Simple mathematical problem (?)

[boooeee]

Originally Posted by DangerousBeliefs
Appeal to authority fallacy. Belief has nothing to do with fact.

OK, you caught me. I still stand by the original proof, though.

Could you elaborate why it is circular reasoning?

Why shouldn't 10X = 9.999(recurring)?

 

[patoco12]

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.
[/B]

Here is the problem: You can't stop at "any point in time" to compute the difference. .9~ repeats forever. That means that you can't ever take a point in time and compute the difference.

Here is a challenge: Write the number .9~ in fractional notation. That should convince you that it is the same number as 1.

 

[Vorticity]

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.

I admit that it could be that there is something wrong with this proof, but I'm not sure I am following your objection.

I start with the assumption that X=0.999...
Then, I multiply by 10. By the definition of the decimal system, this shifts that decimal point one space to the right, so we get 10X=0.999...

Can you explain in more detail the nature of the circularity?

Also, how about my (poorly typeset) sum proof? Any holes?

 

[Suggestologist]

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.

Exactly. Numbers aren't magic. They are NOT purely objective. They depend on what you are using them for, for their meaning.

Numbers can represent a number of a particular thing: How many apples do you have? Numbers can represent a distance: How far is A from B? But these are entirely DIFFERENT types of measures!

Numbers do not stand alone.

 

[Suggestologist]

Here is the problem: You can't stop at "any point in time" to compute the difference. .9~ repeats forever. That means that you can't ever take a point in time and compute the difference.

Here is a challenge: Write the number .9~ in fractional notation. That should convince you that it is the same number as 1.

Um, do you know the meaning of the term "irrational number"? Here is a challenge: Write the number Pi in fractional notation.

 

[patoco12]

Um, do you know the meaning of the term "irrational number"? Here is a challenge: Write the number Pi in fractional notation.

I sure do. However, .9~ is a rational number, so you should be able to write it in fractional notation.

Maybe you should look up these definitions.

 

[Suggestologist]

I sure do. However, .9~ is a rational number, so you should be able to write it in fractional notation.

Maybe you should look up these definitions.

Your argument is circular. If you assume that .9... is rational then it can be written as a fraction. If you don't assume that, then it can't.

 

[Earthborn]

Ask Dr. Math:

No matter how many 9's
you add, you'll never get all the way to 1.

But that's how it seems if you think about moving _toward_ 1. What if
you think about moving _away_ from 1?

That is, if you start at 1, and try to move away from 1 and toward
0.99999..., how far do you have to go to get to 0.99999... ? Any step
you try to take will be too far, so you can't really move at all -
which means that to move from 1 to 0.99999..., you have to stay at 1.

Which means they must be the same thing!

Suggestologist:

If you assume that .9... is rational then it can be written as a fraction.

You can write it down as a fraction: 3/3, which amazingly equals 1.

 

[Suggestologist]

Ask Dr. Math:Suggestologist:You can write it down as a fraction: 3/3, which amazingly equals 1.

But 3/3 = 1. Not .9999999.....

 

[patoco12]

Your argument is circular. If you assume that .9... is rational then it can be written as a fraction. If you don't assume that, then it can't.

Fair enough.

Irrational numbers are defined as numbers whos decimal expansion both is infinate AND never enters a pattern.

But .9~ has a clear pattern. So .9~ is not irrational.

But the union of the set of rationals and irrationals is the set of reals. So if .9~ is not irrational, then it is rational.

So I'm not assuming .9~ is rational. It IS rational.

 

[Earthborn]

1/3 = .33(recurring, which means an infinite number of 3s)
3/3 = 3 * .33(recurring) = 0.9(recurring, which means an infinite number of 9s) = 1

 

[Suggestologist]

Ask Dr. Math:Suggestologist:You can write it down as a fraction: 3/3, which amazingly equals 1.

As Conway has shown, Dr. Math is wrong. You can move "iota" down from 1 to get to .999999...

Dr. Math should be ashamed of himself for stifling little Emily's and little Jenny's development of their own idea of how numbers work, just so that they can fit into the school system's ideas of what is right and what is wrong. Another cog in the machine......

 

[Suggestologist]

Fair enough.

Irrational numbers are defined as numbers whos decimal expansion both is infinate AND never enters a pattern.

But .9~ has a clear pattern. So .9~ is not irrational.

But the union of the set of rationals and irrationals is the set of reals. So if .9~ is not irrational, then it is rational.

So I'm not assuming .9~ is rational. It IS rational.

Well, it does depend on your perspective. If you believe the definition of rational number that you have described, then obviously .9.... is rational. But that's still a circular argument from definition. My preferred definition of a rational number might go something like: can be UNIQUELY represented as the ratio of two integers. But, If .9.... = 1, then why doesn't 1 divided by 1 produce .9..... ?

 

[DangerousBeliefs]

OK, you caught me. I still stand by the original proof, though.

Could you elaborate why it is circular reasoning?

Why shouldn't 10X = 9.999(recurring)?

You're saying A is B, therefore A is B

Or X = .999(recurring) = 1

10X = 9.999(recurring) is true if X =.999(recurring) NOT 1

Therefore X <> 1

 

[Suggestologist]

Fair enough.

Irrational numbers are defined as numbers whos decimal expansion both is infinate AND never enters a pattern.

But .9~ has a clear pattern. So .9~ is not irrational.

But the union of the set of rationals and irrationals is the set of reals. So if .9~ is not irrational, then it is rational.

So I'm not assuming .9~ is rational. It IS rational.

Let me put it this way. What are the integers required to be placed in the nominator and denominator of the ratio that produces .9999999... ?

The nominator would look something like 99999999........
The denominator would look something like 100000000.......

Which amounts to infinity-1/infinity --- which is UNDEFINED!

 

[Skeptoid]

A sum of rationals is itself rational. 9/10¹ + 9/10² + 9/10³ ... = 0.999... = 1.

 

[patoco12]

If you believe the definition of rational number that you have described, then obviously .9.... is rational.

I believe it because it IS the definition. I have only an objective perspective based on fact.

Let me put it this way. What are the integers required to be placed in the nominator and denominator of the ratio that produces .9999999... ?

The nominator would look something like 99999999........
The denominator would look something like 100000000.......

Which amounts to infinity-1/infinity --- which is UNDEFINED!

The numbers 99999999.... and 10000000000... aren't integers.

 

[Suggestologist]

I believe it because it IS the definition. I have only an objective perspective based on fact.



The numbers 99999999.... and 10000000000... aren't integers.

But they are what would be required to produce the sequence .99999999..... as the result of converting a ratio of two "integers" into decimal form. A rational number.

 

[Suggestologist]

A sum of rationals is itself rational. 9/10¹ + 9/10² + 9/10³ ... = 0.999... = 1.

Sure, but how do you get SUMMA x=1 to infinity (9/10^x) to equal 1? It gets closer and closer .... and closer and closer, but it never actually gets there. Does it?

 

[Suggestologist]

I believe it because it IS the definition. I have only an objective perspective based on fact.

Numbers are not objective. Their meaning, and the ways you can use them, depends on what you're using them to calculate.