This classic clip “proves” how 25/5 = 14, and does it three different ways. Maths is a powerful method for providing proof – but we need to be careful that each step is based on correct assumptions.

One of the most well known fake proofs is as follows:

let a = b

Then a^{2} = ab

a^{2} – b^{2} = ab – b^{2}

(a-b)(a+b) = b(a-b)

a+b = b (divide by a-b )

b+b = b (as a = b)

2b = b

2 = 1

Can you spot the step that causes the proof to be incorrect?

Another well known maths problem that appears to prove the impossible is the following:

This was created by magician Paul Curry – and is called Curry’s Paradox. You can work out the areas of all the 4 different coloured shapes on both triangles, and yet by simply rearranging them you created a different area.

A third “proof” shows that -1 = 1:

Let a = b = -1

a^{2} = b^{2}

2a^{2} = 2b^{2}

a^{2} = 2b^{2} – a^{2}

a = √(2b^{2} – a^{2})

a = √(2(-1)^{2} – (-1)^{2})

a = √(1)

-1 = 1

And finally a proof that 1= 0. This last proof was used by Italian mathematician Guido Ubaldus as an example of a proof of God because it showed how something could appear from nothing.

0 = 0 + 0 + 0 + 0 ……

0 = (1-1) + (1-1) + (1-1) + (1-1) ……

0 = 1-1+1-1+1….

0 = 1 + (-1+1 ) + (-1+1) + ….

0 = 1

So, maths is a powerful tool for convincing people of an argument – but you always need to make sure that the maths is accurate! If you want to see the problems in the above proofs, highlight below (explanation in white text):

1) We divide by (a-b) in the 5th line. As a = b, then (a-b) = 0. We can’t divide by zero!

2) Neither of the “triangles” are in fact triangles – the hypotenuse is not actually straight. This discrepancy allows for the apparent paradox.

3) In the second to last line we square root 1, but this has 2 possible answers, 1 or -1. As a is already defined as a = -1 then there is no contradiction.

4) This is very similar to the Cesaro Summation problem which exercised mathematicians for centuries. The infinite summation of 0 + 0 + 0 + 0 … is not the same as the infinite summation 1 – 1 + 1 – 1 + 1 ….

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December 1, 2013 at 10:15 am

tPenguinLTGReblogged this on `The Penguin' says… and commented:

I always loved doing tricks like these on my friends!