0.9 repeater = 1

[BillyJoe]

xouper,

I got it wrong only because you said it was a trick.
Your answer is that it is irrational because it neither repeats nor terminates. But then where is the trick.

BillyJoe.
(Don't tell me that saying there was a trick was actually the trick part )

 

[rwald]

As I understand it, he expected many people to be fooled into thinking that the "repeating" pattern of increasing zeros counted as a repeating decimal. Unfortunately, it didn't work out that way, and the "trick" part of it only confused people more. Oh well. As I said in my first response, it's very simple if you think about it.

 

[xouper]

BillyJoe: But then where is the trick.

For those who have a weak understanding of how rational numbers are defined, I tried to trick you into thinking this might be a rational number.

The trick is in the interpretation of the phrase "repeating pattern". The number .101001000100001... has a repeating pattern, but is it the kind that would make it a rational number? The trick is a test of how well you understand the relationship between decimal expansions and rational numbers.

Your answer is that it is irrational because it neither repeats nor terminates.

Correct. The trick is in the definition of "repeats". Rational numbers have a very narrow definition of what qualifies as a repeating pattern, whereas the number I gave has a more loosely defined repeating pattern.

 

[BillyJoe]

Fair enough.

But for me the trick would have been if, against all
appearances, the number turned out to be rational.

Oh well.

 

[xouper]

BillyJoe: But for me the trick would have been if, against all appearances, the number turned out to be rational.

OK. I apologize that my trick question didn't live up to expectations. Let the tomato throwing commence.

 

[bjornart]

Hey! Unca Cecil is copying us! Straight dope

You gotta love him though:

Doesn't that enhance your quality of life? Of course it does. Not that body paint doesn't have its place, but there's just no substitute for the pleasures of an infinite series.

 

[BillyJoe]

BODY PAINT CONTEXT.....

Q: Why are we talking about this incredibly esoteric topic when there are so many more interesting things to discuss, like whether Rebecca Romijn-Stamos is wearing anything besides blue body paint in the new X-Men movie?

A: Because it's cool, you wanker. Here, have a banana while the rest of us discuss.

BTW, bjornart, someone linked to this previously.

 

[bjornart]

Re: BODY PAINT CONTEXT.....

BTW, bjornart, someone linked to this previously.

Ah. Right. I should have remembered... it's just I didn't see it at the webpage until today. Shoot.

 

[atomx]

I agree that 0.9 repeating does not equal 1.

It is a number that approaches 1.

The coneptual abilities of our mind allow for an us to concieve of an infinitely miniscule difference between 0.9rep and 1.

As such as a calculation that approaches 0. There is a diffence there but miniscule it may be it still exists.

 

[gnome]

I agree that 0.9 repeating does not equal 1.

It is a number that approaches 1.

The coneptual abilities of our mind allow for an us to concieve of an infinitely miniscule difference between 0.9rep and 1.

As such as a calculation that approaches 0. There is a diffence there but miniscule it may be it still exists.

Ok, here we go again. Zombie thread!!!!!!

The number doesn't approach anything. It's not moving, or changing its value.

Please refer to this for some really good info:

http://qntm.org/pointnine

Or read the entire thread and make sure that your point has not already been addressed.

 

[marplots]

This should be moved to philosophy now. Because the mathematics has been well elucidated, but the human element hasn't. For instance, what does your rejection of well stated arguments tell me about you and how you evaluate the world?

I'm assuming the arguments were understood and then tossed aside in favor of an "inner instinct". I get that sometimes, the flash of insight. It's a springboard of creativity. But then comes the drudgery of seeing if the intuition will fit into a formal structure. Unfortunately, more often than not the flaws jump out and the great feeling of discovery gets swept into the dustbin. It can't just feel right, it has to be right.

Just sayin' though -- we've moved away from math into humanities.

 

[John Jones]

Why did I click on this thread?

 

[Soapy Sam]

"aaa..." = "b"

...for a certain meaning of "=" .

 

[This is The End]

This should be moved to philosophy now. Because the mathematics has been well elucidated.....

So I assume close only counts in horseshoes, hand grenades, and .9 repeating = 1??

Some (most?) of the proofs I see on the first page of this thread assume 1/3 = .3 repeating... as far as I remember from math (a long time ago, mind you) that is not so. .3 repeating is only an approximation of 1/3, though it is a very good one.

Imagine you have 3 centimeters (or 3 of anything of equal size), you can divide that into 3 whole centimeters, no repeating decimal needed.

Basically, what I'm saying is, you can't use .3 repeating = 1/3 in a proof of .9 repeating = 1. That's using the exact same assumption of what you are trying to prove.

 

[Molinaro]

I agree that 0.9 repeating does not equal 1.

It is a number that approaches 1.

The coneptual abilities of our mind allow for an us to concieve of an infinitely miniscule difference between 0.9rep and 1.

As such as a calculation that approaches 0. There is a diffence there but miniscule it may be it still exists.

False and it's easy to prove with just words.

For 2 numbers to not be equal, there must be some other number that lies between them.

No number lies between 0.99999... and 1.0 therefore they are the same number.


Also your statement that it is a number that approaches 1 is patently false. If you do anything other than consider an infinite number of repeating 9s, you are considering some other number and not the one given: 0.9999...

I find it amazing how often people do that when looking at this question. Not considering the number in question, and then saying ahah, is just plain silly.

 

[MetalPig]

Some (most?) of the proofs I see on the first page of this thread assume 1/3 = .3 repeating... as far as I remember from math (a long time ago, mind you) that is not so.

Can you do long division? Try it, and see if you don't get 3s all the way down.

 

[soylent]

I agree that 0.9 repeating does not equal 1.

It is a number that approaches 1.

The coneptual abilities of our mind allow for an us to concieve of an infinitely miniscule difference between 0.9rep and 1.

As such as a calculation that approaches 0. There is a diffence there but miniscule it may be it still exists.

It doesn't have to be harder than this: 1 - 0.999... = 0.000...

It might be tempting to write 0.000...1 or some-such; but what would that even mean? It's an infinite procession of zeroes, it can't be followed by a 1 at the end because there is no end, just more zeroes.

The gut feel that they are different stems from viewing 0.999... as a really long procession of 9's that must never the less terminate at some point. No, it's not just a really long chain of 9's, it doesn't end; it's infinite.

There's nothing approaching anything else, because nothing is moving.

 

[Mashuna]

I'm happy with the proof that 0.999...=1

But people who try to tell me that 99.999...=100 are just crazy.

 

[Marquis de Carabas]

0.999... only equals one if you understand math.

 

[sol invictus]

Some (most?) of the proofs I see on the first page of this thread assume 1/3 = .3 repeating... as far as I remember from math (a long time ago, mind you) that is not so. .3 repeating is only an approximation of 1/3, though it is a very good one.

You remember wrong.

(Try dividing 1 by 3 and see what you get.)