# What's your favourite proof in maths?

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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.

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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#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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(Original post by

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

**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

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#4

(Original post by

Euclid was a clever man! Way ahead of his time!

**ThePerfectScore**)Euclid was a clever man! Way ahead of his time!

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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 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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#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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#7

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

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

*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

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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 . For each , is closed in and has empty interior, so, since is a complete metric space, Baire says that the union cannot be countable.

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#10

(Original post by

**JPL9457**)This may have worked somewhere else, but this is the maths thread where people actually know about maths

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#11

(Original post by

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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**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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#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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#13

(Original post by

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

**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

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#14

**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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#15

(Original post by

**JPL9457**)
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#16

(Original post by

I'm pretty sure he was taking the piss, hence the massive troll face.

**You Failed**)I'm pretty sure he was taking the piss, hence the massive troll face.

(Original post by

erm i'll have you know i got a 5 in my maths SAT, i am a very capable mathematician

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

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#18

(Original post by

I second this! Such an original, simple line of thought. Makes you wonder how in the banana he came up with it.

**StrangeBanana**)I second this! Such an original, simple line of thought. Makes you wonder how in the banana he came up with it.

His continuum hypothesis is pretty interesting as well

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#19

(Original post by

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

**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.

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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