fuck yeah proofs!

Mar 10

Condensed Field Axioms

For F a field and a, b, c in F, we have

$\\1.\ (a+b)+c=a+(b+c),\quad (a\cdot b)\cdot c=a\cdot(b\cdot c)\\ 2.\ a\cdot b=b\cdot a\\ 3.\ a\cdot(b+c)=a\cdot b+a\cdot c\\ 4.\ 0+a=a=a+0,\quad 1\cdot a=a=a\cdot 1\\ 5.\ a+-a=0=-a+a,\quad a\cdot a^{-1}=1=a^{-1}\cdot a\ (a\neq 0)$ From these reduced axioms, it is possible to prove several facts commonly labeled as axioms. (See Wolfram for a more general picture.)

Theorem 1:

$0\cdot a= 0 =a\cdot 0$

Proof:

$\begin{align*} a&=0+a\\ a\cdot a &=a\cdot (0+a)\\ a\cdot a &= a\cdot0+a\cdot a\\ 0&=a\cdot 0\\ &=0\cdot a \end{align*}$

Theorem 2:

$(-1)\cdot a=-a=a\cdot(-1)$

(where -1 is the additive inverse of 1)

Proof:

$\begin{align*} 0&=1+-1\\ a\cdot 0&=a\cdot (1+-1)\\ 0&=a+a\cdot(-1)\\ a+-a&=a+a\cdot(-1)\\ -a&=a\cdot (-1)\\ &=(-1)\cdot a \end{align*}$

Theorem 3:

$(a+b)\cdot c= a\cdot c+b\cdot c$

Proof:

$\begin{align*} (a+b)\cdot c&=c\cdot (a+b)\\ &=c\cdot a+c\cdot b\\ &=a\cdot c+b\cdot c \end{align*}$

Theorem 4:

$a+b=b+a$

Proof:

$\begin{align*} (a+b)+-(a+b)&=0\\ (a+b)+(-1)\cdot(a+b)&=0\\ a+b+(-1)\cdot a+(-1)\cdot b&=0\\ a+b+-a+-b&=0\\ a+b+-a+-b+b&=b\\ a+b+-a+a&=b+a\\ a+b&=b+a\\ \\ 0&=-(a+b)+(a+b)\\ 0&=(-1)\cdot(a+b)+(a+b)\\ 0&=(-1)\cdot a+(-1)\cdot b+a+b\\ 0&=-a+-b+a+b\\ a&=a+-a+-b+a+b\\ b+a&=b+-b+a+b\\ b+a&=a+b \end{align*}$

(The above proofs don’t list the axioms when each is used. Online TeX editors dislike fancy math environments.)

Source: CJ Harries

Mar 8

Please?

If someone(s) out there on the internet is willing to be a semi-normal contributor, I’d love to start this back up again. I check it about once a week and start checking through my notes for a decent proof, but later analysis and statistics don’t have easy HTML proofs. I also don’t want to flood the site with the combinatorics proofs I really enjoy.

tl;dr if you’re an interested student, shoot me a message and start typing proofs.

-Edit

TO CONTRIBUTE: drop a message in the Ask box with your email and some ideas. Your email won’t be published, all that, and then the fun begins.

Oct 24

We have disappeared.

If you’re interesting in contributing, shoot me an email and/or message. I have absolutely no time to switch my proofs from TeX to HTML/TeX, so this thing is essentially dead.

Seriously. Proofs are awesome. Contribute. Please. Email me. This thing should not die. I mean, we were on the front page of the science directory when we started. Make that happen again.

—-Edit

And by contributing, I mean editing. As in doing quite a bit of the putting things on the internet.

integrallogic asked: What do you use to input math img's?

The images are created using an online LaTeX editor. If you don’t know TeX, learn it. It’s AWESOME. The benchmark for math papers.

isomorphismes asked: I'd love to see a proof of the Banach-Tarski paradox.

http://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox

I’M SO LAZY

Sep 14

We haven’t disappeared…

School has just started. I personally had to move this weekend, and if you’ve ever had to move while school is in session, you’ll understand me when I say I’m way behind.

That being said, things should be picking up soon. There might even be a new contributor or two.

Also, help keep us on the science page! We promise more will appear soon.

Aug 24

Theorem: If A, B, C are matrices defined over the field F such that the products BC and A(BC) are defined, then so are the products AB, (AB)C, and

Proof: Suppose B is an n × p matrix. Since BC is defined, C is a matrix with p rows, and BC has n rows. Because A(BC) is defined we may assume A is an m × n matrix. Thus the product AB exists and is an m × p matrix, from which it follows that the product (AB)C exists. To show that A(BC) = (AB)C means to show that

for each i, j. By definition,

Source: Hoffman, Kunze, Linear Algebra

Tumbler Tuesday and More…

It’s Tumblr Tuesday. Feel free to recommend us.
Don’t forget to submit good proofs or ask for some interesting ones.
As of right now, the Tumblr Needs a Math Category Petition has 100 online signatures. Feel free to add your e-vote.
I’m currently looking for another (several?) contributors. At the moment, the majority of my time is consumed with school and work. The last two weeks were the first I’ve actually been home for longer than 3 days since May. While I see several proofs a day I’d like to add, they all revolve around Linear Algebra or Real Analysis, or I never get the chance to type them up. Not only would I like to see more on here than a post a week, I’d like to add some diversity.

Theorem: For any positive integer n ≥ 28, the equation 8x + 5y = n has a positive integer solution for x and y.

Proof: Define S := {n ≥ 28 is a positive integer| 8x + 5y ≠ n; x, y positive integers}. Assume, by way of contradiction, that S ≠ ∅. By the Well-Ordering Principle, S has a least element, call it n. Then:
n ≠ 28 = 8 + 4 ⋅ 5;
n ≠ 29 = 3 ⋅ 8 + 5;
n ≠ 30 = 6 ⋅ 5;
n ≠ 31 = 2 ⋅ 8 + 3 ⋅ 5;
n ≠ 32 = 4 ⋅ 8;
n ≠ 33 = 8 + 5 ⋅ 5.
This implies n > 33, but n - 5 ∉ S, as n was, by hypothesis, the smallest element. Hence n - 5 yields positive integers for x and y.
8x + 5y = n - 5
8x + 5(y + 1) = n.
This presents a contradiction, as n ∈ S.

Therefore given any positive integer n ≥ 28, the equation 8x + 5y = n has a positive integer solution for x and y.

Source: Fred Galvin, class notes

Aug 17

http://www.petitiononline.com/tumblrma/petition.html

Tumblr needs a mathematics category. Just saying.

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