Trevor Pythagoras

There are a number of apparent proof that 2=1. An example of one such proof is below

Suppose a=b then
a² = ab
2a²=a²+ab
2a²-2ab = a²+ab-2ab
2a²-2ab=a²-ab
2(a²-ab) = 1(a²-ab)
2 = 1

However this is clearly not true which mean there must be a step in the above algebra which isnt valid.

The invalid step

The invalid step is in fact the very last step in the proof, cancelling both sides by a²-ab. This is because a²-ab = 0 since we initially assumed a=b hence

ab = aa = a²

This then means that the second last line states

2 x 0 = 1 x 0

which is obviously true but you cant “cancel” both sides by 0 since any number divided by zero is meaningless and doesnt have a values. The trick involved in this proof is hiding the divide by zero which most people would recognise as invalid with a more complicated looking expression.

What this means

This example demonstrates the dangers of allowing divide by zeros to occur in your calculations even if its in the form of algebra. It also demonstrates the problems of considering infinty (x/0) as a number that can be used in calculations.