Joe Pagano

In Math, Fractions Are Strange But Decimals Are Even Stranger - Part II



Posted: Saturday, April 30, 2011

by Joe Pagano
Math by Joe

We were told in Part 1 of this series of articles that decimals were indeed a wacky bunch in the numbers arena. Unabashedly we were told that the non-terminating repeater 0.999... was exactly equal to 1, even though it seems that this long decimal always comes up a little short. Here we show how the rigor of mathematics gaps that shortage.

So how can we show that the non-terminating repeating decimal 0.999... is exactly equal to the number 1? That is what we are going to explore right now. To show that 0.999... is precisely 1, even though intuition would indicate otherwise, we use an oft employed mathematical trick which allows us to manage quite well such things as infinite decimals. It would seem that anything infinite would be hard to manage, by its very nature; after all, you cannot contain infinity. Yet this trick allows us to do just that.

Let us set the variable S = 0.999... Herewith the trick. We now multiply S by 10, to get 10S = 9.999... Now both Sand 10S are non-terminating repeating decimals. As such, we can subtract S from 10S with the assurance that all the repeating 9's will cancel out. Thus we get that 10S - S = 9S = 9. To see that more clearly make sure that you position both S and 10S with all the columns of 9's lined up properly, one on top of another, and then perform a traditional subtraction. Now the piece de resistance: 9S = 9 implies indubitably that S = 1, which is what we wanted to show.

If you have not hiccoughed already, what we have just shown-that is to say-proved without doubt is that the non-terminating repeating decimal 0.999...is exactly equal to the number 1. When I was first told this during my undergraduate years, I simply refused to believe it. It was not until I saw the proof that I accepted it, and even then, I was still a bit incredulous. It just seems that when it comes to things like infinity, things can get really---well...strange.

The trick we employed to convert 0.999... to 1 is the same one, albeit with some minor modifications, as the one used to convert any non-terminating repeating decimal into its fractional equivalent. Thus decimals such as 0.1111... and 0.134134134... can be suitably transformed into their corresponding forms a/b, in which a and bare both integers, or whole numbers.

Yes indeed. Fractions are strange but decimals are even stranger.
This Article has been viewed 117 times. (Not updated in real-time.)
No comments yet.
We want your comments! If you can read this, you don't have javascript enabled, so you can't use this comment system. Please enable javascript.