What is .99 repeating as a fraction?
So your argument is that 1/3 != .333...
This is what I want you to do.
Using the old methods of long division, take 1/3 and represent the number as a decimal.
When you are done, post here and let me know what you get.
Quote: MarkAbeI think that YoDiceRoll11 is channeling Zeno of Elea. His analysis of why you can never walk to a wall (because you have to walk half-way first, then half-way again,etc.) will generate the binary fraction .111......, which also equals 1. It took around 2000 years for mathematics to come up with limit theory, which finally explains both the decimal .9999..... and the binary .111.....
Oh hell no. I hate that so called "paradox" problem where you have to complete an infinite series of steps to get to a goal therefore never being able to attain it. Bleghhh.
But good guess.
Quote: TriplellI was responding to the other guys post, which he claimed otherwise.
So your argument is that 1/3 != .333...
This is what I want you to do.
Using the old methods of long division, take 1/3 and represent the number as a decimal.
When you are done, post here and let me know what you get.
The point of the side I agree with, is that you CAN'T represent 1/3 as a decimal, at least a rational one. Let me break this down further.
1/3 = .333... assuming this number forever gets extremely close to 1/3 but never reaching it (irrational decimal notation)
Since the numbers repeat forever, to quantify it, you have to stop time (in a sense, depending on how you look at it) and quantify it as 1/3= .3333333.......................................................................................................................................................4
It's hard to take in, I know. And this is where the disagreement lies. You can disagree, that's fine, like I said, I understand WHY most people would disagree with me, that's ok. Does that make me stupid? I think not.
It is a two step process, and most people disagree. Fine by me.
then what is 1/3 x 3? Is it .99999.... or 1?
All mathematical theorists say that it's 1.
So you are free to write your own mathematical theorems... after all, it's only math.
Quote: boymimboIf 1/3 = .33333....
then what is 1/3 x 3? Is it .99999.... or 1?
All mathematical theorists say that it's 1.
So you are free to write your own mathematical theorems... after all, it's only math.
Its neither...the correct answer is 3/3
Quote: boymimboIf 1/3 = .33333....
then what is 1/3 x 3? Is it .99999.... or 1?
All mathematical theorists say that it's 1.
So you are free to write your own mathematical theorems... after all, it's only math.
You misunderstand.
1/3 *3= 1. Duh.
1/3 != .333... depending on what rules you are using.
That is what my side of the argument states.
Quote: TriplellSo you disagree with the definition of an irrational number, which is a number that cannot be represented as a terminating or repeating decimal.
No.
From my math book:
Quote: MathBookan irrational number is a real number that can't be expressed as a ratio: (a/b), where a and b are integers, b being a non-zero, and is therefore not a rational number.
Basically, an irrational number can't be represented as a simple fraction.
The point being made, is that .333...is irrational and therefore can't be expressed as 1/3.
Quote: YoDiceRoll11No.
From my math book:Quote: MathBookan irrational number is a real number that can't be expressed as a ratio: (a/b), where a and b are integers, b being a non-zero, and is therefore not a rational number.
Basically, an irrational number can't be represented as a simple fraction.
The point being made, is that .333...is irrational.
but it's not, it's represented as the ratio of a = 1 and b = 3
Quote: TriplellQuote: YoDiceRoll11No.
From my math book:Quote: MathBookan irrational number is a real number that can't be expressed as a ratio: (a/b), where a and b are integers, b being a non-zero, and is therefore not a rational number.
Basically, an irrational number can't be represented as a simple fraction.
The point being made, is that .333...is irrational.
but it's not, it's represented as the ratio of a = 1 and b = 3
Right, but the argument is made in reverse, not that 1/3 can't be expressed as .333..., but that .333... can't be expressed as 1/3.
No.Quote: Triplellso basically, you are saying certain fractions cannot be represented as decimals, when really, all fractions can be represented as decimals...
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. (Example: .30 cents *3 = .90 cents, .33 cents *3 = .99 cents, .333 cents (for the sake of the argument) equals .999 cents. You can't ever extrapolate this infinite line to 1 dollar, unless you stop the infinite line and round up. I think I've repeated this enough.
It's ok to disagree, really. I've said it before, and maybe you are internet deaf (blind) but, I UNDERSTAND the math that says .999... = 1. It is logical in the sense that it operates under a certain set of rules. Change those rules however and .999... != 1.
Look up Philosophy of math, report back how disgusted with it you are.
Quote: YoDiceRoll11Quote: MathBook
an irrational number is a real number that can't be expressed as a ratio: (a/b), where a and b are integers, b being a non-zero, and is therefore not a rational number.
Basically, an irrational number can't be represented as a simple fraction.
The point being made, is that .333...is irrational and therefore can't be expressed as 1/3.
So here is the first thing I don't follow. How does your MathBook's definition of an irrational number lead to the conclusion that .333... is an irrational number "and therefore can't be expressed as 1/3"? It appears that you concluded that it can't be expressed as 1/3 prior to (and as a part of) determining that it is irrational. 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?
Quote: YoDiceRoll11The 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.
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? I'm not talking about writing down all of the digits, just that an infinite number of them already exist in the number which we represent as .333... or one of the others. This relates to all of your examples like .333333.........................4. That does not represent the same thing as .333... , which in fact never terminates or gets rounded up or down. 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?
Separate topic:
Quote: YoDiceRoll11Look up the philosophy of math, wow, it is worse than math.
I haven't looked it up, but I suspect Rene Descartes would be a good place to start.
Quote: DocCan 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?
Well, I'm still drunk, perhaps a little drunker than earlier tonight, but, I'm going to guess that it is because you can't have infinite digits immediately. You're going to have to come to an agreement on the terms of infinity or else you'll have to count to infinity.
Did I clarify the point, or do you care to count to infinity for us all?
Somebody already gave some nice formula of how an infinite sum of smaller and smaller getting numbers equal to a finite number.
So, if i have 1/3 (one part from 3) as a decimal i can only represent it as 0.3 recurring but that is 1 part out of 3. Lets say you dont have a line with 3 segments, but an urn with 3 balls, out of which one is black. How do you write this? is it not 1/3? and in decimals it would be 0.3 recurring. Oh, then i dont have a whole ball right?
Quote: edwardAs far as i remember infinity is not a number, it is a mathematical convention so you dont have to count anything. If you take a line and divide it into 3 equal segments, you can argue that that is impossible because you have an infinite number of points from A to B then from B to C and C to D.
Right, so don't count to anything, or in other words "nothing". Same as "infinity". Yes, divide the line into 3 segments and you will never come to a whole of one that you started with. Why is that? Maybe because of that 0.9999999....9 thing we were talking about. There's some molecules and protons and neutrons and electrons that will get lost. At least from the equation that we are talking about. Get it?
Now, on the other hand, to not get too serious about this stuff, and especially not to drive anyone completely nuts, if you combine those 3 segments together they will equal 1 whole segment. Get it? There will still be a line, just not the line you started with.
But, to answer your question, yes there will be an infinite number of points from A to B then from B to C and C to D. It's the difference between reality and probabilities. Probabilities are guessed at, reality just happens.
Okay, so I'm a lot more drunk then I was earlier tonight, but I still think i'm right :)
Quote: JyBrd0403Right, so don't count to anything, or in other words "nothing". Same as "infinity". Yes, divide the line into 3 segments and you will never come to a whole of one that you started with. Why is that? Maybe because of that 0.9999999....9 thing we were talking about. There's some molecules and protons and neutrons and electrons that will get lost. At least from the equation that we are talking about. Get it?
Now, on the other hand, to not get too serious about this stuff, and especially not to drive anyone completely nuts, if you combine those 3 segments together they will equal 1 whole segment. Get it? There will still be a line, just not the line you started with.
But, to answer your question, yes there will be an infinite number of points from A to B then from B to C and C to D. It's the difference between reality and probabilities. Probabilities are guessed at, reality just happens.
Okay, so I'm a lot more drunk then I was earlier tonight, but I still think i'm right :)
But what if i would break the first segment into another half and combine it ?
It's that 0.999...9% thing we were talking about. You have the same thing, but something got lost.
Quote: JyBrd0403Did any protons or neutrons or electrons or anything else get lost in the process. My guess is that in reality they did, mathematically I'd say you'd have the same 1.
It's that 0.999...9% thing we were talking about. You have the same thing, but something got lost.
What if i have 3 beers out of which one is heineken? how do you represent this in decimals?
Quote: YoDiceRoll11Incorrect assumption. I don't think they are approximations. I was merely summarizing the initial portion of the argument I agree with Please don't mistake that for the entire reasoning.
What's your reasoning then?
Quote: YoDiceRoll11The point of the side I agree with, is that you CAN'T represent 1/3 as a decimal, at least a rational one.
1/3 = .333... assuming this number forever gets extremely close to 1/3 but never reaching it (irrational decimal notation)
,,,
The argument says that for 1/3 to equal .333..., infinity has to be stopped, which it can't.
You are making two mistakes. First, as I said before, a number is not the same thing as the process of writing it down. The number does not "go" anywhere, forever or otherwise, does not attempt to "reach" anything, or "get close" to it. It just is.
Similarly, infinity is just a cardinal quantity, it cannot be "stopped", since it is not going anywhere. It just is, just like, say, number "1". Can you "stop" a 1?
Second, you seem to mistakenly think that all infinitely long decimals are non-rational, that is not true. Again, as I said before, 0.333... is a rational decimal, as is any other periodic fraction. In fact, the phrase "rational decimal" does not really make very much sense at all, as rationality is the property of the number itself, not of any particular way to express it. The only sensible way to use the phrase "rational decimal" is to mean "a decimal fraction expressing a rational number". All "rational decimals" are either finite or periodic, but that, again, is not a fundamental property of rational numbers, that they must have a finite representation, but rather an artifact of our (rational-based) positional system. I mentioned it before, there are some exotic positional systems where all rational numbers (including integers) require infinite number of digits to be expressed.
Quote:It's ok to disagree, really. I've said it before, and maybe you are internet deaf (blind) but, I UNDERSTAND the math that says .999... = 1. It is logical in the sense that it operates under a certain set of rules. Change those rules however and .999... != 1.
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?
I would suggest that, if you want to discuss a set of rules, different from that used in math, you should change the terminology as well to avoid confusion. You see, when you use things like "0.999...", "rational", "decimal", "!=", or "1" etc., people are going to assume you are talking about math. If you just said something like "^%$#* .. !#@$# because you can't stop foo and express bar as baz", it would become immediately clear, that you are not talking about math as we know it, and under the rules you are using you might very well be right (some could even be persistent enough to try to find out from you what those rules you are using really are, and why in the world you think they are useful :))
Quote: edwardWhat if i have 3 beers out of which one is heineken? how do you represent this in decimals?
LOL That's easy .333 LOL
Care to argue me on this point. LOL We've already agreed to this point LOL
Quote: JyBrd0403LOL That's easy .333 LOL
what if you would have 6 beers out of which 5 is heineken?
Quote: JyBrd0403Out of 6 beers which 5 is heineken? Need more information to answer the question, with any degree of certainty. For instance, how can you have 6 beers at one point, and then only have 5 beers to choose from how many are heinekens. Did somebody drink one of the beers?
how do you represent in decimals that you have 5 heinekens from a total of 6
the whiskey bottle is somehwere else
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?
I would suggest that, if you want to discuss a set of rules, different from that used in math, you should change the terminology as well to avoid confusion. You see, when you use things like "0.999...", "rational", "decimal", "!=", or "1" etc., people are going to assume you are talking about math. If you just said something like "^%$#* .. !#@$# because you can't stop foo and express bar as baz", it would become immediately clear, that you are not talking about math as we know it, and under the rules you are using you might very well be right (some could even be persistent enough to try to find out from you what those rules you are using really are, and why in the world you think they are useful :))
This was kind of the point I was trying to make earlier. Mathematics is a science...it is not up to debate, and you look like an idiot arguing against it because 1) It's not someone's opinion and 2) .999... = 1 is the correct answer. By saying you disagree with this is just foolish.
It would be the same as if I killed someone in this country and then argued with the judge that under my definition of the law, I am innocent of any crimes.
Quote: JyBrd0403It'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: JyBrd0403It'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: YoDiceRoll11No.
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: DocUsing 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.
Quote: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.
Quote:like .333333.........................4. That does not represent the same thing as .333... ,
Exactly. They aren't the same.
Quote: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.
Quote: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: MathExtremistI 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.
Quote: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 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: YoDiceRoll11LOL, you know very well that isn't what I mean.
No, as a matter of fact, I don't.
Quote: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.
Quote: MichaelBluejay
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.
... which is the same thing as 0.5 apparently :)
Quote: YoDiceRoll11:),
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.
Maybe it's not using any "alternative number system" (kinda like the rest of us who assume, that when you talk about "numbers" without any further qualification, you mean the "mainstream" interpretation assumed by default)?
Quote: weaselmanNo, as a matter of fact, I don't.
Which system in particular do you have in mind that would support your view?
You already said if you change the rules of math, "all bets are off". So I won't even bother since you clearly don't believe in alternative number theory.
Quote: weaselman... which is the same thing as 0.5 apparently :)
:)
Quote: weaselmanMaybe it's not using any "alternative number system" (kinda like the rest of us who assume, that when you talk about "numbers" without any further qualification, you mean the "mainstream" interpretation assumed by default)?
Well, you could dive very deep into this, and say that it is using alternative number system. One where you can just quantify an irrational, infinite number, and dumb it down to equal a rational integer. Prove that wrong.
Quote: weaselmanWhich system in particular do you have in mind that would support your view?
I'll go ahead and be nice, and say that a hyperreal number system supports the idea that an infinite decimal is irrational and never fully reaches the next full integer.
Quote: YoDiceRoll11You already said if you change the rules of math, "all bets are off". So I won't even bother since you clearly don't believe in alternative number theory.
A scientific theory is not a God for me to believe in. There are several alternative number theories I know of, and neither of them AFAICS supports the point of view you are defending as such much less violates or discards any of the mathematical "rules".
In any event, when you are using mathematical terms without further qualification, the assumption is that you are operating in the realm of the "default", standard theory. If you really want to invoke one of the (many) alternatives, you have to explicitly specify which one, thus my question to you.
Do you have any particular alternative theory in mind that you think supports your view, or do you simply believe that there must be one out there, because why not?
Quote: YoDiceRoll11You have an infinite digit repeater of .3333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333...........
let me know when this equals 1/3.
(Confession: I truncated the number in the quotation because the expression is screwing with the formatting of this page, at least on my screen.)
Let you know when this equals 1/3?
Right now. And it always has, given that you state that it is an infinite repeater of the digit.
Quote: weaselmanA scientific theory is not a God for me to believe in. There are several alternative number theories I know of, and neither of them AFAICS supports the point of view you are defending as such much less violates or discards any of the mathematical "rules".
Really, read my above post.
Quote: Doc(Confession: I truncated the number in the quotation because the expression is screwing with the formatting of this page, at least on my screen.)
Let you know when this equals 1/3?
Right now. And it always has, given that you state that it is an infinite repeater of the digit.
But the fact that it is infinite, in my argument, means it can never attain the next integer. Read my above post on hyperreal numbers.
