0.9 recurring

1

For those readers who went missing sometime before their own GCSE in maths, 0.9 recurring means 0.99999 . . . with an infinite number of further 9s added on the end. Every time an extra 9 is added, you get closer to 1, but you never quite get there. So, it seemed clear to me that 0.9 recurring could not be equal to 1, though it would approach 1 in the limit of the recurring series.

In fact, though, it seems that most number theorists would argue that both numbers are exactly the same. There are many sophisticated proofs of this proposition, but you may prefer the following intuitive proof. One third, written as a decimal, is 0.3 recurring. Now multiply this by 3, and you get 0.9 recurring. But 3 times one third can only be equal to 1. Therefore 0.9 recurring must be equal to 1

Another way of proving this, suggested by my daughter's maths teacher, is this. Think of two numbers, x=0.9 recurring and y=1 Can you think of a number that is higher than x but less than y? No, you can't. You might be tempted to claim that 0.99 recurring is greater than 0.9 recurring, but it obviously is not - it is exactly the same. Since you cannot think of any number between x and y, the two numbers must be identical.

I think that one reason why people, including myself, find this proposition so hard to swallow is that human beings have a lot of trouble with the meaning of infinity. Infinity is obviously the largest number anyone can think of, but why can't you add a 1 to infinity to make it bigger? Because if you could, then the first number would not have been infinity in the first place.

Think about the difference between 1 and 0.9. That is easy, 0.1. Then take the difference between 1 and 0.99. Also easy, 0.01. Every time you add another 9 on the end of 0.99, you add another before the 1 in the number 0.01. So if you add an infinite series of 9s, you also add an infinite series of s, which means that the difference between the two numbers becomes infinitesimally small, which to a mathematician means zero.

Being somewhat chastened, I asked my daughter the following (courtesy of the writer Malcolm Gladwell). Imagine you have a very large piece of paper, and you fold it in half. Then you fold it in half again, and you continue folding it until you have done so 50 times. How thick will the resulting wadge of paper be? My daughter and her friends reckoned it would be about as thick as a telephone directory, which is a common answer. (Try your friends.) But Gladwell says that it would in fact take up the distance between earth and the sun, because if you repeatedly double something very small, you soon end up with something almost unimaginably large. That's called a geometric series. So there.