What is .99 repeating as a fraction?

Quote: JyBrd0403

It's whatever the hell 5 divided by 6 equals. I've really enjoyed this tonight, but I have to go to bed it's 5am. I really need to get up tomorrow. Good night. Oh to answer the question its .8333333.

This is incorrect. It's 0.83333... the recurring symbol is important...

I'd like to see the counter proof that 1/3 != 0.3333.... (or that 0.99999.... != 1, which ever you desire) in a succinct way. I'm having trouble understanding how 0.3333.... is claimed not to be 1/3 but an approximation.

Quote: JyBrd0403

It's whatever the hell 5 divided by 6 equals. I've really enjoyed this tonight, but I have to go to bed it's 5am. I really need to get up tomorrow. Good night. Oh to answer the question its .8333333.

This is incorrect. It's 0.83333... the recurring symbol is important...

I'd like to see the counter proof that 1/3 != 0.3333.... (or that 0.99999.... != 1, which ever you desire) in a succinct way. I'm having trouble understanding how 0.3333.... is claimed not to be 1/3 but an approximation.

The counter "proof" is "sorry I do not understand the meaning of the word 'infinity'".

Quote: YoDiceRoll11

No.

The problem is with the infinite nature of .333... and .666... and .999... and any other repeater. If it goes on forever, it won't equal it's "usual ratio" that most people express it with (like 1/3= .333...).

The argument says that for 1/3 to equal .333..., infinity has to be stopped, which it can't.

Real extrapolation of real integers into infinity can be accomplished until the end of time.

I think I see the problem. You're assuming that the notation 0.333... involves a temporal dimension. That is, if you wait a bit longer, 0.333... has more threes in it than 0.333... did a few seconds ago. That's incorrect.

By definition, the notation 0.333..., or alternately 0.3(3) or 0.3 with a bar over the 3 (I wish I knew how to format that here) means "zero followed by a decimal point followed by an infinite number of threes." It doesn't mean an increasing number of threes, and it isn't growing. If you attempted to write down the number, you would never stop (hence infinity), but that's why the notation exists -- so you don't have to. The sole purpose of the written notation is to codify a concept for communication purposes, so if you disagree with the meaning behind the notation, there's not much to discuss. It would be like arguing that 1 != 1 because your notation for 1 doesn't mean the same thing as everyone else's notation for 1. The appropriate reply is: that's not what I mean when I write "1". Similarly, what you apparently mean when you write 0.333... isn't what I mean when I write 0.333... If we're not talking about the same concept, the notation won't help.

If you can accept what 0.111... or 0.333... or 0.999... mean -- respectively, a zero followed by a decimal point followed by an infinite number of 1s, 3s, or 9s -- then here is a simple transformation that demonstrates 0.999... = 1.

First, presumably you accept that 0.111... + 0.111... = 0.222...
Then, presumably you also accept that 0.444... + 0.555... = 0.999...
Then, presumably you also accept that 0.111... - 0.111... = 0
Then, presumably you also accept that 1.111... - 1.0 = 0.111... and 1.111... - 0.111... = 1.0

Now consider that 0.999... + 0.999... = 1.999... (try writing out the first few digits if you don't believe me)
Then subtract 0.999... from both sides of the equation, resulting in
0.999... + 0.999... - 0.999... = 1.999... - 0.999...
reducing to
0.999... = 1.0
Q.E.D.

Quote: Doc

Using that thought/belief/conclusion that it can't be expressed as 1/3 to prove that it can't be expressed as 1/3 doesn't strike me as following sound logic. What did I miss there?

Circular logic is never a good thing, especially in math.

I'll try to make it simpler: Just allow for the sake of this argument to get rid of the idea of infinity, I know it's tough, now tell me that .3333 exactly equals 1/3. You can't. Now bring back infinity, .333...., tell me when the numbers stop and it equals exactly 1/3. You can't, without making some interesting assumptions, again based on what number set, and rules you use.

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And here are the other thing(s) that I don't follow. What's with "infinity has to be stopped"? And your "until the end of time" comment seems to be suggesting that a for a number to have an infinite number of decimal digits consumes time. Why?

:), this is more of a simplistic way of viewing it, philosophically if you will. Pretend that it takes one second to add .3 to .3 to get .33. Now extrapolate. You will always have exactly one more .3. Now I know the math majority will jump on that and say that infinity +1 is a terrible idea and doesn't exist. I say prove that it doesn't with a real example instead of an irrational proof.

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like .333333.........................4. That does not represent the same thing as .333... ,

Exactly. They aren't the same.

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Can you clarify why you believe that the infinite repeating decimal must be truncated or stopped or some such or that we must go to the end of time rather than having infinite digits immediately?

Very good question. The answer is you don't have to wait until the end of time to have infinite digits. You can have infinite digits of .333.... right now. It is infinite. But even infinity has it's own limit (or non-limit) that you can graph. You have an infinite digit repeater of .3333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333........... let me know when this equals 1/3.

The reason why I use the stop time analogy, and the reason the proponents of the argument I agree with do also, is that you have to stop that sequence and round up by a quadzillionth, to get to 1/3. Otherwise you will never get there, because infinity is.......forever.

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Separate topic:

The Philosophy of Mathematics via Stanford.

[See also Wikipedia Article on Philosophy of Math

Quote: weaselman

Yeah, that "certain set of rules" is called "mathematics". :) If you change the rules of math, all bets are off. You can even say, that under your new rules 1 does not exist at all. But who cares?

LOL, you know very well that isn't what I mean.

Read, alternative number systems in applied mathematics.

Quote: MathExtremist

I think I see the problem. You're assuming that the notation 0.333... involves a temporal dimension. That is, if you wait a bit longer, 0.333... has more threes in it than 0.333... did a few seconds ago. That's incorrect.

I was simplifying it. Yes, you are correct. I'm not assuming .333...involves any time at all. I was looking to explain it a different way.

Quote:

By definition, the notation 0.333..., or alternately 0.3(3) or 0.3 with a bar over the 3 (I wish I knew how to format that here) means "zero followed by a decimal point followed by an infinite number of threes." It doesn't mean an increasing number of threes, and it isn't growing.

Correct. And so is the rest of your explanation. I already said, over four times, I understand the math behind this.

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Now consider that 0.999... + 0.999... = 1.999... (try writing out the first few digits if you don't believe me)

This is another spot where we disagree, even though the numbers go on to infinity, to add them together would be impossible under my argument. Since they cannot be quantified, because they go on FOREVER, it isn't rational to add them together at any certain point in time. If you did, you would get something that looked like this: 1.9999998...

My argument (which I will remind everyone, isn't just mine) is that you literally can't add an infinite decimal, because in the number system I accept, for this argument doesn't allow the computation of a non quantified infinite number.

Edit: MathExtremist, dude much respect to you btw, I just can't allow what I believe to be cutting a corner by allowing one to add an infinite decimal. It's like saying that you can add pi plus pi and give me a rational number.

I didn't read this whole thread to see if this was already included, so apologies if it's already been discussed, but if not... (and that's not "not" repeating, by the way)

I wanted to see how the calculator in my MacBook Pro would interpret 0.9... divided by 2. So here's what I did:

1 ÷ 3 = (0.333333333333333...)

x 3 = (0.99999999...)

÷ 2 = (0.5)

Yes, the answer was 0.5! Apparently my Mac knows that 0.9... is equal to 1.

On a consumer calculator, the answer will be 0.4999999..., though.

Quote: YoDiceRoll11

LOL, you know very well that isn't what I mean.

No, as a matter of fact, I don't.

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Read, alternative number systems in applied mathematics.

Which system in particular do you have in mind that would support your view?

:),
My ghetto calculator shows the following:
1/3 = .3333333333333333333.....

x 3 = 1

It's my assumption and belief that this calculator, and the MacBook pro, are rounding up for simplicity.