Favourite Math Theorem.

Pre-College Topics High School Life
21

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<p>omg, I can’t believe I’ve never even heard of this before</p>

<p>n p = {1+2(n-1) for all n of positive integers}
phi(n) = 1p+2p+3p+…np
phi(n) = 1 + 1 + 2 + 1 + 4 + … + 1 + 2(n-1)
= n + 2(1+2+…n-1)</p>

<p>f(n) = 1 + 2 + … n-1
2f(n) = n-1+1 + n-2+2 + …
2f(n) = n(n-1)
phi(n) = n + 2f(n) = n + n(n-1) = n^2</p>

22

<p>Really? It seems pretty easy to figure it out. I did when I was bored in Algebra I. I actually wrote one of my college essays about realizing that since (n+1)^2 - n^2 = 2n + 1 then the difference between each square is the sum of the two roots (which is always odd) and you get the sequence of odd numbers.</p>

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<p>To prove it, it’s just series. Find the convergence of the series similar to how Gauss does it.
That is,
<a href=“http://alt2.artofproblemsolving.com/Forum/latexrender/pictures/b/8/6/b862c799410542f36700c577b6e353af07fa7d3e.gif[/url]”>http://alt2.artofproblemsolving.com/Forum/latexrender/pictures/b/8/6/b862c799410542f36700c577b6e353af07fa7d3e.gif&lt;/a&gt;
It’s a cool “theorem” that I found after doing some homework in complex analysis. Granted, it’s much easier than complex analysis though.</p>

24

<p>And what does the sum of the first n even numbers equal?</p>

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<p>I don’t remember it off the top of my head, but I think it’s [n(n+1)]/2 or something like that. Or maybe there’s no 1/2. I think there’s no 1/2.</p>

<p>edit: yeah, theres’ definitely no 1/2. it’s simply the sum of the first n odd integers + 1 to each odd integer. By that, I mean n^2 + n, which is also n(n+1).</p>

26

<p>Well the sum of the first n integers is [n(n+1)]/2.</p>

27

<p>Fundamental Theorem of Calculus. FTC FTW!</p>

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<p>Central Limit Theorem… aka the only one I actually understand.</p>

29

<p>idgaf theorem</p>

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<p>The law of equivalent exchange…</p>

31

<p>It isn’t exactly a “theorem”, but Goldbach’s conjecture is just awesome.</p>

<p>I will second the recommendation of the Central Limit Theorem. Statistics is actually wonderfully rigorous if you take the time to examine the mathematical underpinnings. There’s a good discussion of the CLT at [Central</a> Limit Theorem – from Wolfram MathWorld](<a href=“http://mathworld.wolfram.com/CentralLimitTheorem.html]Central”>Central Limit Theorem -- from Wolfram MathWorld)</p>

32

<p>LOL, the Central limit theorem was introduced in class today. What a coinky-dink</p>

33

<p>lagrange remainder theorem for me here
being able to estimate the error bound for taylor series is AWSOME XDDD
besides, if it wasnt for this theorem, all our buildings today would be collapsing and stuff since the engineers may not have used enough orders for their taylor series estimation.</p>

34

<p>(e^ix) = Cos(x) + iSin(x)</p>

<p>It’s just pretty…</p>

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<p>bahh CC is the only website which can make me feel like a non-nerd. Hahha my fav theorem would be something totally middle school like:</p>

<p>(a + b)^2= a^2 + 2ab + b^2</p>

<p>or a^2 - b^2 = (a + b)(a - b)</p>

36

<p>^^And it makes sense! <3 <3
I prefer it when you substitute x=Pi, but that is not a theorem.</p>

37

<p>^ No but it is a gorgeous identity…</p>

<p>The “fundemental theorems” are also very very pretty, especially FT of Arithmetic and FT of Linear Algebra. But still nothing holds a candle to the classic: e^(ix) = Cos(x) + iSin(x)</p>

38

<p>The sum of the cube of the first n consecutive positive integers will always be a perfect square, try to prove this assertion.</p>

39

<p>Cauchy Shwarz’s Inequality is pretty cool</p>

<p>So is Chicken McNugget Theorem</p>

40

<p>

[quote]
bahh CC is the only website which can make me feel like a non-nerd. Hahha my fav theorem would be something totally middle school like:</p>

<p>(a + b)^2= a^2 + 2ab + b^2</p>

<p>or a^2 - b^2 = (a + b)(a - b) [\quote]</p>

<p>man, the good days of 6th grade math. So much fun.</p>