At a conference this week, Marty Ross talked to us about talking about maths –  a meta-talk. But one thing that he brought up was that almost all high school students refused to believe that 1=0.9999… In fact, it’s even worse than this though; the teachers didn’t really believe it themselves. So today’s post will be my attempt to convince you that it is indeed true, several different ways. After all, you don’t really understand anything unless you can explain it three different ways.

You’re happy to believe that 0.1, 1/10 and 10% are the same thing, aren’t you? Is there any difference between an apple, ein Apfel or une pomme?

What’s in a name? That which we call a rose
By any other name would smell as sweet

I didn’t think I’d ever see the day I’d be quoting Shakespeare, but Juliet had a point.

So let’s convince you (or your audience/students) that 0.999…, is indeed another way of writing the number 1. Since writing 0.999… all over the place isn’t very aesthetically pleasing, let’s give it a name! Throughout the rest of this post, we will call 0.999… the number “Batman”. We will prove that 1=Batman.

1. If you are happy to believe that one third (1/3) is equal to 0.3333…, then surely this is the easiest way to see that 1=Batman.
if 1/3=0.333… then you can simply multiply both sides of this equation by 3 to see that three thirds equals one which equals three times 0.333…, which is Batman. Easy done.

2. This is the standard proof. Notice that 10 times Batman is 9.999… Since there are still infinitely many 9s after the decimal point, this is also 9 + Batman. And at that point it just comes down to some very simple algebra. We have 10 times a number is equal to 9 plus that same number; the only possible solution to this is the number 1.

$10X=9+X$
$10X-X=9$
$9X=9$
$X=1$

That is, Batman is equal to 1.

3. If 1 was NOT equal to Batman then there must be some number between them, right? After all, we can always find the average of two numbers.
If you are yet to be convinced, I challenge you to tell me what the average of 0.999… and 1 is.