What's your favourite proof in maths?

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

I so hope you're joking here...

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

The issue is step 8. You are right that both sides equal zero, but there's nothing wrong with any of the equations up until the division of both sides by

a^2 - ab

i.e. the division of 0 (since a=b,

a^2 - ab = a^2 - a^2 = 0

)

Essentially the proof is saying that

0 \times a = 0 \times b \ \Rightarrow \ a = b

which is obviously nonsense since the equation holds for any finite

a,b

Reply 21

thewagwag

6

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.

he divided by 0. a^2 - ab = 0

Reply 22

EconFan_73

3

Archimedes proof that the area of a circle equals πr² (or at least that it doesn't equal anything else :P) is absolute genius.

Reply 23

interstitial

18

Original post by EconFan_73

Archimedes proof that the area of a circle equals πr² (or at least that it doesn't equal anything else :P) is absolute genius.

I prefer the integration proof of this, just another example of one of the many uses of calculus :biggrin:

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Reply 24

Juichiro

18

Original post by the bear

indeed. he has brought pleasure to untold millions of school children.

What a perv! :mad:

Reply 25

EconFan_73

3

Original post by majmuh24

I prefer the integration proof of this, just another example of one of the many uses of calculus :biggrin:

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Noooo! Nowhere near as impressive! Archimides proved the previously undefined area of a circle geometrically using just a pencil, straight edge and compass, by comparing it to the area of regular polygons inscribed and circumscribed about the circle, in a time of very limited mathematical understanding! Yet you find plugging some numbers into a pre-existing formula is more interesting? :angry:

Reply 26

interstitial

18

Original post by EconFan_73

Noooo! Nowhere near as impressive! Archimides proved the previously undefined area of a circle geometrically using just a pencil, straight edge and compass, by comparing it to the area of regular polygons inscribed and circumscribed about the circle, in a time of very limited mathematical understanding! Yet you find plugging some numbers into a pre-existing formula is more interesting? :angry:

I find the whole concept of calculus fascinating, especially how an integral can be thought of as the limit of a sum of infinitely small values, which is why I find this way better :tongue:

It's not the plugging in numbers which interests me, it's the fact that this method works and can be applied to anything which I find interesting :biggrin:

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Reply 27

ThePerfectScore

OP

Original post by Noble.

I so hope you're joking here...

Or he just missed that the term is zero. Easy to do.

Reply 28

ThePerfectScore

OP

Original post by Hodor

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.

At what year do you do this?

Reply 29

Technetium

18

I reallt like the proof that

\sqrt2

is irrational, it was the first proof that I learned and the first proof that got me interested in Mathematics. :rolleyes:

Reply 30

user2020user

16

The infinitude of primes.

Original post by ThePerfectScore

...

Have you seen the proof of the sum of a geometric series?

(edited 10 years ago)

Reply 31

HelloWorld!

The proof that

[br]\cos nx+i\sin nx=\left(\cos x+i\sin x\right)^n=e^{nix}

Reply 32

user2020user

16

Original post by HelloWorld!

\underbrace{\cos nx+i\sin nx= \left(\cos x+i\sin x \right)^n}_{\alpha} = e^{nix}

From what I've learnt so far, the first equality

\alpha

can be proved for all integers via induction and the second is merely a consequence of adding the two Maclaurin expansions.

How would you go about proving

\alpha

for all real numbers?

Reply 33

ThePerfectScore

OP

Original post by Khallil

The infinitude of primes.

Have you seen the proof of the sum of a geometric series?

Yes, it uses a variation of the Gauss trick, doesn't it?

Reply 34

Hodor

3

Original post by ThePerfectScore

At what year do you do this?

What, Baire's theorem? I first met it in my second year, although this proof is from a third year functional analysis course.

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Reply 35

user2020user

16

Original post by ThePerfectScore

Yes, it uses a variation of the Gauss trick, doesn't it?

It's not really a variation. It's a different method entirely :smile:

There's no rearranging of terms, simply multiplication and subtraction!

\displaystyle \ \sum_{j=0}^{n-1} x^{j} = 1 + x + x^2 + \dots + x^{n-2} + x^{n-1} \\ \\ \implies \ x \sum_{j=0}^{n-1} x^{j} = x + x^2 + \dots + x^{n-1} + x^{n} \\ \\ \implies (x-1) \sum_{j=0}^{n-1} x^{j} = x^{n} - 1 \\ \\ \implies \sum_{j=0}^{n-1} x^{j} = \dfrac{x^n - 1}{x-1}

Reply 36

123formyabc

12

I do maths and further maths at a2 level and my head is hurting after reading all this

Reply 37

Chloe_P

1

Original post by 123formyabc

I do maths and further maths at a2 level and my head is hurting after reading all this

Aww diddums

Reply 38

Wesbian

1

Proof that 2^(1/3) is irrational.

Suppose not. Then there exist integers p and q such that 2^(1/3)=p/q, so p^3=q^3+q^3, which contradicts Fermat's Last Theorem

So elegant!

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