The Ultimate “0.999… = 1″ Guide
Monday, November 16, 2009
I did a wonderful persuasive post on the 0.999… = 1 topic before with lots of lovely (and convincing) pictures, but some people still don’t “get it”. So I’ve decided to re-do the post and produce this — the most comprehensive 0.999.. defying post on the entire internet! Or something like that!
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 an infinite amount 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.
If you’re still unconvinced that nothing like 1/infinity exists, consider what would happen if we did math with infinity.
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 believe this, your problem lies in challenging the principle of long division, not in challenging me. 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. If you did the long division an infinite amount of time, you would get all infinite digits.
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. In math, statements are provable. If you think 0.999… does not equal 1, you should be able to prove it.
A Note on Rounding
One last note: I’ve noticed a decent amount of people agreeing with me that “Yes, 0.999… = 1, because 0.999… rounds to 1“. While I applaud you for getting the right answer, you just got it for all the wrong reasons. This makes no sense.
Rounding to something does not make it equal to something. 3.1415926… rounds to 3, but it definitely does not equal 3. Rounding actually means “close, but not quite”. There is a difference between equals (like how 6/2 equals 3) and close (like how 6.02 is close to 6).
0.999… is instead actually and authentically equal, no rounding involved, to 1. It’s just another way of writing 1, just like “3-2″ and “17/17″. Why? Well, just read this post, silly!
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.
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Terrific job! You covered everything!
I love this and tried to prove that all numbers to not exist as we know them.
For example. If 1/9= 0.111111111111111111…. then it is impossible to write down this decimal number. If it is impossible write down, then it is just a concept and to me, it is infinity.
Therefore 1/9 = infinity. As every number has a reciprocal ie (1/9)^-1 = 9, but 1/9 = infinity so therefore 9/1 is infinity. The number 9 is a concept. The same can be said for 3/9 and 9/9. Hence the numbers 1,3 and 9 are just concepts. If they are concepts, they are infinite.
We were all taught that 1 + 1 = 2, but as 1 is infinity 1 + 1 = infinity too.
Any number equals itself multiplied by 1, so 2 = 2 * 1. But 1 is infinity, so therefore 2 is equal to infinity as well.
It seems that maths has been manipulated so that certain values conform to the laws of mathematics. So when it was found that 0.1111111111111111111…. multiplied by 9 actual equals 0.9999999999999………. and not 1, it was decided to give them to same value (which was 1 in this case) to prove that any number divided by itself equals 1, when this is not the case. When we write 9 * (1/9) = 1, we actually mean that it is approximately equal to 1 as 9 * (1/9) has no value.
In theory 1/0 should equal infinity as well as in the laws of mathematics this is impossible to do.
1
- = Inifinity
0
1 = 0 * infinity
1 = infinity
as I have shown that the value of 1 is infinity above, therefore 1/0 = infinity.
Maths is an exact science we we taught. 1 + 1 = 2 and nothing else. But as 1 + (9/9) or 1 + (3/3) or 1 + (11/11) equals 1.99999999999999…… then it is not an exact science.
I’m going to have to stop you right there.
First of all, “1/9″ is not impossible to write down; it is only impossible to write down in decimal notation. You can always write it down as “1/9″ or “2/18″ or “0.111…”, or in many other representations.
Second of all, if it is impossible to write down, that does not make it infinity. It makes it “1/9″. “1/9″ is not an abstract concept at all, it is a specifically defined number.
Third of all, not all concepts are the same concept just because they’re abstract. “Infinity” and “Justice” are both concepts, but they are not the same concept.
“for example. If 1/9= 0.111111111111111111…. then it is impossible to write down this decimal number”
Not in a non-base 10 counting system.
“Not in a non-base 10 counting system.”
Please post for me in binary, octal or hexadecimal or whatever base you care to think of, the true equivalent of 0.1111111111111111111… base 10.
My calculation of the binary equivalent is 0.000111000111000111 recurring. Therefore a physically impossible number to write down.
But surely 1/9 is a representation of 0.11111111111… When this is written, it means to divide 1 by 9. If it is impossible to write down the true decimal value of 1/9 then how can 1/9 or 2/18 be a representation. Or is it a presentation of something which does not exist.
Yes there are different concepts and yes infinity and justice are not the same thing, but if something is impossible to write down due to the infinite number of 1′s then surely this is a representation of infinity.
I’d say 0.11111111…is a decimal representation of 1/9, it’s clear that 1/9 does exist, get one of something divide into 9, you can do it despite the clumsy recurring 1 in the base 10 counting sustem.
In base 9 it would be 0.1 as I understand it.
In base 3; 0.01.
Base 27, 0.3.
Heres a link.:
http://www.easysurf.cc/cnver17.htm#batob10
Base 10 is just the framework we use because we have 10 fingers, previously base 8, base 12, base 20, base 60 have all been used.
It’s a bit like saying because english has no exact match for the japanese word….ummmm….yarigai…that we lack any real concept (hard working, determined)
Joseph, I fixed the typos in your comment for you.
~
@Andy:
You still have not defended this statement. It is false — just because you cannot write a number in base 10 decimal notation does not make it infinity, as I have previously pointed out.
Cheers!
Proving 0.999… is not equal to 1
Why 0.999… is not equal to 1? Because:
1. 0.999… is a number in the infinite world, but 1 is a number in the finite world.
For example, 1 represents an apple. But then 0.999…? We don’t know. Because we live in a finite world, and then we develop our mathematic system.
We can use “infinite world” and “finite world” to resolve some of zeno’s paradox, too.
2. Because of indeterminate principle.
Because of indeterminate principle, 1/9 is not equal to 0.111….
For example, cut an apple into nine equal parts, then every part of it is 1/9. But if you use different measure tools to measure the volume of every part, it is indeterminate. That is to say, you may find the volume could not exactly be 0.111…, but it would be 0.123, 0.1142, 0.11425, etc.
This standpoint tells us, our world is only a sample from a sample space.
But that doesn’t follow. There’s no such thing as an “infinite world”, and a number isn’t infinite just because it has an infinite number of numbers after the decimal point. 0.999… is finite.
~
“1″ represents an apple, but “2/2″ represent an apple, “4-3″ represents an apple, and “negative i squared” represents an apple. Same with “0.999…”, that represents an apple too, for every reason I’ve said here.
~
That’s not how we resolve Zeno’s Paradoxes. We solve them because while the distance is infinitely divisible, the time needed to cross each division also decreases infinitely.
~
That’s also false. Also, the “indeterminate principle” isn’t even a thing. It exists nowhere in mathematics. I’m pretty sure you just made it up.
~
If the parts were equal, they would have the same volume. Yet 0.123*9 = 1.107, which is more than one apple.
If you cut an apple into nine equal parts, there’s no way the volume could be anything but 0.1111… If part of the volume is 0.123, then you didn’t cut the apple into nine equal parts. You just cut it into nine parts that look pretty really definitely close to equal, but aren’t actually equal.
【1 paradox】Why 0.999… is not equal to 1?
Written in 2012
The current mathematic theory tells us, 1>0.9, 1>0.99, 1>0.999, …, but at last it says 1=0.999…, a negation of itself. So it is totally a paradox, name it as 【1 paradox】. You see this is a mathematic problem at first, actually it is a philosophic problem. Then we can resolve it. Because math is a incomplete theory, only philosophy could be a complete one. The answer is that 0.999… is not equal to 1. Because of these reasons:
1. The infinite world and finite world.
We live in one world but made up of two parts: the infinite part and the finite part. But we develop our mathematic system based on the finite part, because we never entered into the infinite part. Your attention, God is in it.
0.999… is a number in the infinite world, but 1 is a number in the finite world. For example, 1 represents an apple. But then 0.999…? We don’t know. That is to say, we can’t use a number in the infinite world to plus a number in the finite world. For example, an apple plus an apple, we say it is 1+1=2, we get two apples, but if it is an apple plus a banana, we only can say we get two fruits. The key problem is we don’t know what is 0.999…, we can get nothing. So we can’t say 9+0.999…=9.999… or 10, etc.
We can use “infinite world” and “finite world” to resolve some of zeno’s paradox, too.
2. lim0.999…=1, not 0.999…=1.
3.The indeterminate principle.
Because of the indeterminate principle, 1/9 is not equal to 0.111….
For example, cut an apple into nine equal parts, then every part of it is 1/9. But if you use different measure tools to measure the volume of every part, it is indeterminate. That is to say, you may find the volume could not exactly be 0.111…, but it would be 0.123, 0.1142, or 0.11425, etc.
Now we end a biggest mathematical crisis. But most important is this standpoint tells us, our world is only a sample from a sample space. When you realized this, and that the current probability theory is wrong, when you find the Meta-sample-space, you would be able to create a real AI-system. It will indicate that there must be one God-system in the system, which is the controller. Look our world, there must be one God, as for us, only some robots. Maybe we are in a God’s game, WHO KNOWS?
More info, three other download points(written in Chinese):
yourfilelink.com/get.php?fid=780934
http://www.axifile.com/en/69D108FD7F
localhostr.com/file/3LtuSLb/the%20end%20of%20the%20world.rar
Briefly, but no rudeness intended:
“Because math is a incomplete theory, only philosophy could be a complete one.” – non sequitur
“But we develop our mathematic system based on the finite part” – Cantor didn’t.
“0.999… is a number in the infinite world, but 1 is a number in the finite world” – 1 can be written as 1.000…. is it now part of the infinite word?
“0.123, 0.1142, or 0.11425″ then you failed to cut it into 9 equal parts.
Let us say the cake consists of 54 x 10^23 cake atoms (which Tony Stark will invent in the “Avengers” movie). Cut it into 9 equal parts, each part now contains 6 x 10^23 cake atoms, because this is represented by 0.1111…. (if a cake is represented by 1) in a base 10 counting system says nothing about whether you truly have 1/9th or not.
This is not actually a paradox. Consider this instead “10>7, 10>8, 10>9, 10=10″ is not a negation of itself.
~
0.999… represents an apple as well. Just like how 4/2 represents an apple.
~
This is just an assertion. Why do you actually think that?
We actually do know what is 0.999…, in fact it is the limit the limit of (1 – 1/(10^n)) as n goes to infinity.
What do you think of all the proofs I included in this article that involve the use of 0.999… as an actual mathematical object?
~
What does “lim0.999…” that “0.999…” doesn’t?
~
Again, if you find the volume is not exactly 0.111…, then it is not true that you cut the apple into nine equal parts, because not all the parts would be equal.
~
This doesn’t make any sense.