Introduction
I did a wonderful persuasive post on the .999… = 1 topic before with lots of lovely (and convincing) pictures, but some people still don’t “get it”. So I’ve decided to produce the most comprehensive 0.999.. defying post on the entire internet!
Doubters, read and weep!
Disclaimer
1.) This post is not a joke. All math in this article is correct and verifiable.
2.) This post assumes all numbers to be used in the common every-day base 10 system. For most people, there is no other system, so don’t worry about it. This part of the disclaimer is just to keep the math-geeks from whining at me.
Definition Questions
What is a number?
A number is a mathematical unit used for counting and measuring. All numbers are infinite decimal expansions and can be rewritten in thousands of different ways. The number “2″ is a representation for the infinite decimal expansion “2.0000000000000…”, which is a “2″ followed by an infinite number of zeros. We just write “2″ because it’s easier and means the same thing. However, “2″ is not just “2″, but it’s also “6/3″, “1+1″, “the square root of 4″, etc. All of these representations equal “2.00000000000000000…”
What does “0.999… = 1″ mean?
“0.999… = 1″ is a mathematical statement that declares the number “0.999…” and the number “1″ are the same number. This means that the notations are interchangeable and just like you can use “12/6″ to mean “2″, you can use “0.999…” to mean “1″.
Of course, you can also say “3 = 4.2″ and “6 = pi”. Math police don’t come and tear those statements down, which is why we have to prove statements.
What is “0.999…”?
“0.999…” is also a number. Just like “2″, it can be written in thousands of different ways. “0.999…” is the same as “point nine repeating”, “.999…”, “0.9_”, 
, 
, “0.99999999 and so on”, etc. It’s a 0 followed by a decimal point followed by an infinite number of nines, or (9/10) + (9/100) + (9/1000) + (9/10000) + … all the way to infinity.
What is 1-0.999…
Subtract “point nine repeating” from one and tell me what you get. The answer should be zero. If two numbers subtract to equal zero, they’re the same. 61-(122/2)=0, so 61 = 122/2.
1-0.999… isn’t 0, stupid.
Oh, really? Then what is it?
Really? “0.999…1″? We’ll discuss why that can’t be true in the next question.
What does “an infinite number of nines” mean?
It means that the trailing nines after the “zero point” go on forever. When we say “0.99999999999999999…” we are talking about nines that never end. There is no “last decimal” at which you can add more decimals. “0.516398″ has a last decimal of “8″, and you could easily add “.0000002″ to make “0.5163982″. However, we can’t do this with “0.9999…” because the nines occupy every single decimal available.
This means, while you can write out numbers like “0.999…1″ or “0.999…5″, they have no meaning because they don’t correlate to any number on the number line. “0.999…” has meaning because it represents the number equal to the infinite sequence “(9/10) + (9/100) + …”.
Every decimal has this absolute definition. 2.569 is 2 + (5/10) + (6/100) + (9/1000). If you want to make the number “0.999…1″, you also need a corresponding sum of fractions. What does the digit “1″ go over when you add it to “0.999…”? It has to go over something, or it can’t exist. Because the number 0.000…1 has no sequence that it is equal to, it doesn’t stand for anything at all, and thus has no meaning.
Also, if a number like “0.999…1″ existed, then we could also have “0.999…9″, which would be the exact same as “0.999…” as both numbers have the same amount of nines — an infinite amount.
The number 1 goes over infinity! 1/infinity is surely bigger than 0. Maybe it’s “0.000…1″ after all.
There’s another problem. “1/infinity” is also not a number, because “infinity” is not a number. Infinity is a concept to mean “without end”. Dividing 1 by infinity is like dividing 1 by justice. You won’t get an answer, no matter how hard you try to pack the Supreme Court.
What infinity can be used in is to describe limits. We can discuss the sequence “y = 1/x” and talk about what happens as x gets increasingly large. We notice that as x increases, y decreases getting closer to 0 with each increase in x. The limit of “y = 1/x” is what y is approaching as x increases forever. That number appears to be 0.
We can think about limits when we think about what 0.9999… is approaching.
Imagine this sequence:
0.9
0.99
0.999
0.9999
0.99999
…
The sequence is gradually approaching 0.999…, however it is also gradually approaching 1. A sequence can’t have two limits, it can only have one. This means that the two limits we found must really be the same number, and therefore 0.999… = 1.
But the sequence is also getting closer and closer to 2, and 4, and 67. There’s definitely more than one limit.
This shows a misunderstanding of what limits really are. When a number is approaching it’s limit, it will get almost exactly there, and theoretically would be exactly there when you finally get to infinity. But you can’t ever get to infinity, you can only approach it. For example, 0.99999999999 is very close to 0.999… by our practical purposes, and is also very close to 1. It is not close to 2 — it’s actually 10,000,000,000,000% closer to 1 than it is to 2. It’s not approaching anything but 1.
Proving “0.999… = 1″
Now that we know what “0.999… = 1″ means, we have to prove it to be true. We already have the subtraction proof (1-0.999…) and the limit proof in the above definition questions, but here’s more:
For any two different real numbers, there is always a number between those two numbers. Between “3″ and “4″ is “3.5″. Between “62″ and “62.62″ is “62.37″. Between “1″ and “1.01″ is “1.001″. Between “0.999…” and “1″ there is… nothing.
But what about 0.999…5?
We’ve already established that “0.999…5″ is an impossible number, and because it doesn’t exist, it can’t be between 0.999… and 1.
Math just can’t notate a number between 0.999… and 1, but a number exists.
That’s not true. A real number is mathematically defined to be writable as a series of decimals. The decimals don’t have to be predictable (3.1415926…), but they have to be there. Every real number can be written.
The Famous Fractions Proof
I know what you’re thinking: “9/9 = 0.9999999…, so what?”. That’s the point. “9/9″ also equals 1. 9 goes into 9 exactly one time. You don’t even have to do it with a number divided by nine, it works with any fraction:
Or…
But “1/3″ doesn’t equal “0.333…”
That’s not true. If you do long division, you’ll find that three goes into one 0.333… times. I’ve solved a convincing portion of the long division in my previous post, but feel free to do the long division yourself.
No, really. The longer I do the long division, the closer I get to “0.333…”, but “0.333…” never reaches 1/3, therefore they’re not equal.
You’re misunderstanding what “0.333…” really means. The longer you do the long division, “1/3″ isn’t getting any closer or farther from “0.333…”, you’re just adding more threes. 1/3 is further away from “0.3″ than it is from “0.33″, but all three of the numbers lie on a number line somewhere. All the numbers are fixed, and they don’t move — they don’t get closer or farther away from anything.
I’m serious. 1/3 doesn’t equal 0.333…, it equals something else!
Okay. What does it equal?
You can’t write out 1/3 as a decimal.
You most certainly can. “1/3″ is a real number and all real numbers — even “pi” or the “square root of 2″ — can be written out as decimals. “1/3″ is written out as “0.3333…” or a zero followed by a point followed by an infinite amount of threes.
The Famous Algebraic Proof
That’s a false proof! I can just say x = 0.5, make 10x = 9.5 and prove 0.5 = 1!
No, because if you did, 10x would not be 9.999…, it would be 5. This proof is verifiable by arithmetic.
If you multiply 0.999… by 10, you get 0.999…0!
As in the earlier definitions, a number like 0.999…0 is ridiculous, and is the same as 0.999…
The Real Proof
Quasi-Proof by Lack of Counterproof
Just pointing out that there is not a single proof that “point nine repeating” does NOT equal one. Anywhere. On the internet… While having no counterproof is not a proof in itself, it is still good enough to cast doubt on the subject.
That’s All, Folks!
If you have any questions or doubts, leave a comment and I’ll be sure to follow it up in another post. If you need more, there is some great information in this post by somebody else with even more refutations of misguided arguments, as well as a lot more on the 1/3 = 0.333… thing, and the 1/infinity thing.
Until then, 0.999… = 1, and by consequence 1.9999… = 2, and 56.345869999… = 56.34587. It’s just another way to write the number, like 20/5 is another way to write 4.
Terrific job! You covered everything!