View Full Version : Two reasons why .999...=1

Normandy6644

2005-Mar-16, 02:17 AM

I wanted to post this in one of the other threads, but I couldn't decide which and I it didn't really fit in either. I know these have both been mentioned several times now (including a few by me!), but it can't hurt to look at them again, especially if you are new to the discussion. Here they are, in all their beauty.

*Note: If it seems excessive to start yet another thread about this, by all means delete it and I can post this in one of the others.

http://img.photobucket.com/albums/v506/Normandy6644/ninerepeating.jpg

W.F. Tomba

2005-Mar-16, 03:03 AM

My math is very rusty, so I'm probably making a mistake here, but shouldn't that say 9/10^n in step 5 of the second one?

Severian

2005-Mar-16, 03:08 AM

Yeah, there are a couple of mistakes in the second one. The sum should be 9*(1/10^n). Otherwise, your line 7 doesn't make any sense; 1-9/10 = 1/10

Normandy6644

2005-Mar-16, 03:10 AM

Yep, fixed. Sorry, LaTeX can be screwy if you're in a hurry! :oops:

Severian

2005-Mar-16, 03:18 AM

Still not quite right.

You still have the 9/10 everywhere in line 7. You should have 9 \sum\limits_{n=1}^\infty \left ( \frac{1}{10} \right )^n = 9 \frac{\frac{1}{10}}{1-\frac{1}{10}}} = 1

or something

edit-heh I think you fixed it while I was typing this ;)

SciFi Chick

2005-Mar-16, 03:24 AM

The way I look at it, we have an entire forum dedicated to the lunar conspiracies. Until this math issue is resolved, I don't care how many threads it takes.

W.F. Tomba

2005-Mar-16, 03:33 AM

It still looks wrong to me. Look at step 7. You've got r=9/10 in the numerator and r=1/10 in the denominator.

r should be 1/10, shouldn't it? The sum is 9 times a geometric series with r=1/10.

That gives you 9*((1/10)/(1-(1/10)))

=9*((1/10)/(9/10))

=(9/10)/(9/10)

=1

Captain Kidd

2005-Mar-16, 03:34 AM

Aaaaaghhh! I mentioned these threads to my wife.

Her: Well, it doesn't

Me: Yes it does

Her: No, it doesn't

Me: Yes it does

Her: No, it doesn't

Me: Yes it does

Her: No, it doesn't

Me: Yes it does

.

.

.

You get the picture.

So, now I'm really going to follow this simple solution once those kinks get worked out I'll show it to her.

Severian

2005-Mar-16, 03:37 AM

It still looks wrong to me. Look at step 7. You've got r=9/10 in the numerator and r=1/10 in the denominator.

r should be 1/10, shouldn't it? The sum is 9 times a geometric series with r=1/10.

That gives you 9*((1/10)/(1-(1/10)))

=9*((1/10)/(9/10))

=(9/10)/(9/10)

=1

His line 7 has 9/10 in the numerator...

Yours has 9*(1/10)

He just multiplied the 9 in

W.F. Tomba

2005-Mar-16, 03:39 AM

OK, but in line 6 he still says that r=9/10. I guess that was confusing me.

Severian

2005-Mar-16, 03:40 AM

Oh yeah, he sure does. He doesn't mean it ;)

EvilBob

2005-Mar-16, 03:41 AM

Her: Well, it doesn't

Me: Yes it does

Her: No, it doesn't

Me: Yes it does

Her: No, it doesn't

Me: Yes it does

Her: No, it doesn't

Me: Yes it does

:lol:

I think you've just summed up the previous 3 threads on this subject! By the way, I haven't the math to follow the second method, but the first has completely clinched it for me. That's something I can show someone who doesn't agree....

Normandy6644

2005-Mar-16, 03:42 AM

Oh yeah, he sure does. He doesn't mean it ;)

Argh!! I will fix this thing eventually!

Inferno

2005-Mar-16, 03:43 AM

The way I look at it, we have an entire forum dedicated to the lunar conspiracies. Until this math issue is resolved, I don't care how many threads it takes.

We should add this to the file of "Who killed kennedy" and "How were the pyramids built". We're just never going to be able to solve it.

SciFi Chick

2005-Mar-16, 03:45 AM

The way I look at it, we have an entire forum dedicated to the lunar conspiracies. Until this math issue is resolved, I don't care how many threads it takes.

We should add this to the file of "Who killed kennedy" and "How were the pyramids built". We're just never going to be able to solve it.

I'm not sure I agree about the "who killed Kennedy", but I'm with you 100% on the "how were the pyramids built" :lol:

WaxRubiks

2005-Mar-16, 03:48 AM

9*(1/10) is the same as 9/10..

Normandy6644

2005-Mar-16, 03:49 AM

There, it should be okay now.

Note to self: Before posting something, make sure to doublecheck it! #-o

um3k

2005-Mar-16, 03:49 AM

"How were the pyramids built"

http://www.badastronomy.com/phpBB/viewtopic.php?p=435932#435932

Severian

2005-Mar-16, 03:51 AM

There, it should be okay now.

Note to self: Before posting something, make sure to doublecheck it! #-o

8-[ er..you still say r=9/10 between lines 5 and 6

WaxRubiks

2005-Mar-16, 03:56 AM

I suppose rule 6 is proven somewhere else, or is that common sense?

W.F. Tomba

2005-Mar-16, 03:56 AM

There, it should be okay now.

Note to self: Before posting something, make sure to doublecheck it! #-o

8-[ er..you still say r=9/10 between lines 5 and 6

Actually, he's fixed it; I see r=1/10 there. You might have to re-reload the page or something.

Severian

2005-Mar-16, 03:57 AM

ah, my bad :oops:

WaxRubiks

2005-Mar-16, 03:58 AM

RULE 6 ? 8-[

W.F. Tomba

2005-Mar-16, 04:03 AM

Note to self: Before posting something, make sure to doublecheck it! #-o

It's not your fault. There is a tiny malicious gremlin who lives between .999... and 1 and perpetuates the myth that the two numbers are not equal by sneaking errors into proofs of their equality. It is necessary for him to do this because if everyone agreed that .999...=1, he would be left homeless.

WaxRubiks

2005-Mar-16, 04:07 AM

can't be much of a home as it is.... 8-[

what with all the noise...

Nowhere Man

2005-Mar-16, 04:09 AM

Rule 6: There is NO ... rule six!

Rule 7: No poofters!

Bruce

WaxRubiks

2005-Mar-16, 04:20 AM

I'm glad that's been cleared up... :-?

Disinfo Agent

2005-Mar-16, 11:31 AM

I suppose rule 6 is proven somewhere else, or is that common sense?

Here you go: geometric series (http://www.answers.com/topic/geometric-progression).

farmerjumperdon

2005-Mar-16, 01:42 PM

Is the first version (the simple one, the non-calculus one), correct?

SciFi Chick

2005-Mar-16, 01:44 PM

http://img.photobucket.com/albums/v506/Normandy6644/ninerepeating.jpg

In that first equation, how do we decide that n=.9999...

Nicolas

2005-Mar-16, 01:50 PM

First line: "let n = 0.99999999..."

Disinfo Agent

2005-Mar-16, 01:50 PM

It's just a label we assign to the number denoted by "0.999...", to make the steps in the proof clearer. You could do the same without using a variable.

SciFi Chick

2005-Mar-16, 01:50 PM

First line: "let n = 0.99999999..."

Dear me. :oops: I better go get that second cup of coffee now. :o

It's always the obvious stuff that trips me up. :lol:

Disinfo Agent

2005-Mar-16, 01:56 PM

Is the first version (the simple one, the non-calculus one), correct?

I see nothing wrong with it. Of course, you could say that it "begs the question", in a sense. It assumes that the expression "0.999..." represents a unique real number, and that it is defined in such a way that the usual properties of real numbers that we all know and love remain valid. But then, why define it in any other way?

If we're very picky, then the first proof also assumes some knowledge of calculus, since infinite decimals like 0.333... can only be rigorously defined as limits. However, most people accept such expressions intuitively without protest (at least until we tell them that 0.999...=1 :wink:).

ToSeek

2005-Mar-16, 02:30 PM

RULE 6 ? 8-[

There is NO Rule 6!

WaxRubiks

2005-Mar-16, 02:57 PM

Perhaps this is where we are going wrong? :wink:

Eroica

2005-Mar-16, 04:21 PM

Is the first version (the simple one, the non-calculus one), correct?

I don't think it's a rigorous proof. How do we know that 10x0.9999.... = 9.9999.... ? Intuitively it looks right, but how would you actually do the calculation?

10xn means n+n+n+n+n+n+n+n+n+n. So 10x0.9999.... means:

0.9999....

0.9999....

0.9999....

0.9999....

0.9999....

0.9999....

0.9999....

0.9999....

0.9999....

0.9999....

__________

????????????

Since there's no right-hand column in which to start your addition, you can't even begin to do the sum.

Nowhere Man

2005-Mar-16, 04:26 PM

ToSeek: I beat you! :D

In the summation part, one must be careful how one simplifies the series:

9 * ((1/10) / (1 - 1/10))9 * ((1/10) / (9/10))9 * 1/9

At this point you can either say9/9 = 1which is cool,

or9 * 0.111111111... = 0.999999999...and you're right back where you started! ](*,)

Fred

W.F. Tomba

2005-Mar-16, 04:28 PM

Is the first version (the simple one, the non-calculus one), correct?

I don't think it's a rigorous proof. How do we know that 10x0.9999.... = 9.9999.... ? Intuitively it looks right, but how would you actually do the calculation?

10xn means n+n+n+n+n+n+n+n+n+n. So 10x0.9999.... means:

0.9999....

0.9999....

0.9999....

0.9999....

0.9999....

0.9999....

0.9999....

0.9999....

0.9999....

0.9999....

__________

????????????

Since there's no right-hand column in which to start your addition, you can't even begin to do the sum.

You don't have to do addition from right to left. That's just a convenience. You can go in any order you choose because of the associative property.

Also, I'm not sure that multiplication needs to be defined as a series of additions.

That said, it's not a rigorous proof, just a demonstration.

Disinfo Agent

2005-Mar-16, 04:30 PM

Eroica, multiplying by ten moves each digit one place to the left (it's one of those nice "properties of the real numbers we all know and love" :wink:). [Edit: See Frog march's post below.]

Edited to simplify and correct.

WaxRubiks

2005-Mar-16, 04:33 PM

The 9 in 0.9 is in the tenths column and so represents 9/10(nine tenths) and so when multiplied by 10 becomes 9.

The 9 in 0.09 is in the hundredths column and so represents 9/100(nine hundredths) and so when multiplied by 10 becomes 9/10 ie 0.9

.

.

.

.

.

etc

every thing just moves left by one place.

W.F. Tomba

2005-Mar-16, 04:34 PM

ToSeek: I beat you! :D

In the summation part, one must be careful how one simplifies the series:

9 * ((1/10) / (1 - 1/10))9 * ((1/10) / (9/10))9 * 1/9

At this point you can either say9/9 = 1which is cool,

or9 * 0.111111111... = 0.999999999...and you're right back where you started! ](*,)

Fred

But that in itself implies that .999...=1.

What these explanations really show is that .999... must be 1 as long as .111...=1/9, .333...=1/3, etc.

PyroFreak

2005-Mar-16, 04:40 PM

Actually, Nowhere Man, you just cleared it up for me.

Isn't it true that 9 * 1/9 = 1? and isn't 1/9 = .111111...... ?

Well, there you go. I already figured it equalled 1, but sometimes you got that little voice in your head that says : wait, you know its wrong!

Like my test I just took yesterday, how I got the right answer but then thought on it for the remainder of the test time, changed the answer, and ended up getting it wrong.....

kinda sucks.

Edit: ugg, Tomba you beat me to a reply to Fred's comment......

Nowhere Man

2005-Mar-16, 05:20 PM

But that in itself implies that .999...=1.

Which is the question that started the whole thing.

What these explanations really show is that .999... must be 1 as long as .111...=1/9, .333...=1/3, etc.

It works for me. Some people have trouble with infinity, as in the infinite 9s after the decimal point. I can't picture infinity in my mind, but I can still work with it as a concept or tool.

Fred

W.F. Tomba

2005-Mar-16, 05:24 PM

But that in itself implies that .999...=1.

Which is the question that started the whole thing.

Right, and it's exactly what the demonstration was trying to show. I'm saying that what you identified is not a problem, but another way of showing that .999...=1.

Disinfo Agent

2005-Mar-16, 05:27 PM

ToSeek: I beat you! :D

In the summation part, one must be careful how one simplifies the series:

9 * ((1/10) / (1 - 1/10))9 * ((1/10) / (9/10))9 * 1/9

At this point you can either say [...]

9 * 0.111111111... = 0.999999999...

Which leads to the trivial conclusion 0.999... = 0.999...

[or] 9/9 = 1

Which proves what we wished to prove.

No problem there. 8)

Normandy6644

2005-Mar-16, 07:04 PM

Yeah, the first one is definitely a non-rigorous way, but I always thought it was quite elegant and simple. The second one is rooted in more complex mathematics, but is certainly rigorous provided that you accept what calculus has shown about infinite series and things like that.

WaxRubiks

2005-Mar-16, 07:32 PM

I don't know if that IS calculus!!!

If you want another problem, try proving whether or not prime numbers ever end, or do they just keep finding higher and higher prime numbers, the more powerful computers get...?

8)

Infinite Series

"In his first paper on the Calculus (1669), Newton proudly introduced the use of infinite series to expedite the processes of the calculus... As Newton, leibnitz, the several Bernoullis, Euler, d'Alembert, Lagrange, and other 18th-century men stuggled with the strange problem of infinite series and employed them in analysis, they perpetuated all sorts of blunders, made false proofs, and drew incorrect conclusions; they even gave arguments that now with hindsight we are obliged to call ludicrous."

>From "MATHEMATICS: The Loss of Certainty" by Morris Kline.

Ok, I'm wrong again....!

jfribrg

2005-Mar-17, 07:41 PM

If you want another problem, try proving whether or not prime numbers ever end, or do they just keep finding higher and higher prime numbers, the more powerful computers get...?

The infinitude of primes has been known for thousands of years. And its an easy proof. In general, for any positive integer n (prime or otherwise), the prime factors of n!+1 are all > n.

Examples:

n=5, n!+1 = 121, Factors are 11,11

n = 6, n!+1 = 721, factors are 7,103

n=10, n!+1 = 39916801, prime

n = 12 , n!+1 = 479001601 , factors are 13,13,2834329

If the primes are finite, then there is a largest one. Call it p. Then p!+1 is not divisible by any number <= p. Therefore the prime factors of p!+1 are all > p, which contradicts the notion that p is the largest prime.

VTBoy

2005-Mar-17, 11:30 PM

Try to prove or disprove that R and R^2 and C are the same size, were R is the real number line, R^2 is the Plane of all real numbers, and C is the plane of complex numbers. Prove the set R, R^2, and C are all the same size.

Then prove, That for all n,x,y,m size of C^n = size C^x = size R^y = size R^m where n,x,y,m can be any number.

SciFi Chick

2005-Mar-17, 11:38 PM

Try to prove or disprove that R and R^2 and C are the same size, were R is the real number line, R^2 is the Plane of all real numbers, and C is the plane of complex numbers. Prove the set R, R^2, and C are all the same size.

Then prove, That for all n,x,y,m size of C^n = size C^x = size R^y = size R^m where n,x,y,m can be any number.

Why?

worzel

2005-Mar-18, 01:54 AM

Just curious Normandy6644, why didn't you include:

1/3=0.33..

3/3=0.33... + 0.33... + 0.33... = 0.99...

This is rather similar to your first proof I know, but it may just clinch it for some.

WaxRubiks

2005-Mar-18, 02:51 AM

Is Scifi chick realy banned? :o :o :o

:o

The Supreme Canuck

2005-Mar-18, 02:52 AM

Yep. I'm afraid so.

Normandy6644

2005-Mar-18, 02:52 AM

Just curious Normandy6644, why didn't you include:

1/3=0.33..

3/3=0.33... + 0.33... + 0.33... = 0.99...

This is rather similar to your first proof I know, but it may just clinch it for some.

I just about that today. I've mentioned it before, but I forgot to include it!

mickal555

2005-Mar-18, 03:24 AM

Yep. I'm afraid so. :cry:

VTBoy

2005-Mar-18, 07:54 AM

Try to prove or disprove that R and R^2 and C are the same size, were R is the real number line, R^2 is the Plane of all real numbers, and C is the plane of complex numbers. Prove the set R, R^2, and C are all the same size.

Then prove, That for all n,x,y,m size of C^n = size C^x = size R^y = size R^m where n,x,y,m can be any number.

Why?

Because it is something which seems impossible, but it is true. The idea that the space spaned by R^10 is the same size as the one spaned by R^1, seems like nonsense, but it is true.

Disinfo Agent

2005-Mar-18, 11:28 AM

It's rather OT for this thread, though. :-?

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