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**Grantingriver****Member**- Registered: 2016-02-01
- Posts: 95

Regarding the puzzle which you titled "Is this really true" the argument in fact is a fallacy to see why, lets suppose that the recurring number is finite say X=0.9999 in the light of this assumption lets repeat the calculation which you posted:

10X=0.9999×10 ⇒ 10X=9.999

⇒ 10X-X=9.999-0.9999

⇒ 9X=8.9991

⇒ X=8.9991÷ 9

⇒ X=0.9999 (which is the original value)

Therefore when we work with any finite recurring of 0.9 the result will be less then one. The problem only occurs when we let the recurring tend to infinity, but if we trace the behavour of the result as the recurring tend to ∞ we find:

X=0.99999 ⇒ X=8.99991÷9=0.99999

X=0.999999 ⇒ X=8.999991÷9=0.999999

... ... ...

∴ X=0.999... ⇒ X=8.999..÷9=0.999... (which is the correct answer).

What does equal "1" is not the number 0.9̅ instead it is the sum of the following infinite geometric series:

∑0.9/10ᵐ as m → ∞ which is:

S= 0.9+0.09+0.009...= 0.9/(1-1/10)=0.9/(9/10)=10/10=1

One fallacious argument which could lead to a similar paradox may goes as follow:

1=9/9=9×1/9=9×0.1̅=0.9̅ ⇒ 1=0.9̅ (these results follow since the set of real numbers is a field). However, the wrong hypothesis in this argument is "1/9=0.1̅" which should be "1/9≈0.1̅" since the number "0.1̅" gets closer and closer to the number "1/9" but never reaches it (this is because we can not reach infinity). So the correct angument is:

1=9/9=9×1/9≈9×0.1̅=0.9̅ ⇒ 1≈0.9̅

Lets take "1/64" as an example to illustrate this problem. We know that 1/64=0.015625 in this case we can not say 1/64=0.015 or 1/64=0.0156 or 1/64=0.1562 the correct propositions are 1/64≈0.015, 1/64≈0.0156 and 1/64≈0.01562 respectively. On the other hand the proposition "1/64=0.015625" is definitely true. But as we have mentioned it is impossible to reach an end to the number "0.1̅" therefore it allways approximates but never equals the number "1/9". The value which does in fact equal that number is the sum of the infinite series ∑0.1/10ᵐ as m → ∞. So the corrcet argument should be:

1=9/9=9×1/9=9×(∑0.1/10ᵐ as m → ∞)=9×(0.1/(1-1/10))=9×(0.1/(9/10))=9×10/10×1/9=1 ⇒ 1=1

Which is the expected and correct value.

Note: when we do math we should not apply the rules without reasoning, we should follow the logic since all math is logic.

Q.E.D

*Last edited by Grantingriver (2016-02-02 10:08:35)*

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**anonimnystefy****Real Member**- From: Harlan's World
- Registered: 2011-05-23
- Posts: 16,037

Hej!

This topic has been discussed many times here and you'll get different opinions on it. Since opinions should not affect absolute truth, the solution is to either take 0.(9)=1 by convention, or just define decimal represntation so that 0.(9) is not a possible decimal representation, which is the variant I support.

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

The knowledge of some things as a function of age is a delta function.

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**Grantingriver****Member**- Registered: 2016-02-01
- Posts: 95

Dear anonimnystefy,

I respect your opinion and absolutly agree with you "it is a mutter of convention". But the convention has been made by mathematicians to build a consistent body of knowledge that could help us to solve the real-life problems more Efficiently, so they distinguish between the sum of an infinite sieres (which is defind as the limit of the sequence of partial sums) and the decimal number of infinite repetition that is:

They are compeletly different concepts "by convention between mathematicians" for the reasons mentioned above "to build a coherent conceptual system". Also for the same reason:

Since the rational numbers are equivalent calsses so any members of a certain class can represent that class, but numbers of infinite repetition are not rational numbers ,in fact, this is one of the reasons that lead to the extension of the rational numbers into the real numbers.

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**anonimnystefy****Real Member**- From: Harlan's World
- Registered: 2011-05-23
- Posts: 16,037

Grantingriver wrote:

Since the rational numbers are equivalent calsses so any members of a certain class can represent that class, but numbers of infinite repetition are not rational numbers ,in fact, this is one of the reasons that lead to the extension of the rational numbers into the real numbers.

Wait what? Am I misunderstanding or do you also believe that 1/3≠0.(3)...?

Also, I see no inconsistencies in defining 0.(9) to be equal to 1, and thus see no argument against such a definition... What's more, defining 0.(9) to be anything other than 1 would be inconsistent, cause then the difference 1-0.(9) would be nonnegative, but also less than 1/n, for every natural number n, which would make it *have* to be 0. So, it's either that, or making it invalid notation.

*Last edited by anonimnystefy (2016-02-25 18:46:17)*

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

The knowledge of some things as a function of age is a delta function.

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**bob bundy****Administrator**- Registered: 2010-06-20
- Posts: 8,382

It's a property of the real number system that, between any pair of real numbers, there is another one. So it's reasonable to ask 'What comes between 0.9 recurring and 1?

Also all numbers are just defined to have values; there is no absolute authority that says what any number is worth. For example 2^(1/2) means what? Mathematicians have investigated the properties of powers and decided that square root of 2 is a sensible definition. In the same way many agree that 0.9 recurring = 1 is a sensible definition. You can reject it if you want but why bother ?

Once Isaac Azimov was giving a lecture (back in the days of blackboards) and a critic made fun of him for believing in 'imaginary numbers'. He handed the critic a stick of chalk and asked to be given back a half stick of chalk. His point: One half doesn't exist either. Numbers mean what you want them to mean. I feel that Alice might have said that.

Bob

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**Grantingriver****Member**- Registered: 2016-02-01
- Posts: 95

At first let's put it clear: I totally agree with both of you "it means what we define it to mean", but the definition has already been made. However, about the suspicions which you produced in your arguments, the mathematically consistent treatments for both arguments would be as stated below:

Infinity is not a number, infinity is a special concept, even if we use the symbol "∞" to represent it we don't mean it is an ordinary number, since it has properties which ordinary numbers are lack of. For this reason mathematicians created a coherent conceptual system to deal with it. So the first argument (which anonimnystefy has presented) should, according to the mathematical practice, go as follow:

So we have to use the concept of "limit" when we deal with this situation. the other argument (which is persented by bob and which, in fact, is another aspect of the first argument) treat "∞" (the number of 9s) as an ordinary number which we can define "precisely", but infinity is not an ordinary number and can not be reached, so for any natural number n threre will be a number between 1 and also there will be a difference between them. However, as n approaches ∞ (but could not reach it) the difference between these two numbers shrinks more and more (but will not be Zero) and since the limit of a function at "∞" is defined as the number the function approaches more and more, no matter how large the number is, as the independent variable appoaches (but not reach) ∞, therefore "by the definition" the difference between them will equal "zero" as a "limit".

Note: Someone may wonders "why we have to distinguish between these two concepts?, that is why:

The reason is simply because if we did not, we would be misleaded!! To see why, let's follow the reasoning of those who prefer to adopt the definition "" on the ground of that the more we add digits to ""the more we get close to "1" and since "∞" is so large then the difference will be Zero at the end. If we accept this argument we must accept also that for the same reason!! However, if we distinguish between them and adopt the "limit" concept we will reach the correct answer because the definition of the "limit" at "∞" states that the limit exists if for any ε>0 there is N>0 such that:

Hence applying this condition to these two cases we will have respectively:

And

Which are the correct results. These results hold since the infinite seires in both cases are geometric with general ratio less then one.

Q.E.F

*Last edited by Grantingriver (2016-02-26 05:44:51)*

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