Figuring How To Get There From Here 

By Mike O’Hern, Center Director of Mathnasium of West Knoxville

Have you ever tried to wrap your head around something that just doesn’t seem possible, but it is?  I remember a couple of summers ago fighting with what’s referred to as “The Monty Hall Problem” until it seemed like I was going to go nuts.  Since I don’t want you to go nuts, I won’t go into it here. Maybe in the future, I’ll feel a little less magnanimous and tell you anyway.

I was recently asked about one of those things that just doesn’t seem right, but it really is.  Here we go.  First, you know that 0.9 is really close to 1, right?  But 0.99 is much closer.  So 0.999 is getting crushingly close, right?  But what about 0.99999999999?  Could you fit your feeler gauge between that number and 1?  Yes, if you can find a small enough feeler gauge.  But if we put enough nines in there, can we get to 1?  That brings us to the kicker.  What about 0.999…?  How close is that?  The “…” after the nines means that the nines go on forever, never stopping.  (If I were writing by hand, I would simply write 0.9 and put a bar over the 9. However, I’m not smart enough to get my computer to do that.)

What if I told you that no, you can’t fit your feeler gauge between 0.999… and 1 because 0.999… = 1?  We got there!  “That’s not possible!” you cry.  Okay, I’ll admit that I felt the same way, and because I’m such a sweetheart I’m not going to just say it and leave you to figure it out.  Actually, there are multiple ways to show this to be true, but I’ll stick with the ones that are simplest and don’t require knowledge of high-level math.  (I’m not trying to “dumb anything down,” mind you.  It’s just that I prefer the simplest way to get somewhere myself!)

What if I told you that no, you can’t fit your feeler gauge between 0.999… and 1 because 0.999… = 1?

Think for a minute about 1/3.  To write this as a decimal fraction you would write 0.333…  Yes, the threes go on forever.  Now let’s add up three of them:

1/3            0.333…

1/3            0.333

+ 1/3       + 0.333…

1             0.999…

Okay, so 0.999… equals 1 because 3 x (1/3) is the same as 3 x (0.333…).  Not convinced yet?  Too simple?  I thought so.  You’re so stubborn.  Let’s try it another way.

On this one, we will start with the statement that something is equal to our nasty little mystery number like this:  x = 0.999…, then use some simple algebra to figure out what is equal to it.  We’ll multiply the whole equation by ten, then subtract the initial equation from that.  Looks like this:

10x = 9.999…

–   x = 0.999…

9x = 9

1x = 1, or simply x = 1

Now I know you’re saying, “but if you multiply a number by ten, then there would be a zero at the end of it!”  But in this case there isn’t, because the nines go on forever, so all the multiplying does is move the decimal place.

The bottom line is that if you have 0.9999 etc with a finite (limited) number of nines, you’ll get really, really close to 1 without hitting it, but if the nines never stop, you’re there.  Turns out you can get there from here!

Mike O’Hern, Center Director of Mathnasium of West Knoxville, earned his Bachelor’s Degree in Metallurgical Engineering at the University of Tennessee, Knoxville in 1988. He pursued graduate studies in Materials Science & Engineering while on the Research Staff at Oak Ridge National Laboratory. Mike has had a life-long love of mathematics and teaching, and feels that math is not about learning to be ready for the next math class – it’s about learning to think.