In a past blog, I provided a very simple proof to show that .999… does not equal 1, and that was simply to subtract .999… from 1 to get a non-zero number. This hasn’t convinced any believers (that I know of) that this equation isn’t true, because those people have already considered this and discarded it for one reason or another… usually because the result is an infinite string with a single different digit on the end.

After a few months of tossing around this idea, I was finally struck by an even better proof that is just as simple as the other one and yet more profound. As far as I can find, it hasn’t yet been explored, and yet it’s so incredibly simple. Not only do I feel stupid for not seeing it sooner, but it’s shocking that no one else has either. Of course, no one saw that energy and mass were directly correlated until Einstein came along and pointed out the obvious.

What is: .999… minus 1? Naturally we all see the 1 as the larger number (if we see either of them as larger), so we’re biased to do the subtraction in only one direction. But if they’re equal, shouldn’t the subtraction work in either direction? When you subtract X from X, it literally doesn’t matter which X comes first because the answer will be zero either way. So let’s see if that’s the case here… but first let’s consider a simpler problem for context in analyzing the result.

What is 2 minus 4? A mathematician will tell you to invert the problem and then make the answer a negative (for an answer of -2). Why? Because subtraction freaks out when it tries to subtract larger numbers from smaller ones. Here’s what I came up with:

You’ll notice that it keeps going to the left… infinitely. The result is an infinite number of 9’s followed by a single 8, which is clearly not the answer to 2 minus 4. This isn’t a special case, either… any time that you subtract a larger number from a smaller number you’ll get an infinite series of 9’s before a wrong answer. Try it yourself by hand with any two numbers.

Now for the problem at hand. What do you get when you subtract 1 from .999…?

There’s something familiar about this result, isn’t there? The answer is important for 4 good reasons:

1. The result is a non-zero number, which proves that .999… does not equal 1. X – X always equals zero.

2. The math is all based around the decimal point, just as other problems that are “doable” such as .888… minus .111… or like .999…. minus .1 — nobody can complain that solving this requires finding the last digit of the series of 9’s (unless they deny the double-standard of doing similar problems that I’ve mentioned here).

3. The pattern of repeating 9’s going infinitely to the left is exactly like the pattern we observe whenever we subtract a larger number from a smaller one, which means that .999… is smaller than 1 as a person would suspect.

4. The answer is internally consistent with my answer to 1 minus .999… as I’ll explain in the next blog.

So there you have it. .999… cannot equal 1 because, no matter how you believe you proved that it worked, the answer is wrong. To quote Richard Feynman, the famous physicist:

“It does not make any difference how beautiful your guess is. It does not make any difference how smart you are, who made the guess, or what his name is – if it disagrees with experiment it is wrong.”

To use your own Feynmann quote against you, if your answer doesn’t match reality (ie, 2-4 is clearly not infinitely large, that’s clearly nonsense), your theory must be wrong. In other words, all you’ve demonstrated here is that you’re doing subtraction very very wrong. We can’t take anything from what you’ve done with trying to subtract 1 from 0.999… because your method of subtraction is clearly flawed.

Obviously it’s flawed — that’s the point. You can’t subtract 4 from 2 using normal methods because it creates a “wrong” number that is clearly not the right answer. This is what happens every time that we try to subtract a larger number from a smaller number. But if a person is claiming that 1 and .999… are equal, subtracting one from the other shouldn’t give us a “wrong” number but rather 0. They don’t… thus, 1 is larger than .999…

The problem here is, you’re using your flaw the wrong way. All you’ve done is show your method of subtraction isn’t working. It’s like making a hammer out of soft butter, then blaming the nail when you can’t hammer it in.

Also, to make another quick wee post, the standard method of flipping the numbers and making the result negative is a valid method. Consider, say, x1 and x2, with x1 > x2. We then define y = x1 – x2. This is all good, you’ll get a positive number out of this.

Now we try x2 – x1. x2 < x1, so we've got a minor problem here.

So, let's define z = x2 – x1. First, multiply both sides by -1:

-z = -(x2 – x1)

Expand out the bracket to get:

-z = -x2 + x1, which is equivalent to -z = x1 – x2. We already know this is y, so we substitute:

-z = y

Or z = -y.

In other words, subtracting the smaller number from the second is the same as swapping the two numbers and negativing the result.

So bring this into the concept of 0.999…-1, the answer is going to be -(1 – 0.999…), which, if you think the answer is 0, you'll get -0. Or 0.