuitive at that.

What's an order of magnitude among friends?

Magnitutes get overestimated. Imagine you are 10m from a nuclear exploding bomb. Do you think it would make a difference for you if its a 10kt or 100kt device?

#### Hatto

##### Active Member
Magnitutes get overestimated. Imagine you are 10m from a nuclear exploding bomb. Do you think it would make a difference for you if its a 10kt or 100kt device?
Textbook definition of "overkill".
Literally too...

Forum Donor

#### Hatto

##### Active Member
HAH!

And also, that looks like a nice book.

#### Hatto

##### Active Member
Not so hard to prove if you are a Redditor:
There's also proof by MS Paint.

#### NTK

##### Major Contributor
Forum Donor
HAH!

And also, that looks like a nice book.
Free too.

OP

#### jsilvela

##### Senior Member
Forum Donor
So, Galois Theory is certainly not "within easy reach", but the core insight that you can study objects by associating to them algebraic structures that somehow keep a simplified essence is very well explained in this video. I'm really liking this channel, Aleph 0. He has a gift for exposition.

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