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 a2 = ab
a2 – b2 = ab – b2
(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:

curry's

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
a2 = b2
2a2 = 2b2
a2 = 2b2 – a2
a = √(2b2 – a2)
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 ….