Ricky wrote:pretty much as soon as you define 0 and 1, you define all of the real numbers.

wait... what? care to elaborate

I can certainly try.

Imagine we define what 0 and 1 are. These are the identity for addition and multiplication respectively. This simply means:

0 + a = a + 0 = a

and

1 * a = a * 1 = a

For all a in the "field" (a set with special properties). But from here on out, we won't call these numbers because it turns out that they don't have to be. The following will apply to anything with these properties (as well as a lot which haven't been mentioned). So we will call the additive identity (0) e, and the multiplicative identity (1) u.

Now we can define all the positive numbers, simply by adding up e's.

2 = e + e

3 = e + e + e

and so on...

Note that I use "2" and "3" here as symbols. All "2" stands for is the identity added with itself. "3" just means the identity added 3 times.

I now call all these things positive elements. I define negative numbers to be:

if a is positive, then -a is an element of F such that:

a + -a = e (0)

With numbers, this means 2 + -2 = 0. With a field (as the reals are defined), we are guaranteed that such elements exist. So now we have all the "integers".

Now we define elements 1/a to be such that:

a * 1/a = u

Again, in a field, we are guaranteed that such elements exist. We also define a/b to be:

a * 1/b

And now you'll notice that we get all the rationals. An offset of this is that u/e (1/0) does not exist, for various reasons.

Now we get to the reals and here is where things get complicated. The reals can be defined by sets of rational numbers, known as a Dedekind cut. However, it would take too much to explain how this works.

So hopefully it is a bit clear how we can define every single number with 0 and 1. I believe this can be done with 1 alone, but it requires a few little tricks, and I'm not sure if these tricks apply to all complete fields or not.

]]>bad notation, but you know what i mean

]]>As I suspected, first place went to e^iπ + 1 = 0. But he also gave some runner-ups (or runners-up?), which were quite interesting.

In no particular order,

1.

2.

3.

That last summation I particularly liked because it was so complicated. And for every term in the series you add on, you get around another 8 decimal places of π, so it converges incredibly quickly.

]]>pretty much as soon as you define 0 and 1, you define all of the real numbers.

wait... what? care to elaborate

]]>That a formula like that produces whole numbers is amazing enough, but **Fibonacci Numbers**?

is my favorite as well, it is just so beautiful, displaying 5 of the most important numbers in mathematics, addition, multiplication, exponentiation, and equivalence. Also it is quite simple, which is a main factor in it being well-known.

Other results I am particularly fond of are

and

(of course taking the second expression too literally will give rise to dispute; it is actually an asymptotic relation)

Also the fact that

which shows that i[sup]i[/sup] is in fact a real number, is quite astonishing.

I enjoy infinite sums as well, but usually none stick out significantly from the rest. Here is *Zhylliolom's identity* for φ (please submit this one, don't forget to mention me ):

(actually, I need to rederive my identity to make sure it is actually correct here, so wait on that!)

]]>I mean, it is a fairly interesting result, how you can combine 3 seemingly unrelated numbers, 2 of which are transcendental and one of which is imaginary, together and get such a nice result. But I also think it's possible that there's a better one out there.

Three? That equation contains the five most important numbers in math. 0 and 1 are two of the most important numbers, and as it turns out, pretty much as soon as you define 0 and 1, you define all of the real numbers.

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