### Brain Teasers

# Disappearing Numbers

For every number that exists between 0 and 1 (say for example 2/3) there exists exactly one other number called its reciprocal (in this case 3/2) that is bigger than 1.

Also, for every number that exists between 0 and 1 (say 0.5) there is exactly one number that exists, found by adding 1 (1.5) that must lie between 1 and 2.

Since every number between 0 and 1 corresponds to exactly one number between 1 and 2 and at the same time corresponds to a number bigger than 1 then there must be the same number of numbers that lie between 1 and 2 as there are bigger than 1.

That means that all of the numbers bigger than 1 lie between 1 and 2 so there aren't any numbers bigger than 2.

This is clearly false, so where is the error?

Also, for every number that exists between 0 and 1 (say 0.5) there is exactly one number that exists, found by adding 1 (1.5) that must lie between 1 and 2.

Since every number between 0 and 1 corresponds to exactly one number between 1 and 2 and at the same time corresponds to a number bigger than 1 then there must be the same number of numbers that lie between 1 and 2 as there are bigger than 1.

That means that all of the numbers bigger than 1 lie between 1 and 2 so there aren't any numbers bigger than 2.

This is clearly false, so where is the error?

### Hint

Reversing the reasoning will give you the same correspondence. Every number bigger than 1 has a reciprocal less than 1 and every number between 1 and 2 has a corresponding number less than 1 found by subtracting 1. No error here.### Answer

The problem lies in how many numbers there are. There are an infinite number of numbers between 0 and 1 and hence an infinite number of numbers between 1 and 2 and also greater than 2. Surprisingly there can be different sized infinities so you cannot say that two infinite quantities are equal. (Take 1 away from infinite and you still have infinite).Hide Hint Show Hint Hide Answer Show Answer

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