## 20 March 2011

### Secret behind 0.999... = 1

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 Wikipedia: 0.999...

Searching a solution we often forget about the problem behind it. I'm familiar with the wrong focusing on 'how' instead on 'why', and what's more I've seen people doing the same mistakes as well. For example, let's just look at law regulations as synonym of complication paired with uncleanness. However, I'll drop that subject and introduce the similar problem occurring in maths. Do not expect any great proof, because I'll do that to explain it as simple as possible.

##### `0.999... = 1`

The proof for the equation above is the above derivation:

```       x = 0.999...     10x = 9.999... 10x - x = 9.999... - 0.999...      9x = 9       x = 1 ```

If you are interested on it, you can find more about it on Wiki: 0.999... or at once read quite fine explanation by Cecil Adams. The fact is that he will not going to get Nobel price for that but in real numbers field his explanation is mostly accurate.

I wanted to explain the reason behind it rather than submitting next proof for it. For this purpose I'll introduce the proof that every person earns the same amount of money, of course using the same method as above.

``` Let's:   m - best earnings on earth.   x - earnings some man.   y - earnings some other man. We have then:   x >= 0   y >= 0   x <= m   y <= m Let's assume:   x = y We can divide by nonzero:   x/2m = y/2m Let's bring it to integer numbers: using (x <= m)   x/2m = 0 using (y <= m)   y/2m = 0 So we have proved equality:   0 = 0 ```

Thus, I proved that each of us earns the same amount of money, so it's proved the validity of the communist system.

##### Explanation

Now, it's high time to end the torment and explain the point. Each of us earns the same amount, 'but' in accuracy to double amount of the best salary possible to be earned. The same thing happens in the case of `0.999 ... = 1`. Except the fact that in the latter case the accuracy of the integers is 1, the precision of real numbers is (roughly) 1/∞. So we can safely confirm equality between 0.(9) and 1, 'BUT' with an accuracy of 1/∞, which shows that in the same time there is (roughly) the difference between these two numbers.

##### Summary

All proves (including those the Wiki) are accurate, but in the same time they do not carry too much sense.

##### This is how it is

Proper infinity algebra is not used with some fairly simple reasons (in general):

• Infinitely small values are 'relatively' meaningless, so for quite a while there was no need to include them in the model.
• Algebra with ∞ numbers begins to exhibit vector traits.
• The calculation results are interesting only if it is lifted upwards.
• Inclusion of infinity in the model is quite tiring.
• A few existing formulas would have to be corrected in order to properly fit infinity algebra.

##### More on...

If somebody wants to make some calculation, he/she may read about surreal numbers, that contain the infinite and infinitesimal numbers (greater than zero but less than any positive real numbers). The mathematical description you can find on wiki as well: 0.999...: Alternative number systems

1. I think I can see your point, but unfortunately your 'proof' doesn't actually support it. Your proof that all men earn the same amount is flawed by the fact that you make the assumption x = y. Therefore your proof is actually 'two men who earn the same amount of money, earn the same amount of money' and it isn't extensible to the set of all men (or all women).

Contrast that to the 0.999... = 1 proof, which makes no assumptions beyond those of the real number system (which as you point out, imposes some assumptions and restrictions), making it a much more general proof of something that isn't already a tautology ;)

2. Your proof has multiple errors causing the entire article to fall apart.

Let's:
m - best earnings on earth.
x - earnings some man.
y - earnings some other man.
We have then:
x >= 0
y >= 0
x <= m
y <= m

Let's assume:
x = y -- Really? You plan to prove everyone makes the same income by assuming everyone has them same income except the highest paid man?
We can divide by nonzero:
x/2m = y/2m
Let's bring it to integer numbers:
using (x <= m)
x/2m = 0 -- This only proves that man x does not make more than m, which was already stated as m is the highest paid man.
using (y <= m)
y/2m = 0
So we have proved equality:
0 = 0 -- The only thing proved is that nobody makes more than the person who makes the most, which is meaningless.

0.9999.... does in fact equal 1.

3. Hello,

Your proof that all men earn the same amount is flawed by the fact that you make the assumption x = y.
Since when assumption have to be truth?
Aunt Wiki says:
In mathematics and logic, a direct proof is a way of showing the truth or falsehood of a given statement by a straightforward combination of established facts, usually existing lemmas and theorems, without making any further assumptions.

But if you really need, you are allowed to start reading it from the bottom (0 = 0)

All I wanted to show is that same as 0.1 do not belong to integers, the do not belong to real numbers. With all consequences behind it for both real and integer numbers.

4. @Lasu

My point is that your 'proof' is nothing more than showing a tautology (given that two men earn the same amount, then two men earn the same amount), which shows nothing (and is not what you stated you were showing), as opposed to the one you're contrasting to, which is quite a deep result regarding the nature of infinity and infinitesimals within the system of real numbers.

5. Hello,
@Anonymous

"Really? You plan to prove everyone makes the same income by assuming everyone has them same income except the highest paid man?"
There is '<=' so it will be done including 'highest paid man'.

"The only thing proved is that nobody makes more than the person who makes the most, which is meaningless."
1. It's meaningless in same matter as 1 = 0.999...
2. I wrote: "Each of us earns the same amount, 'but' in accuracy to double amount of the best salary possible to be earned"

Now I can see where you can have the doubts, so let me help you with those.

"My point is that your 'proof' is nothing more than showing a tautology (given that two men earn the same amount, then two men earn the same amount), which shows nothing (and is not what you stated you were showing)"
You mistakenly interpreted first line of prof:
x = y
Do not mean same as:
x = x
Let me use sample:
m = 4\$
x = 0\$
y=3\$
Now we have:
0\$=3\$               x = y
0\$/8\$=3\$/8\$      x/2m = y/2m
0\$/8\$ = 0         x/2m = 0
3\$/8\$ = 0         y/2m = 0
0 = 0

So as you can see this if accurate for different values for x and y, therefore I proved equality 0\$ and 3\$ using rounding errors.

7. Cranks are always entertaining :)

8. @Anonymous
"Cranks are always entertaining :)"

That makes life more interesting, isn't it?

9. 0.999999 = 1 proof is not about rounding, but the limit of infinite sequence...

10. Your logic: f(x) = f(y) => x = y. This is not true for all possible f (like round in Your case)
In case of .9999 proof, the f(x)=10x, but in this situation implication 10x = 10y => x = y is true! That's why 0.9999 proof is working fine, and Your "proof" is just a bullshit...

11. Dear @Anonymous
"0.999999 = 1 proof is not about rounding, but the limit of infinite sequence..."
This proof works fine because of rounding error.

Your "proof" is just a bullshit...
Once more I'll underline that my post is not a prof / it's just explanation. Prof would be beyond most people understanding because it require advanced mathematics knowledge.

Any way if you are so sure about your opinion you should correct wikipedia as well starting with Infinitesimals section in 0.999...

"implication 10x = 10y => x = y is true!"
Yes it it. But the problem is that 0.999... * 10 is not 9.999...
It's like that because 0.999... is recipe for number rather than just number.
To be precise 0.999... = 1 - speedOfAimingZero
so:
10*0.999... will be 10 - 10*speedOfAimingZero

Hope that it will help!

12. You are wrong, 0.999999 = 1 proof is not working because of rounding error.

0.999... * 10 is indeed 9.999...

13. 1) 0.9999(9) is a real, existing number, not "a recipe for the number". The same way as PI=3.1415... is a exising number.

2) 0.99(9)* 10 = 9.99(9) the same way as 1/3 * 10 = 3+1/3 (0.333(3) * 10 = 3.333(3))

3) 0.9999=1 proof is a real proof, not a semi-proof based on rounding errors....

4) If You think 0.999(9) is not equal to 1, please give me a number (or recipe for the number) which is between 0.999(9) and 1...

5) According to Your understanding, is 0.333(3) equal to 1/3 or no?

14. @Anonymous

Sorry for late response, but i'm busy enough to do not have enough sleep.

"1) 0.9999(9) is a real, existing number, not "a recipe for the number". The same way as PI=3.1415... is a existing number."
"2)..."
"3)..."
All those statements depends on the space we talk about. real:'0.(9)' can correspond to many values in infinitesimals same way as integer:'1' can correspond to multiple values in real numbers. But if we want to be precise 0.(9) is same number as 'one', 'six' or any other literal. Therefore 0.(9), one and 1 mean the same in real numbers, but it do not equal 'physically' 0.999... sequence.

"5) According to Your understanding, is 0.333(3) equal to 1/3 or no?"
In real numbers yes it is, as i said earlier 0.333(3) is just other notation of 1/3.
In infinitesimals it is not.

"4) If You think 0.999(9) is not equal to 1, please give me a number (or recipe for the number) which is between 0.999(9) and 1..."
It's really great question, so I'll try to give you the most simple answer that I have.

0.(9) = ({0, 9/10,99/100,999/1000,...},{1})
is less than:
0.(9) = ({9/10,99/100,999/1000,...},{1})
is less than:
1 = ({1},{1})

0.(9) < (1-0.(9))/2 < 1

More 'real' example:
Lets assume that we have three friends MrL, MrO and Mr1, and they compete to gain money. Each day one of them that have the most resources gain 1\$ and the one of them that have the least will have to give back 1\$.
Mr1 have 1# at start get 0# each day
MrO have 0.0# at start and get 9/(10^day)# each next day (0.9, 0.09, 0.009, ...)
MrL have 0.0# at start and get 9/(10^(day-1))# each day since starting with second day ((0,) 0.9, 0.09, 0.009, ...)
Day 0:
MrL: 0#
MrO: 0#
Mr1: 1#

Day 1:
MrL: 0#
MrO: 0.9#
Mr1: 1#

Day 2:
MrL: 0.9#
MrO: 0.99#
Mr1: 1#

As we see MrL will always have less than MrO (including infinity) and will lose 1\$ each day, and Mr1 will always have more MrO and gain 1\$ each day.

15. Infinitesimals are just another exotic number theory, not very useful IMHO. If You are not talking about real number You need to mention this at the beginning. If no assumption were made, everyone consider real numbers. You cannot jump from one algebraic structure to another without warning, that makes Your consideration pointless...

0.999(9)=1 proof is true in real numbers, and has nothing in common with rounding errors

(Infinitesimals are not subset of real numbers...)

"Therefore 0.(9), one and 1 mean the same in real numbers, but it do not equal 'physically' 0.999... sequence."
What does "physically" mean? ;) This is math, there is no reality or physics here. There are assumptions and implications. There is no superior world or physics where math is working...

16. @Anonymous

Infinitesimals are just another exotic number theory, not very useful IMHO.
You cannot say like that if u ever want to use integral.

0.999(9)=1 proof is true in real numbers
You got that wrong: 0.(9) equals 1, but proof for that is invalid. And I'm telling you this because using maths to prove that is the same as going to the university and proving that 1 = 1.2 in integer numbers, what would be at least funny for audience.

Precision of real number is ~ 1/(10^∞+1).
So I'll say it once again 0.(9)=1 in real numbers but proving it doesn't have much common sense, because difference between those numbers is far lower than real numbers precision.

What's more, the number system where 0.(9) can be written clearly say that 0.(9)<1.

What does "physically" mean? ;)
For me maths is very real, mostly when I pay in store ;).

17. This is not rounding. For example if You are working with integers, and want to find value of 6/5, You cannot do it, as 1.2 simply does not belongs to integers. The result is beyond Your domain. You cannot say: 6/5 is equal to 1 in integers, but equal to 1.2 in domain of real numbers...

The question is: Is 0.999(9) real number or it is not? If it is real number, and is not equal to 1, then You should be able to find a real number between them, as between any two different real numbers there exist a number between them. (real number, not inifinitesimal). Can You do it?

18. @Anonymous
The question is: Is 0.999(9) real number or it is not?
I thought that it's clear already.
As number/sequence '0.(9)' do not belong to real numbers.
As literal it mean the same as 'one', '1' so yes, it does belong to the real numbers, so there is no point in searching for number between '1' and '1'.

19. So why the proof is "invalid"? To make it invalid You have to change the meaning of infinity, You have to change assumptions of real numbers theory. Proof is 100% OK, because it's author assumed we are working with real numbers.

Of course You can build a theory where this equation is not true, but You cannot say it's invalid in the world of real numbers just because it does not work with inifinitesimals...

20. @Anonymous
Proof is 100% OK, because it's author assumed we are working with real numbers.
That's the exact reason why proof is invalid. Using number in wrong space effect in unpredictable result in case of any operation. Same as assuming that (imaginary number)^2 will be positive.

21. Well, if number does not belongs to particular space, You cannot make any operation as operation is undefined. The result is not "unpredictable", the result does not exists. In this proof all numbers belongs to real numbers, and result is OK.

22. @Anonymous

Someday in a free time I'll prepare update for Wiki and extend current explanation.

When you notice that equality of 0.999(9) with 0.999(9)*10 -9 is just assumption then maybe you will understand the problem.

So in the proof mentioned above we have actually two assumptions (while there should be only one):
0.999(9) = 0.999(9)*10 -9
and
0.999(9) = 1

23. According to a definition of "(x)" symbol used in this notation (with real numbers), the equality of 0.999(9) and 0.999(9)*10-9 is quite obvious and easy to proof. The problem is, that in case of infinitesimals some basic concepts (like period notation) are redefined, so results may be diffrent...

24. Here is the final say on this matter: