Someone help me with the 0.9 repeating = 1 proof

[Eeezee]

I tried it two different ways

The pattern method
1/9 = 0.1 repeating
2/9 = 0.2 repeating
3/9 = 0.3 repeating
...
9/9 = 0.9 repeating = 1

And then I tried it this way
3/9 = 0.3 repeating
6/9 = 0.6 repeating
3/9 + 6/9 = 0.3 repeating + 0.6 repeating
9/9 = 0.9 repeating = 1

She still wouldn't believe me though! She keeps saying that since there is always something at the end of the string of numbers (despite being infinite) then it can be added together with something else to get 1 and is therefore not 1. I'm certain that you can't add anything to 0.9 repeating and get 1, but how can I prove this to her?

 

[byosys]

Ask her what number could possibly exist between .9 repeating and 1. If you can't fit a number between two other numbers (.9 repeating and 1.0), the numbers must be equal.

Edit: I reread your orginal post. My idea is essentially just another way of saying "what plus .9 repeating = 1," which was in your post. Hopefully it works for you.

 

[imported_inspire]

The first method you tried wasn't very good - the induction is funny. The real numbers are not countable, so that's where her problem is coming in. Algebra, in my opinion, lacks the vocabularly to sufficiently explain this. But, your Algebra is correct. You may be able to do it differently job if you can represent 1/3 and 2/3 as either continued fractions or infinite sums.

EDIT: The concept of showing that nothin plus .9 repeating is not an easy task. You're basically stuck with:

Let x=.9 (repeating). Let a+x=1 for some nonzero real, a. Now you have to use this and draw a contradiction (I think you'll invariably end up using your algebra arguement).

That, or some crazy elegant combinatorics-style proof. The algebra one you had was very succinct, but the confusion lies in the subtleties. I'll think about this some - I couldn't not think about it anyway...

Double-Edit: Got something for you - my pretzels looked like infinity and it got me thinking. The expressions are in LaTeX format.

\sum_{i=1}^{\infty} 10^{-i} = .1 (repeating) = 1/9
Thus:
9*(\sum_{i=1}^{\infty} 10^{-i}) = 9*(1/9) = 1

In english, the infinite sume from i=1 of ten raised to the negative i-th power is simply .1 (repeating), which is also expressed as 1/9. Multiply both sides by 9. Viola.

Pretty much a complicated version of what you're saying, but maybe it'll quiet her down.

 

[Born2bwire]

Originally posted by: Eeezee
She still wouldn't believe me though! She keeps saying that since there is always something at the end of the string of numbers (despite being infinite) then it can be added together with something else to get 1 and is therefore not 1. I'm certain that you can't add anything to 0.9 repeating and get 1, but how can I prove this to her?

Sounds like she is having trouble with the underlying concept of infinity. This is the kind of logical fallacy that people get caught up into whenever it is discussed at large on digg or ATOT. People truncate the problem to better visualize it but by doing so invariably change the number. I suggest getting a new girlfriend rather than trying to explain this further, it will be easier.

 

[imported_inspire]

Originally posted by: Born2bwire

Originally posted by: Eeezee
She still wouldn't believe me though! She keeps saying that since there is always something at the end of the string of numbers (despite being infinite) then it can be added together with something else to get 1 and is therefore not 1. I'm certain that you can't add anything to 0.9 repeating and get 1, but how can I prove this to her?

Sounds like she is having trouble with the underlying concept of infinity. This is the kind of logical fallacy that people get caught up into whenever it is discussed at large on digg or ATOT. People truncate the problem to better visualize it but by doing so invariably change the number. I suggest getting a new girlfriend rather than trying to explain this further, it will be easier.

Before you do that, though, you should go ahead and present Zeno's paradox to her and watch what she does. That would be funny.

 

[Aluvus]

Algebraic:

Let A = 0.999 (repeating)

Let B = 10 * A = 9.999 (repeating)

B - A = 10 * A - A = 9 * A
B - A = 9.999 (rep) - 0.999 (rep) = 9

Therefore,
9 * A = 9
A = 1
0.999 (repeating) = 1

 

[evilbix]

The proof is not true. It doesn't work.

You're simply doing a proof of 1 = 1. However, you end up with .999... because you chopped the number into a repeating decimal you approximate your result for 1/9. The problem with any electronic or paper method to trying to express 1/9th is the painfully obvious truth that at some point you're just gonna have to stop and approximate. It just comes down to the question of how precise you want your answer to be.

 

[stardrek]

Originally posted by: Eeezee
I tried it two different ways

The pattern method
1/9 = 0.1 repeating
2/9 = 0.2 repeating
3/9 = 0.3 repeating
...
9/9 = 0.9 repeating = 1

And then I tried it this way
3/9 = 0.3 repeating
6/9 = 0.6 repeating
3/9 + 6/9 = 0.3 repeating + 0.6 repeating
9/9 = 0.9 repeating = 1

I dislike the use of a fraction proof for this because it is, I think, a bad example. I feel it is a bad example because the decimal expression of a fraction is an estimate. But I do know a algebraic way of expressing this. A math prof in highschool showed this to me a while ago.

10x = 9.99 (repeating)
- x = .9999 (repeating) <---That is 10x minus x but it dosn't keep the spacing I used.
------------------
9x = 9

This shows x being equal to 1, because 9 * 1 = 9 while at the same time having x = .999999 (repeating) in the problem.

Mabye this will help.


Woops, looks like Aluvus already wrote this, sorry...

 

[imported_inspire]

Originally posted by: evilbix
The proof is not true. It doesn't work.

You're simply doing a proof of 1 = 1. However, you end up with .999... because you chopped the number into a repeating decimal you approximate your result for 1/9. The problem with any electronic or paper method to trying to express 1/9th is the painfully obvious truth that at some point you're just gonna have to stop and approximate. It just comes down to the question of how precise you want your answer to be.

Who were you talking to?

 

[Eeezee]

Originally posted by: byosys
Ask her what number could possibly exist between .9 repeating and 1. If you can't fit a number between two other numbers (.9 repeating and 1.0), the numbers must be equal.

Edit: I reread your orginal post. My idea is essentially just another way of saying "what plus .9 repeating = 1," which was in your post. Hopefully it works for you.

She says .0 repeating with a 1 on the end for however many 9s are in 0.9 repeating. I don't think she understands how there is no end to the 9s!

 

[BrownTown]

you can use google, or the search button on this forum to find many other proofs of this and choose the one you think works best. Personally I just consider it one of those silly little quirks in math that people think they are smart because they can repeat it. Kinda like that proof that 1=2.

 

[Eeezee]

Originally posted by: Born2bwire

Originally posted by: Eeezee
She still wouldn't believe me though! She keeps saying that since there is always something at the end of the string of numbers (despite being infinite) then it can be added together with something else to get 1 and is therefore not 1. I'm certain that you can't add anything to 0.9 repeating and get 1, but how can I prove this to her?

Sounds like she is having trouble with the underlying concept of infinity. This is the kind of logical fallacy that people get caught up into whenever it is discussed at large on digg or ATOT. People truncate the problem to better visualize it but by doing so invariably change the number. I suggest getting a new girlfriend rather than trying to explain this further, it will be easier.

Your suggestion of getting a new girlfriend made me laugh pretty hard. One of my friends suggested I prove it by typing 0.9 repeating in a TI-83 to see if it equaled one, and that made me laugh slightly harder

 

[Aluvus]

Originally posted by: Eeezee

Originally posted by: byosys
Ask her what number could possibly exist between .9 repeating and 1. If you can't fit a number between two other numbers (.9 repeating and 1.0), the numbers must be equal.

Edit: I reread your orginal post. My idea is essentially just another way of saying "what plus .9 repeating = 1," which was in your post. Hopefully it works for you.

She says .0 repeating with a 1 on the end for however many 9s are in 0.9 repeating. I don't think she understands how there is no end to the 9s!

The number she is describing is 1/infinity, which is identically zero.

 

[Eeezee]

Originally posted by: evilbix
The proof is not true. It doesn't work.

You're simply doing a proof of 1 = 1. However, you end up with .999... because you chopped the number into a repeating decimal you approximate your result for 1/9. The problem with any electronic or paper method to trying to express 1/9th is the painfully obvious truth that at some point you're just gonna have to stop and approximate. It just comes down to the question of how precise you want your answer to be.

Agreed, who were you talking to? 1/9 = 0.1 repeating infinitely without approximation. Same with 3/9 and 6/9, they repeat infinitely without approximation.

We're talking math here, so it's not the real world of finites

 

[Eeezee]

Originally posted by: inspire

Originally posted by: Born2bwire

Originally posted by: Eeezee
She still wouldn't believe me though! She keeps saying that since there is always something at the end of the string of numbers (despite being infinite) then it can be added together with something else to get 1 and is therefore not 1. I'm certain that you can't add anything to 0.9 repeating and get 1, but how can I prove this to her?

Sounds like she is having trouble with the underlying concept of infinity. This is the kind of logical fallacy that people get caught up into whenever it is discussed at large on digg or ATOT. People truncate the problem to better visualize it but by doing so invariably change the number. I suggest getting a new girlfriend rather than trying to explain this further, it will be easier.

Before you do that, though, you should go ahead and present Zeno's paradox to her and watch what she does. That would be funny.

I think you're right, that might actually get her to come to 0.9 repeating = 1 because it's another infinite set

 

[Eeezee]

Originally posted by: BrownTown
you can use google, or the search button on this forum to find many other proofs of this and choose the one you think works best. Personally I just consider it one of those silly little quirks in math that people think they are smart because they can repeat it. Kinda like that proof that 1=2.

I searched for infinite, infinity, 0.9 repeating, repeating, and 0.9 with no results

 

[byosys]

Originally posted by: Aluvus

The number she is describing is 1/infinity, which is identically zero.

1/infinity actually indeterminate, not zero.

She says .0 repeating with a 1 on the end for however many 9s are in 0.9 repeating. I don't think she understands how there is no end to the 9s!

OK, then ask what number could exist between 0 and 0.0(rep)1. This is the only way I can think of, but if she dosn't really get the consept of repeating decimals and how they do not end, I doubt this approach will work.

 

[Aluvus]

Originally posted by: byosys

Originally posted by: Aluvus

The number she is describing is 1/infinity, which is identically zero.

1/infinity actually indeterminate, not zero.

Any real (and non-infinite) number divided by infinity is zero. Infinity divided by infinity is undefined.

 

[imported_inspire]

That's not a good idea, byosys, since the betweeness of the real number line analogs to the uncountability of the real numbers. So, if you concede that such a number exists, you can always find one smaller.

 

[imported_inspire]

Originally posted by: Eeezee

Originally posted by: inspire

Originally posted by: Born2bwire

Originally posted by: Eeezee
She still wouldn't believe me though! She keeps saying that since there is always something at the end of the string of numbers (despite being infinite) then it can be added together with something else to get 1 and is therefore not 1. I'm certain that you can't add anything to 0.9 repeating and get 1, but how can I prove this to her?

Sounds like she is having trouble with the underlying concept of infinity. This is the kind of logical fallacy that people get caught up into whenever it is discussed at large on digg or ATOT. People truncate the problem to better visualize it but by doing so invariably change the number. I suggest getting a new girlfriend rather than trying to explain this further, it will be easier.

Before you do that, though, you should go ahead and present Zeno's paradox to her and watch what she does. That would be funny.

I think you're right, that might actually get her to come to 0.9 repeating = 1 because it's another infinite set

Yeah - sometimes people are more impressed by complicated proofs too - so you might like the one I did with the infinite series expansion of 1/9.

 

[Eeezee]

Originally posted by: Aluvus

Originally posted by: byosys

Originally posted by: Aluvus

The number she is describing is 1/infinity, which is identically zero.

1/infinity actually indeterminate, not zero.

Any real (and non-infinite) number divided by infinity is zero. Infinity divided by infinity is undefined.

What's zero times infinity? If it's zero, which I think it might be, then all of this falls apart.

 

[Eeezee]

Originally posted by: inspire

Originally posted by: Eeezee

Originally posted by: inspire

Originally posted by: Born2bwire

Originally posted by: Eeezee
She still wouldn't believe me though! She keeps saying that since there is always something at the end of the string of numbers (despite being infinite) then it can be added together with something else to get 1 and is therefore not 1. I'm certain that you can't add anything to 0.9 repeating and get 1, but how can I prove this to her?

Sounds like she is having trouble with the underlying concept of infinity. This is the kind of logical fallacy that people get caught up into whenever it is discussed at large on digg or ATOT. People truncate the problem to better visualize it but by doing so invariably change the number. I suggest getting a new girlfriend rather than trying to explain this further, it will be easier.

Before you do that, though, you should go ahead and present Zeno's paradox to her and watch what she does. That would be funny.

I think you're right, that might actually get her to come to 0.9 repeating = 1 because it's another infinite set

Yeah - sometimes people are more impressed by complicated proofs too - so you might like the one I did with the infinite series expansion of 1/9.

Yeah, it was a good proof, I might use it too

 

[BrownTown]

Originally posted by: Eeezee

Originally posted by: BrownTown
you can use google, or the search button on this forum to find many other proofs of this and choose the one you think works best. Personally I just consider it one of those silly little quirks in math that people think they are smart because they can repeat it. Kinda like that proof that 1=2.

I searched for infinite, infinity, 0.9 repeating, repeating, and 0.9 with no results

Google
can't seem to link the forum search, but just search in this forum for the number 9 will work, or the word repeating etc...

But then again, about 90% of the questions asked here can be solved in 5 minutes on google, so clearly people want more than just google to help them.

 

[byosys]

Originally posted by: inspire
That's not a good idea, byosys, since the betweeness of the real number line analogs to the uncountability of the real numbers. So, if you concede that such a number exists, you can always find one smaller.

I see your point. However, the point behind my logic is that .0(rept)1 does not exist.

Any real (and non-infinite) number divided by infinity is zero. Infinity divided by infinity is undefined.

******, I'm gonna have to go look this up. I always thought 1/infinity was undefined.

Edit: well, Maple says 1/infinity = 0, so I guess your right.

 

[TuxDave]

Originally posted by: Eeezee

Originally posted by: Aluvus

Originally posted by: byosys

Originally posted by: Aluvus

The number she is describing is 1/infinity, which is identically zero.

1/infinity actually indeterminate, not zero.

Any real (and non-infinite) number divided by infinity is zero. Infinity divided by infinity is undefined.

What's zero times infinity? If it's zero, which I think it might be, then all of this falls apart.

Zero times infinity is also undefined.