What's your favourite proof in maths?
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ThePerfectScore
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#1
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.
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.

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the bear
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#2
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
http://www.math.utah.edu/~pa/math/q2.html
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ThePerfectScore
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#3
(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'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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the bear
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#4
Noble.
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#5
- Fundamental theorem of algebra via Liouville's theorem
- Sub-additivity of the Lebesgue measure by constructing pairwise disjoint sets
- Proving any open subset of
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.
- Sub-additivity of the Lebesgue measure by constructing pairwise disjoint sets
- Proving any open subset of

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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#6
Proof that
is irrational when
, but
Gotta love that infinite descent 
And also the proof that
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And also the proof that

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FireGarden
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#7
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#7
I have a fair few proofs I'm very fond of, but here's a real nice one:
Theorem: There exist two positive irrationals
such that
is rational.
Proof: If
is rational, take
.
If
is irrational, take
.
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
Theorem: There exist two positive irrationals


Proof: If


If


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

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Hodor
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#8
The proof that the real numbers are uncountable, using Baire's theorem:
Write
. For each
,
is closed in
and has empty interior, so, since
is a complete metric space, Baire says that the union cannot be countable.
Write





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JPL9457
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#10

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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#11
(Original post by majmuh24)
Division by zero
This may have worked somewhere else, but this is the maths thread where people actually know about maths
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Division by zero

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

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thewagwag
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#12
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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username1258398
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#13
(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
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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JPL9457
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(Original post by majmuh24)
Division by zero
This may have worked somewhere else, but this is the maths thread where people actually know about maths
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Division by zero

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

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Autistic Merit
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#16
(Original post by You Failed)
I'm pretty sure he was taking the ****, hence the massive troll face.
I'm pretty sure he was taking the ****, hence the massive troll face.

(Original post by JPL9457)
erm i'll have you know i got a 5 in my maths SAT, i am a very capable mathematician
erm i'll have you know i got a 5 in my maths SAT, i am a very capable mathematician


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The Polymath
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#18
(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.
I second this!


His continuum hypothesis is pretty interesting as well

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JPL9457
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#19
(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.
Not going to lie, I graduated with a first in maths but have no idea how I would go about contradicting this proof.
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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#20
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