An age old question has been asked several times over the history of mathematics. Mostly to gullible high-school students who are stereotypically scared of math and numbers, but it has been asked nonetheless. This diabolical question seems simple, but don’t be fooled by its simplicity, or any other -plicity.
Countless people have fallen into the clever trap set by… whoever first proposed the question, I guess. Few have cleared this hurdle in their life, and some have even descended into a chronically deeply disturbed state, unable to come to terms with the answer that everyone else seems to have come to terms with. (alright, maybe I’m exaggerating a tiny little bit)
But what is the correct answer to this question? Think for a moment before you read on, although I know you’re not going to do it anyway.
- +The answer, to put it simply, is
Yes. They are perfectly equal. Not happy? Read on.
Now, for the math. Here are three ways:
First off, something that will get you thinking. What’s the decimal form of the fraction 1/3 ?
1/3 = 0.333…
So, multiply both sides by 3. On the left, you get 1. And on the right, you just multiply all the threes by 3 to get all nines, right?
1 = 0.999…
You could also do the same with 1/9, or 1/11, and more. Still not convinced? Method 2 coming up.
How about a (slightly) stronger argument? As you might be knowing, every recurring decimal is a rational number. That is, has a fractional representation. What’s that for 0.999…? Well, being the geniuses you all are, I’m sure you can do that yourselves. But for the record:
x = 0.999…
10x = 9.999…
Subtracting the first from the second,
9x = 9 or x = 1
And since we took x = 0.999… initially, we get that 0.999 . . . = 1
This last method should have a better chance of winning you guys over. Note that this isn’t rigorous, it’s just an aid to understand why.
So you say 1 and 0.999… aren’t equal. Well, if that’s true, then their difference must be non-zero. What is their difference, anyway?
1 – 0.9 = 0.1
1 – 0.99 = 0.01
1 – 0.999 = 0.001
Looking at this series and extrapolating a long way ahead (all the way to infinity), we get:
1 – 0.999… = 0.000…1
Which is zero point (infinite zeroes) one. Technically, it probably isn’t correct to write this, but just go with it.
Now, you say that this isn’t zero, huh? (even though it’s kind of obvious now, since there are infinite zeroes, but oh well).
If so, find me a smaller positive number than this. Note that you can’t go “Oh, just add another zero,” because there’s still an infinite amount of zeroes.
So you can’t find a positive number smaller than this. What does this mean? You have a number (supposedly non-zero) that is the smallest of all the positive numbers. Surely that should strike you somewhere as being absurd. You know that you can always get a smaller number by halving. There is no smallest number.
Since there is no smallest positive number (remember this, it’s important), and yet this number is smaller than all other positive numbers, this number must be zero. (just let it sink in for a moment)
And if the difference is zero, that means that 0.999… is indeed, equal to 1.
Note, there are also a few other methods on how to prove this. A simple one is based on geometric progressions, and you can find a more rigorous one using limits. I’m not gonna do that here.
All in all, I hope you’re finally convinced (if you weren’t before) that 0.999… = 1.
In fact, it’s just another way of writing a number with a finite decimal representation. For example, you know that
1/4 = 0.25
But, you can also write this as:
1/4 = 0.24999…
It’s exactly the same thing. Turns out it’s just a second notation, with minor differences. You learn something new everyday.