Simple mathematical problem (?)

[Zep]

Think of it this way:

What is the difference between 1 and 0.999(recurring)? The difference is infinitely close to 0, or in other words, the two are indistinguishable.

If 0.999(recurring) isn't 1 what on earth could it be? Is there a difference?

Well, it ISN'T 1 for a start. 1 is 1, that's all. 0.999(recurring) is INFINITELY CLOSE to 1 but it is NOT ACTUALLY 1.

Note also that when you multiply real numbers by integers that there is an increase in the inaccuracy in the lower order places. For example, 0.999 x 9 = 8.991. And 0.999999 x 9 = 8.999991. And so it is reasonable to take it that the "far end" of 0.999(recurring) will suffer a similar loss in accuracy as a result of the same arithmetical operation.

Furthermore, let's try applying the "proof" given to a more wacky example:

X = 12345
10X = 123450
10X - X = 123450-12345 = 111105
9X = 9 x 12345 = 111105

So the "proof" actually works with ANY real numbers, as it logically should, really. Leading to the question: What's the point?

 

[geni]

This assumes that you believe that 1/3 = .3333 (recurring)

You've just moved the problem to a different set of numbers

 

[DangerousBeliefs]

Here is a much better proof:

This assumes that you believe that 1/3 = .3333 (recurring)

1 = 3/3
= 3 * (1/3)
= 3 * .33333 (recurring)
= .999999 (recurring)


1 and .9999 (recurring) are the same number. They aren't close, they are the same.

You're assuming 1/3 = .333(recuring)

.333(recurring) is an approximation of 1/3

1 = .333(recurring)/.333(recurring)

1 = 1

 

[geni]

0*0.999(recuring)=0
1*0=0

therefor 0.999(recurring)=1

 

[Zep]

For an "approximation" of zero, think of the Cantor Set...

 

[boooeee]

Here's the thread:

Here we go again

 

[Vorticity]

I think the proof you are looking for is:

X = 0.999(recurring)
10X = 9.999(recurring)
10X - X = 9X
But, by virtue of the first two equations, 10X - X also equals 9 (the recurring 0.999's cancel out)
So, since 10X - X = 9X and 10X - X = 9,
9X = 9
X = 1

Yes, that's much better. The proof in the opening post had a typo in it. I think an even simpler way to put it is:

X=0.999(rec)
10X=9.999(rec)
10X-X=9.999(rec)-0.999(rec)
9X=9
X=1

As far as I can tell, this proof is valid, and X is (rigorously) equal to 1.
If you don't mind sums, another proof goes:

0.999(rec)
= Sum_(n=1)^Infinity 9*(1/10)^n
= 9*( Sum_(n=0)^Infinity (1/10)^n - 1 )
= 9*( 1/(1-(1/10)) - 1 )
= 9*( 10/9 - 1 )
= 9*( 1/9 )
= 1

and so on. I'm sure there are more rigorous proofs that start from the very basic axioms that define arithmetic and the decimal expansion system.

0.999(rec) = 1, with no qualifications or reservations.

 

[Suggestologist]

The difference is 1 infintesimel (which had better exist otherwise calculs has a serious problem)

Calculus is actually much easier to understand in terms of infinitessimals rather than limits, IMO. If I remember my mathematics history correctly, limit-based calculus came later than infinitessimal-based calculus.

 

[Earthborn]

If you Square both numbers an infinet number of times on goes to 0 and the other goes to 1 therefore they are different numbers.

No, they wouldn't. Don't be silly.

From a mathematical standpoint, they are unique values.

No, they are just two different ways of writing down the same number.

The difference is 1 infintesimel (which had better exist otherwise calculs has a serious problem)

I am sure it exists. And you'll also find that 0.(and infinite amount of zeroes)01 = 0

0.999(recurring) is INFINITELY CLOSE to 1 but it is NOT ACTUALLY 1.

What is the difference between 'infinitely close' and 'exactly the same' ? Nothing right?

And so it is reasonable to take it that the "far end" of 0.999(recurring) will suffer a similar loss in accuracy as a result of the same arithmetical operation.

And that 'loss in accuracy' is infinitely small. In fact, you'll never encounter it, no matter how low you take searching for it. Because it isn't there.

X = 0.999999
10X = 9.99999 {note the loss of decimal place already}
10X - X = 9X = 9 x 0.999999 = 8.999991
therefore X = 9X/9 = 8.999991/9 = 0.999999
Or, in other words, you have simply proven the identity.

But if you take 0.999(recurring) you'll never lose a decimal place, 'cause there is an infinite amount of them!

 

[Vorticity]

Here's an interesting proof that 0.999... = 1 at the bottom of this link:

http://mathcentral.uregina.ca/QQ/database/QQ.09.00/joan2.html

 

[boooeee]

Originally Posted by patoco12
Here is a much better proof:

This assumes that you believe that 1/3 = .3333 (recurring)

Here's an even better proof:

This assumes you believe me to be the unquestioned authority on all subjects:

0.999(recurring) = 1

QED

 

[patoco12]

You've just moved the problem to a different set of numbers

No...

I'm working in rational numbers.

 

[patoco12]

If you are having trouble coming to grips with the fact that .9 (recurring) is the same number as 1, then here is a tip:

Don't use a calculator. Calculators approximate real math. They will deceive you if you aren't careful.

 

[Suggestologist]

What is the difference between 'infinitely close' and 'exactly the same' ? Nothing right?

Wrong, if you go by Conway's surreal number system then the difference is "iota". But perhaps right in a naive real number system.

 

[Dymanic]

Originally posted by geni

The difference is 1 infintesimal (which had better exist otherwise calculus has a serious problem)

And the degree of precision to which such differences can be calculated had better be less than infinite, or chaos theory has a serious problem.

 

[DangerousBeliefs]

Yes, that's much better. The proof in the opening post had a typo in it. I think an even simpler way to put it is:

X=0.999(rec)
10X=9.999(rec)
10X-X=9.999(rec)-0.999(rec)
9X=9
X=1

But this proof does not work.

X=0.999(rec)
10X=9.999(rec)
10X-X=9.999(rec)-0.999(rec)
9X=9
X=1

So then 10X=9.999(rec) should be true if X=1 is true and it is not.

The only way this can work is by using circular reasoning.

 

[DangerousBeliefs]

Here's an even better proof:

This assumes you believe me to be the unquestioned authority on all subjects:

0.999(recurring) = 1

QED

Appeal to authority fallacy. Belief has nothing to do with fact.

 

[Suggestologist]

And the degree of precision to which such differences can be calculated had better be less than infinite, or chaos theory has a serious problem.

Maybe it just needs to use a more sophisticated number line, then.

 

[T'ai Chi]

3 * (1/3) = .999... = 1.

Or, for any tiny e>0, I can find the number of 9's to the right of the decimal point such that 1-.999... < e. Just tell me how close you want to get, and I'll give you .999.. with enough 9's to satisfy the inequality.

 

[Zep]

earthborn, I edited my previous post because I had a better idea instead! Sorry to make you read what I wrote first...

What is the difference between 'infinitely close' and 'exactly the same' ? Nothing right?

Wrong. There IS a difference, it IS infinitesimal, but it EXISTS.

Consider the two numbers 0.9 and 1.0. There exists on the real number line a number that is 90% of the way from 0.9 to 1.0, which is 0.99. That still leaves 10% of that gap to 1.0. There now exists a number 90% of the way from 0.99 to 1.0, 0.999, still leaving 10% of that gap to 1.0. And so on ad infinitum. So that, at any point in time, there remains that "gap" of 10% between whatever number 0.999(rec) at the level of accuracy you can conceive of now and the actual number 1.0.

Fun with scaling....!

And that 'loss in accuracy' is infinitely small. In fact, you'll never encounter it, no matter how low you take searching for it. Because it isn't there.

But it IS there. Although in practical terms, like if we were building bridges, angstrom units don't often come into the equations when measuring the length of steel beams, that's true.

Have you ever explored deep into those visual fractal patterns? The deeper you go into them, the higher the arithmetical accuracy required to calculate them. But at some point you run up against the accuracy of the computer itself, but NOT the accuracy of the fractal image. Similar concepts apply here.