A Simple Proof
x = 0.999…
10x = 9.99…
10x-x = 9.99… – 0.999…
9x = 9
x = 1
0.999… = 1
I Think, Therefore I am Confused
If you are like me, you look at this equation and scoff. Common sense and years of math intuition tell you that what is on the left is not the same as what is on the right. If you are like me, you are wrong. But if you are like me, you will look it up on wikipedia or at wolfram.com, or at Better Explained, or perhaps even watch vihart’s video explanation and you will see that there are rigorous proofs… but you will yet still have doubts, unsatisfied.
You are not alone. This is one of those bedeviling problems that has worn many a thinker – from Pythagoras forward – bald with head scratching.
Welcome to wonderland.
You see, it is not our fault. When you and I look at that equation, we imagine two distinct, finite numbers. This is what the form of these numbers want us to do. But this formulation is a visual shortcut to a deep, counter-intuitive concept that, if followed, will take you all the way down the rabbit hole; all the way back to Pythagoras and thousands of years of mathematical struggle and strife.
It turns out that mathematicians are not really that much different from you and I. They do not like all that rigor any more than you and I do. Well, maybe some of them do! But the truth is that the rigor came later, after years and years of intuition failure. And this is the interesting thing: the rigor is only there because we got stuck really bad.
But strangely, it turns out that Logic has the creative power to build Grand Tapestries that Imagination could never fathom – not even it it’s wildest dreams!
The first thing that trips us up when we see the repeating decimal is we think of a discreet number that we can extend to the precision of our will. We imagine that we can subtract this number from 1 and measure the gap between them. But when we do imagine this, we must be imagining a discreet, finite number, which is to say we are NOT imagining 0.999… but some number infinitely smaller!
Ah, the joys of being a finite being trying to conceive and measure the infinite!
The problem, you see, is that this infinite decimal cannot be contained by the mental image we have of what a number is. It leaks out of the box. The mere utterance of the existence of such a number could incite murder in Pythagoras’ day!
We were taught to think of multiplication as repeated addition but one day that model, or image, or idea, stopped working for us. You may not even remember this. If you are like me, your intuition of number was already set and to this day you think of multiplication as repeated addition. But this is because when you and I think of number we do not think of sqrt(2).
From Better Explained:
In school, first we are told numbers are “counts” of something, then we learn to add, or “combine our counts.” Next, we learn that multiplication is just repeated addition and this interpretation works well for round numbers like 2 and 10 but concepts like -1 and sqrt(2) don’t work. Why? Our “model” or “analogy” or “picture” was incomplete. Numbers aren’t just a count; a better viewpoint is a position on a line. Now we see arithmetic as ways of transforming the location of this position to a new location. Addition is now seen as sliding and multiplication as scaling.
The moral of the story is that this is what happens when we reach the brink, or limit, of an idea. Our thoughts are created in images and sometimes those images just don’t work in certain situations and we have to find new images.
This is what is happening with the infinite series. The image we have in our head of what a number is breaks down. It no longer works. We have to throw intuition away and trust the rigors of logic. It was only because our intuition, and our images, failed us in places like this that we had to back up to the beginning and start to lay down the mathematical rigor and build up ever so carefully from the bottom up, making sure along the way that we were leaving absolutely no wiggle room in our logic and in our proofs. This took thousands of years and it is a pretty impressive accomplishment. Some problems literally stopped us in our tracks for hundreds and hundreds of years.
I admit, I have gone back and forth on this and many other similar issues, trying to fight what the mathematicians have proven. I myself do not know how to picture a “number” that can never be pinned down, one that is the sum of a process that never ends! I just can’t do it! Every time I try to imagine this “number” I am really imagining that the process has stopped. But it hasn’t and down the rabbit hole I go again!
But everything I am telling you has been proved in the most rigorous way. The only escape is to postulate a new kind of number and that leads to worse problems. But it has in fact been done. I don’t think the results are very pretty, though.
This is called an “infinite series.” All such infinite series represent exactly one real number (in this case, 1) or run off to infinity. There are (possibly infinitely) many “infinite series” that represent (or converge to) any given real number. Which is to say, given any real number, there are many (possibly infinitely many) infinite series that correspond to it.
In other words, all real numbers can be “represented” as a repeating decimal. There is no debate about this. Incidentally, don’t you think this is a curious thing to say, that a number can be represented as another number!
The problem begins when we ask the question, “what is the sum of this series,” which is to say: “add this to this to this to this and never stop, and when you are done, tell me what the sum is!” The contradiction should jump out at you at this point. The problem is that no one has been able to posit an escape from this contradiction.
The notation 0.999… is just a representation that we came up with so we could express the idea of “9(1/10) + 9(1/100) + 9(1/1000) … forever” more succinctly. The problem is that when we see 0.999… we think of a static number and imagine we can add digits to it to make it more and more precise at will. But in reality, the repeating decimal is just what we write down when we write down the “end product” of something that has no end.
The end product of something that has no end.
This is a contradiction. It does not make sense. But we created a map for it anyway, so that we could do arithmetic with it. As it turns out, we can do arithmetic with things that do not make sense in themselves. Nonsense squared = q; and q times q times q equals q3, which is nonsense cubed. As long as we forget the meaning of nonsense, we have no problem calculating nonsense cubed. My goal here is to point out the nonsense that we have glossed over in these last thousand years….
The source of the confusion in this case is that we want pictures, so we make a picture of 0.999… and then when we try to use that picture, it fails us because it leads us to believe that the equation, “0.999… = 1” is false. But this is because our intuition about what 0.999… means is wrong.
I agree with you: we intuitively imagine that there must always be a gap between 0.999… and 1, but that is because we can only imagine the process that is behind the symbol in finite terms. We think, “no matter how long you allow the process to run, it never reaches that for which it approaches.”
Seems rock solid, but it is in the phrase, “no matter how long” that we stumble, for in this phrase we are really postulating a finite length or duration – a “how long” of the process – at which point we stop the process and say “at this certain POINT in the series” there is a gap between our sum and 1.
In effect, we are saying, “there is a gap here. And here. And here. And here.” And so we conclude there is always a gap, QED. Watertight! Right?
What we mean when we make that argument is that there is a gap at each finite point, each time we stop the process and measure. But this is sort of like the problem you have in quantum physics where the measurement collapses the strange, mysterious thing into the expected phenomenon. In so doing, we destroy the strange mysterious thing itself!
The crux of the problem is that you can never stop to take a measurement to see if there is a gap. Any time that you imagine a gap, you are stopping the addition machine and imagining a finite point. There is no getting around this because there is in fact no gap between 0.999… and 1 even though there is a gap at each and every finite point that is built into what 0.999… means!
Think of the problem in reverse, and ask, if there is a number between 0.999… and 1, what could it possibly be? To imagine such a number is to imagine that 0.999… is finite, that the process of adding up all those fractions has ended. But it never does, and so there is never even a point to point to and measure the gap!
Another way of stating this is to say that 0.999… is infinitely close to 1, which is de facto to say that no other number can exist in the “in between” because the “in between” is infinitely small, which is another way of saying the size of the gap is zero.
Yet another way to crash into this paradox is to ask the simple question, after the number X. what is the next real number? Once you think about it you will realize that there is no such thing as the next real number because there is an infinite number of numbers between any two real numbers. Therefore, the concept of a “next” real number fails. Such a concept is undefined. No matter how hard you try to make it so, there simply is no such thing as a “next” real number. This is so counter-intuitive to our concept of the number line as one point followed by another that we really have to pause and consider if we really understand the number line at all. I suggest that for most of us, the answer to this question is a resounding no.
We are finite and we are trying to measure the infinite! This is why our imaginations necessarily fail us. But we cannot help but to think in this way.
This is why math was grounded in the rigors of logic, because we kept running into these problems of intuition failure.
The History of Number
One of my favorite books is called “Number.” It has been around a while, written by Tobias Dantzig. Einstein said it was the “beyond doubt the most interesting book on the evolution of mathematics which has ever fallen into my hands.” It is one of the books that I would take to a desert Island were I to be stranded there for life. This problem we are discussing is literally thousands of years old. Pythagoras was the first to slam head-on into it. It took us thousands of years to “settle the case” rigorously and it is only because we were able to do so that any engineering beyond that of the ancients is possible, for the mathematics of these infinities is what makes all modern electrical engineering possible.
The history of science and the history of mathematics is one of the greatest stories there is and it is one of our greatest treasures. It should be taught as part of the liberal arts. I can’t really do it justice, but I enjoy reading those who can. If your interest is peaked, start here.