What's your favourite proof in maths?

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ThePerfectScore
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Inspired by the favourite number thread. Feel free to sketch up the details of the proof if you wish to. It doesn't have to be your favourite proof of a main theorem or result. It could be the proof or calculation of some simple equation, or even some cool method that you have seen somewhere (your book etc). For example, deriving the quadratic formula by completing the square. I'll start.

Gauss' derivation of the sum of first n natural numbers.

Let S be the sum. Then

S = 1+2+3+....+(n-1)+n ---------------- (1)

But writing it from the other side

S = n+(n-1)+(n-2)+...+2+1 -------- (2)

Adding (1) and (2) together

2S = n(n+1)

So S = [n(n+1)]/2.
:drool:

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the bear
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Euclid's proof of the infinitude of primes... it is brilliant, yet simple enough for a Higher Level GCSE child to understand.

http://www.math.utah.edu/~pa/math/q2.html
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ThePerfectScore
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(Original post by the bear)
Euclid's proof of the infinitude of primes... it is brilliant, yet simple enough for a Higher Level GCSE child to understand.

http://www.math.utah.edu/~pa/math/q2.html
Euclid was a clever man! Way ahead of his time! :eek:
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the bear
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(Original post by ThePerfectScore)
Euclid was a clever man! Way ahead of his time! :eek:
indeed. he has brought pleasure to untold millions of school children.
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Noble.
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- Fundamental theorem of algebra via Liouville's theorem
- Sub-additivity of the Lebesgue measure by constructing pairwise disjoint sets
- Proving any open subset of \mathbb{R} is the union of, at most, countably many disjoint, open intervals via defining an equivalence relation.

I don't really keep an ongoing list of favourite proofs, but the above are a few recent proofs I've seen that stand out as being particularly neat.
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interstitial
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Proof that \sqrt k is irrational when  \sqrt{k} \not\in \mathbb N , but k \in \mathbb N Gotta love that infinite descent :teehee:

And also the proof that \displaystyle\sum_{r=1}^\infty \dfrac{1}{r^2} = \dfrac{\pi ^2}{6}

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FireGarden
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I have a fair few proofs I'm very fond of, but here's a real nice one:

Theorem: There exist two positive irrationals x,y such that x^y is rational.

Proof: If \sqrt{2}^{\sqrt{2}} is rational, take x=y=\sqrt{2}.

If \sqrt{2}^{\sqrt{2}} is irrational, take x=\sqrt{2}^{\sqrt{2}},\ y=\sqrt{2}.


Another theorem, whose proof once came with advice from the lecturer: "You need to be able to write this proof in the snow with pee even at night when drunk"

There is no surjection :A\mapsto \mathcal{P}(A)
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Hodor
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The proof that the real numbers are uncountable, using Baire's theorem:


Write  \mathbb{R} = \bigcup\limits_{x \in \mathbb{R}} \{x\} . For each  x, \{x\} is closed in  \mathbb{R} and has empty interior, so, since  \mathbb{R} is a complete metric space, Baire says that the union cannot be countable.
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JPL9457
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interstitial
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(Original post by JPL9457)
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Division by zero :hand:

This may have worked somewhere else, but this is the maths thread where people actually know about maths

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You Failed
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(Original post by majmuh24)
Division by zero :hand:

This may have worked somewhere else, but this is the maths thread where people actually know about maths

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I'm pretty sure he was taking the piss, hence the massive troll face.
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thewagwag
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Cantor's diagonal argument. Proof that there are infinities of different sizes, by showing how there must be more real numbers than whole numbers, despite both sets being infinite


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StrangeBanana
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(Original post by thewagwag)
Cantor's diagonal argument. Proof that there are infinities of different sizes, by showing how there must be more real numbers than whole numbers, despite both sets being infinite
I second this! Such an original, simple line of thought. Makes you wonder how in the banana he came up with it.
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JPL9457
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(Original post by majmuh24)
Division by zero :hand:

This may have worked somewhere else, but this is the maths thread where people actually know about maths

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erm i'll have you know i got a 5 in my maths SAT, i am a very capable mathematician
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Autistic Merit
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(Original post by JPL9457)
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Not going to lie, I graduated with a first in maths but have no idea how I would go about contradicting this proof.
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interstitial
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(Original post by You Failed)
I'm pretty sure he was taking the piss, hence the massive troll face.
I know, I was joking

(Original post by JPL9457)
erm i'll have you know i got a 5 in my maths SAT, i am a very capable mathematician
:eek: I only got a level 4, does that make me stupid (I got a level 3 in Year 2 though )

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The Polymath
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(Original post by majmuh24)
Proof that \sqrt k is irrational when  k > 0 and  \sqrt{k} \not\in \mathbb N Gotta love that infinite descent :teehee:
What?

k=\frac{1}{4} > 0 \\\sqrt{k}=\frac{1}{2}\not\in \mathbb N
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interstitial
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(Original post by StrangeBanana)
I second this! Such an original, simple line of thought. Makes you wonder how in the banana he came up with it.
His theorem about power sets was also so simple, yet so obvious, it extended infinities beyond the simple 1,2,3,4.. and gave rise to uncountable infinities as well

His continuum hypothesis is pretty interesting as well

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JPL9457
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(Original post by Autistic Merit)
Not going to lie, I graduated with a first in maths but have no idea how I would go about contradicting this proof.
congrats, what uni?

is there a proper professional way to contradict a proof? because my way would just be saying, in step 5, the whole thing = 0
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interstitial
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(Original post by The Polymath)
What?

k=\frac{1}{4} > 0 \\\sqrt{k}=\frac{1}{2}\not\in \mathbb N
I meant to put when k is an integer I'll change it now

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