Translation Sniper

I loved this little problem from math class.

0.999… (with the ellipses representing ‘repeats forever’) equals one.

The first time most people see this it doesn’t quite compute, since conceptually people think of 0.999… as ‘always getting closer and closer to 1 – but not quite reaching it – for infinity’. Wrapping your head around the idea that ‘infinity’ can equal a finite number can make your stomach feel a bit queasy.

The first proof I learned was straight algebra:

1) Let x = 0.999….
Multiple both sides by 10 (if you do the same thing to both sides of the equal side, the equation stays equal). So:

2) 10x = 9.999…

Now, let’s subtract equation 1) from equation 2). The left side is easy: 10x minus x is 9x. For the right side, we could look at it digit-by-digit. For the ones digit (the number just to the left of the decimal point), 9 minus 0 is clearly 9. For the tenth digit, .9 minus .9 is zero. Continuing, .99 minus .99 is 0, .999 minus .999 is zero, and we can continue into infinity – in other words, the entire right side of the decimal point in equation 1) is exactly the same in equation 2), which means that subtracting one from the other is 0. So we’re left with the left side of the decimal point, which we said above was 9.

So: 
   10x = 9.999…
-      x = 0.999…
————————–
     9x = 9

Divide both sides by 9, and we get x = 1.

And x = 0.999… = 1.

Cool, huh?

But my favorite proof was actually one I didn’t learn about until later, even though it’s much easier.

First - remember that 0.5 is the same thing as 1 divided by 2, or 1/2.  1 divided by 4 (1/4) is 0.25, etc. It turns out that one divided by nine (1/9) is equal to 0.111…

It’s clear that 2 times 1/2 is equal to 1. It’s obvious that 4 times 1/4 is equal to 1. So clearly, 9 times 1/9 must equal to 1.

Likewise, 9 times 0.111… must equal 0.999….

But 9 times 0.111… is the same thing as 9 times 1/9, and we saw above that 9 times 1/9 was equal to one. Ergo, 9 times 1/9 = 9 times 0.111… = 0.999… = 1.

Note that you could to this using thirds: 1/3 is equal to 0.333…. Three times 0.333… is equal to 0.999…, and we know that 3 times 1/3 is equal to 1, so 3 times 0.333… = 0.999… = 1.

Don’t forget to bring this up at your next cocktail party! You’ll be a hit!