What is .99 repeating as a fraction?
Quote: WizardJust out of curiosity, did anybody's position change over the course of this thread?
Mine did, but I'm the math imbecile of this place. To my credit, I at least accepted and understood the answer at Doc's first post on the second thread =p
Quote: WizardHere is what I find to be a more interesting question. A random number is chosen from a uniform distribution between 3 and 4. What is the probability it is NOT pi? If your answer is 1, is that the same as a hard 1?
The first question, the 0.999... or 1, that is a notation question, ie, our way of communicating numbers sometimes has unusual anomalies.
The second question, that is a true math question. I don't know the answer to it without looking it up.
Quote: Wizard
Here is what I find to be a more interesting question. A random number is chosen from a uniform distribution between 3 and 4. What is the probability it is NOT pi? If your answer is 1, is that the same as a hard 1?
If you randomly select a number with an infinite number of digits, you will never select the number 3.5 either. 3.5 exists, but the probability of selecting it out of an infinite amount of numbers is 0.
Quote: weaselmanAre you implying there is more than one number with the value 1? :)
I was hoping to not be put on the spot with that. For the purposes of discussion, let me suggest the following terms:
Soft 1: The probability a random number drawn between 3 and 4 is not pi.
Hard 1: The probability a random number drawn between 3 and 4 is between 3 and 4 (inclusive).
I am saying that hard 1 = soft 1. Can I convince any skeptics who disagree? I doubt it.
Quote: WizardI was hoping to not be put on the spot with that. For the purposes of discussion, let me suggest the following terms:
Soft 1: The probability a random number drawn between 3 and 4 is not pi.
Hard 1: The probability a random number drawn between 3 and 4 is between 3 and 4 (inclusive).
I am saying that hard 1 = soft 1. Can I convince any skeptics who disagree? I doubt it.
If you don't mind being put on the spot, again, I believe your example is a perfect example showing hard 1 and soft 1 are 2 different numbers with 2 different values. In other words, soft 1 has a distinct place in the universe, and is not just another representation for hard 1.
How can I convince you to say soft 1 != hard 1? Would it help if you actually drew Pi from a random draw between 3 and 4? It doesn't have to be a fair game for this, just draw Pi. Tape the number to the side of the hat. LOL.
Seriously, though. If you drew Pi, that would be the difference between soft 1 and hard 1. And, for these purposes, it's rather easy to draw Pi.
Quote: WizardI am saying that hard 1 = soft 1. Can I convince any skeptics who disagree? I doubt it.
In the abstract it is, or rather it's close enough that it makes no difference. In concrete terms, it's also close enough. Could you tell the difference between, say 1 gram and 0.999999999..... grams? Or between a house edge of 0.60% and 0.599999999999999.....%?
The natural asnwer is "No."
But the difference between 1.0 and 0.999999999999...... in quantum mechanics is that between a single electron and the entire universe (or so I've been led to believe) ;)
(I spent last evening stuck in traffic listening ot a lengthy explanation about the origins of quantum mechanics and why electrons don't lose their energy while circling an atom)
Regardless; this is a different number than the 0.9999... number, because that is a notational problem and this is an conceptual problem.
HOWEVER. If I am wrong, I accept. I'm not married to my answer.
Quote: MoscaWithout getting into the numbers themselves, I could ask the question this way: "Is it a certainty that the number chosen is not pi?" Well, no. But how much uncertainty is in there, is infinitesimal.
Regardless; this is a different number than the 0.9999... number, because that is a notational problem and this is an conceptual problem.
HOWEVER. If I am wrong, I accept. I'm not married to my answer.
If it will help to see it with .999... it would be 1 is all the possible numbers between 3 & 4. .999... is all the possible numbers except Pi.
I still think it's a great example. It's a lot easier to see, then 1 - .999... = .000...
sorry for reviving the thread, i saw it near the top and assumed it was still active rather than checked the date :-/
It is easy to show that "hard 1" === "soft 1". Let a be "hard one" and b be "soft one". Then a*b=a and a*b=b, by definition of unit, and therefore a=b. Q.E.D.
Quote: weaselmanYou can say that it is 0.999... if you like, it is not wrong, but since it is just a longer way to say "one", we usually call it 1. :)
I think you're missing the point. The Pi example, shows beyond a reasonable doubt, that .999... has a DIFFERENT value then 1. If .999... really did equal 1 , it would have the same value. So, now I'm wondering why anyone would think that .999... = 1 ? You can clearly see with your examples they are not equal to each other.
I think it was Gamer who pointed out that I said .999... = .999... Looking at the Pi example, I don't know how accurate that is. I think one could say that .999... is all the numbers between 3 & 4 except Pi, and it could also be all the numbers except Pi and 3, or all the numbers except Pi, 3, and 3.5. That would show that .999... can represent different values (none of which are equal to 1).
Thinking about it, I would say .999... is 9's repeating to infinity, would actually be better stated as .999... is 9's repeating past a certain point, a Googolplex for example, therefore, .999... can hold different values. .999... is cut off at a certain limit. For these purposes, .999... could be 1 past a Googolplex, or 2 past a Googolplex, or 3 past a Googolplex etc.
But, does that mean .999... = .999... Mathematically, I think it would be the same number because of the limit placed on it. But, conceptually, it can have an infinite amount of different values.
Quote: JyBrd0403I think you're missing the point. The Pi example, shows beyond a reasonable doubt, that .999... has a DIFFERENT value then 1.
No, it does not. I don't know what made you think that, but you are wrong. If "Pi example" shows anything at all, it is that an event with probability 0 is not necessarily the same thing as an impossible event (the proper term is "almost impossible). It has nothing to do with the peculiarities of positional notation or 10-base numerals.
Quote:If .999... really did equal 1 , it would have the same value.
Yes. And it does. "Equals" and "has the same value" are synonyms.
Quote: weaselmanNo, it does not. I don't know what made you think that, but you are wrong. If "Pi example" shows anything at all, it is that an event with probability 0 is not necessarily the same thing as an impossible event (the proper term is "almost impossible).
Yes. And it does. "Equals" and "has the same value" are synonyms.
Is this for real? 1 is all the numbers between 3 & 4, correct? .999... is all the numbers except Pi. 2 different values. One number includes Pi, one number does NOT. Probability of 0 means the event is impossible (kindergarten stuff). It's impossible to draw the number 8 from the Pi example.
Probability of .999... equaling 1 is 0 (thanks to Pi) .
Quote: JyBrd0403Is this for real? 1 is all the numbers between 3 & 4, correct? .999... is all the numbers except Pi.
2 different values.
No, it is the same value. Both probabilities are equal to 1.
Quote:One number includes Pi, one number does NOT.
So, you insist, that excluding a number should change the probability is that your point?
So, probability of hitting any number between 3 and 4 is 1, and probability of hitting any number except Pi is 0.999... which, according to you, must be different, because a number is excluded, right?
Then how about excluding two numbers? What would you say is the probability of hitting any number except Pi or 3.5?
Or, even better, lets exclude an *infinite number* of numbers. What would be the probability of hitting an *irrational* number between 3 and 4?
In fact, (get this!) the probability of hitting any number, that can be described, expressed or referred to in any way at all is 0.
Quote:Probability of 0 means the event is impossible (kindergarten stuff).
Well ... *in kindergarten* it does :) In the real world, it does not though. Like I said earlier, the proper term is "almost impossible". The probability of hitting *any* particular number out of the continuous interval is 0. Yet, some number certainly *will* get hit. Shows, that it is not really impossible.
Quote: JyBrd0403If it will help to see it with .999... it would be 1 is all the possible numbers between 3 & 4. .999... is all the possible numbers except Pi.
I still think it's a great example. It's a lot easier to see, then 1 - .999... = .000...
Yours is the only solution to this which I would instantly disregard, actually.
Quote: weaselmanSo, you insist, that excluding a number should change the probability is that your point?
So, probability of hitting any number between 3 and 4 is 1, and probability of hitting any number except Pi is 0.999... which, according to you, must be different, because a number is excluded, right?
Then how about excluding two numbers? What would you say is the probability of hitting any number except Pi or 3.5?
Or, even better, lets exclude an *infinite number* of numbers. What would be the probability of hitting an *irrational* number between 3 and 4.
I told you it's a much more compelling arguement. :)
Yeah, the probability of hitting an irrational number between 3 and 4 would be .999... .999... has more then 1 value, conceptually. 1 thank goodness only has 1 value. .999... may have thousands of different values, but your examples clearly shows none of them are the value 1. There's always at least 1 number missing. Someone tried to say just that in a previous post, that there's always 1 number between .999... and 1. The Pi example clearly shows that.
0 in the real world is always 0. If I have a bowl of apples, and no oranges, we have 0 oranges. It might just be for a split second before someone puts an orange in the bowl, but it was 0 oranges none the less. The probability of hitting *any* number out of a continuous interval is != 0. I've given the number .000... if you don't like that one you have to come up with another representation. 0 by definition has no value. By description the probability of hitting any particular number is a value > 0. Some refer to it as an infintesimal.
The point still remains the same. 1 has a different value then .999... 0 has no value , your example shows a distinct value, however infintesimal.
To prove your point, you would have to show 1 is all the numbers between 3 & 4 except Pi. Then 1 = .999...
But, really, once you showed that 1 has a different value then .999... in your examples, it proves they are not the same number. They would have to have the exact same value. .999... would have to be all the numbers between 3 & 4 also. It's not. It's missing Pi (at the very least it's missing Pi).
Quote: MoscaYours is the only solution to this which I would instantly disregard, actually.
That's what turns people into idiots. Disregarding the obvious facts, because they don't like the teacher. Means that person is unteachable. Stuck in stupid.
Quote: JyBrd0403That's what turns people into idiots. Disregarding the obvious facts, because they don't like the teacher. Means that person is unteachable. Stuck in stupid.
Irony, meet thyself.
Quote: MoscaIrony, meet thyself.
I learn from everyone and everything. If you're a moron with just 1 great idea, I'm not going to be apart of the 99.999...% that disregard it. The only thing with this thread is, I just don't buy into B.S. That's all. 1/3 * 3 = 1 so .999... = 1, is B.S. It's a flaw in the math, that's all.
Quote: JyBrd0403I learn from everyone and everything. If you're a moron with just 1 great idea, I'm not going to be apart of the 99.999...% that disregard it.
So 100% of people disregard it?
Quote: PopCanSo 100% of people disregard it?
LOL. Everyone except me, Pop. I'm Pi in this example. LOL.
Please, please, please... agree with this. I know the teacher is hated, but 1 = 1 still, right? Jesus.
Quote: JyBrd0403Can we all agree that 1 only has 1 value, that of 1?
But "2+2=5 for very large values of 2"
:)
Quote: NareedBut "2+2=5 for very large values of 2"
:)
Nareed, I thought you were on my side. LOL. 2 only has 1 value, that of 2. 2.5 + 2.5 = 5. :) Couldn't you have said 3 + 3 = 5 for lower values of 3. LOL>
Quote: JyBrd0403Nareed, I thought you were on my side. LOL. 2 only has 1 value, that of 2. 2.5 + 2.5 = 5. :) Couldn't you have said 3 + 3 = 5 for lower values of 3. LOL>
That last part only stands to reason, doesn't it?
Actually the quotation was a popular tagline in some old time BBSes. I used it as a screen saver once, too, when 3D letters were in vogue.
Quote: NareedThat last part only stands to reason, doesn't it?
Actually the quotation was a popular tagline in some old time BBSes. I used it as a screen saver once, too, when 3D letters were in vogue.
Now, that you say that, I think I recognize the quote. I don't know if the last part stands to reason or not anymore. I guess not if you accept that 2.5 is a lower value of 3. :) or If you believe 3 = 4 or 2, or if you have an equation showing 5 = 6 or if... :)
Quote: JyBrd0403Now, that you say that, I think I recognize the quote. I don't know if the last part stands to reason or not anymore. I guess not if you accept that 2.5 is a lower value of 3. :) or If you believe 3 = 4 or 2, or if you have an equation showing 5 = 6 or if... :)
There was an equation showing that 1=2 somewhere...
Seriously, I am very attached to Aristotle's Law of Identity. So therefore 0.999... < > 1
But, really, on practical matters, the difference between 0.999999..... kilograms and one kilogram is so small they may as well be the same. Likewise if you can carry, say, 45 lbs as checked luggage, it would be unfair to charge you extra if your luggage were 45................00001 pounds, right? Even thou they're also not the same.
Quote: NareedThere was an equation showing that 1=2 somewhere...
Seriously, I am very attached to Aristotle's Law of Identity. So therefore 0.999... < > 1
But, really, on practical matters, the difference between 0.999999..... kilograms and one kilogram is so small they may as well be the same. Likewise if you can carry, say, 45 lbs as checked luggage, it would be unfair to charge you extra if your luggage were 45................00001 pounds, right? Even thou they're also not the same.
You won't get an agruement from me on that, Nareed. I agree, with everything you just said. I really don't understand why it's a difficult concept to grasp. I think I'm more curious of how this is sort of a modern day "is the world flat or round" thing. It's hard to believe people will fight so hard to defend a concept that they themselves can show to be incorrect.
The limit presumably being the highest number we have available, which my understanding is a googolplex (let me know if I'm wrong there). I'm not 100% sure about what the highest number actually is, but I'm sure there's an infinite number of numbers past that limit. My point is at some point, after a googolplex let's say, you give the number the representation .999...
Which kinda disproves that notion that .999... is infinitesimally close to 1 and therefore equals 1. In "reality" it's only infinitesimally small in one of it's forms, the Pi example for instance, in other forms it's infinitely far away from 1.
Quote: NareedThere was an equation showing that 1=2 somewhere...
This uses a divide by zero to make 1 = 2 :
Let a = b.
Let a^2 mean a*a (just to define my notation)
Thus :
a^2 = ab
a^2 + a^2 = ab + a^2
2a^2 = ab + a^2
2a^2 - 2ab = ab + a^2 - 2ab
2a^2 - 2ab = a^2 - ab
Which can be written as :
2 * (a^2 - ab) = 1 * (a^2 - ab)
=> 2 = 1
Quote: JyBrd0403After giving this some more thought, I really do think the ... is better off stated as being the limit of the repeating 9's and not Infinity itself.
The limit presumably being the highest number we have available, which my understanding is a googolplex (let me know if I'm wrong there). I'm not 100% sure about what the highest number actually is, but I'm sure there's an infinite number of numbers past that limit. My point is at some point, after a googolplex let's say, you give the number the representation .999...
Which kinda disproves that notion that .999... is infinitesimally close to 1 and therefore equals 1. In "reality" it's only infinitesimally small in one of it's forms, the Pi example for instance, in other forms it's infinitely far away from 1.
There is "no highest number we have". For example, googleplex times a googleplex is necessarily bigger than a googleplex. Lets call it a Cesspit-Plex.
But a Cesspit-Plex to the power of a Cesspit-Plex is even bigger number... lets call that a JyBrd0403-plex.... and well you get my drift...
Quote: thecesspitThere is "no highest number we have". For example, googleplex times a googleplex is necessarily bigger than a googleplex. Lets call it a Cesspit-Plex.
But a Cesspit-Plex to the power of a Cesspit-Plex is even bigger number... lets call that a JyBrd0403-plex.... and well you get my drift...
JyBrd-plex , I like that. :) I understand your point, but we have a little dilemma with the Pi example. .333 * 3 = .999... that's easy enough to say that the 9's are repeating to Infinity. But, with the Pi example you have a case where .999... has more than 1 value. So, at what point do you say okay this is .999... There has to be a limit there. I'm definitely not going to be the person who counts to infinity to classify all the different numbers. LOL.
I guess I'm looking for the highest number ever counted to. I think they had a computer that was continuously counting to find out what the highest number available was, and presumably to name new numbers. Does anyone know what the highest number counted to by a computer is? I think that number would be the limit I was talking about.
If I did have 1 billion dollars riding on a bet that 2 + 2 = 4, would I win the bet if I answered 3.999... ?
One of my favorite math youtubers posted the following link recently, and I found it strangely ironic, but maybe it will help you :)
9.999... reasons that .999... = 1
And here is her April 1st post :)
Why every proof that .999... = 1 is Wrong
And Nareed, do not ask me who is on first again. I told you, I do not know Who is on first. And somehow, when i tell you I don't know Who is on first, I wind up on third base. Even the "rainman" could not figure out who's on first !!
The pi example doesn't prove anything in your favor. There is an infinite amount of numbers between 3 and 4. If you remove 1 number (pi), you are simply subtracting 1 from infinity, which is still infinite. The Wizard's question poses somewhat of a parodox. There is infinite numbers between 3 and 4. Pi is between 3 and 4. The probability of choosing pi is 1/infinity, which is 0, yet if I asked someone the question: I am thinking of a number between 3 and 4, what is it?, and they said pi, and I was thinking of pi, then it happened, thus not impossible.
The whole googolplex is the largest number thing is wrong. The way numbers exist is the fact they never end, therefore, there will always be a number larger then any number you choose.
A computer counting numbers doesn't change the "largest number". First off, computers only have so much space, so even if you created a computer that counted 1 number for clock cycle, at some point, the numbers would roll over back to 0 and start all over.
The largest number a computed counted to depends on the number of bits that computer uses to store numbers. The bits are controlled by the data lines. Today, it's typical to see processors with 64 data lines, which means the largest number would be 2^64 - 1, which is 18,446,744,073,709,551,616 . But by using different algorithms, computers could store much larger numbers than this, by simply storing the numbers in multiple address spaces. So now the number is limited to the data lines and the address space. The address space is limited to the address lines. Data lines and address lines are multiplexed, so it can be assumed that they have the same number (64)...
So now the largest number a computer could store would be 2^128. But then, it seems pretty foolish to store numbers that way, so number started be stored in different means. This is why if you use your calculator on your computer, the largest number it will probably accept is much larger then 2^128.
But this is all irrelevant, because even if a computer was continously counting a number and storing it correctly, that still doesn't make that number "the largest number"...
Quote: TriplellThere is infinite numbers between 3 and 4. Pi is between 3 and 4. The probability of choosing pi is 1/infinity, which is 0, yet if I asked someone the question: I am thinking of a number between 3 and 4, what is it?, and they said pi, and I was thinking of pi, then it happened, thus not impossible.
So, if I bet 1 billion dollars that you have a 0% chance (0% being no chance at all) of picking pi from a number between 3 & 4, would I win the bet?
0.000... = 0 = No chance at all of picking pi. Would I win the bet?
Quote: TriplellBut this is all irrelevant, because even if a computer was continously counting a number and storing it correctly, that still doesn't make that number "the largest number"...
It would be the largest number ever counted to, though, and that's the question I was asking. Whatever that number is would seem to me to be a good place to put a limit on .999...
Quote: TriplellYes you would win.
One of my favorite math youtubers posted the following link recently, and I found it strangely ironic, but maybe it will help you :)
9.999... reasons that .999... = 1
And here is her April 1st post :)
Why every proof that .999... = 1 is Wrong
I think she just became my favorite too. :)
Quote: JyBrd0403It would be the largest number ever counted to, though, and that's the question I was asking. Whatever that number is would seem to me to be a good place to put a limit on .999...
There is NO LIMIT on 0.999... that's the point of the notation.
Quote: thecesspitThere is NO LIMIT on 0.999... that's the point of the notation.
The traditional view puts a limit of infinity on the number .999... .999... goes to Infinity, and does not go past Infinity. There's a limit of infinity put on the number. My question is what notation would you give for a number that goes past the highest number ever counted to by a computer. Repeating 9's carried out to this number are still infinitely short of infinity, but the number would still be represented as .999... I think it would be better stated that the limit is not infinity for .999..., but rather, the limit is the highest number ever counted to.
In other words, .999... holds the values of all numbers between the highest number ever counted to and Infinity.
Quote: JyBrd0403It would be the largest number ever counted to, though, and that's the question I was asking. Whatever that number is would seem to me to be a good place to put a limit on .999...
You make a program that counts and stores numbers. Here, I will write one for you quick.
main(){
x =0;
while (1){
x++;
}
}
In 10 years, let me know what this number is at. If the operation can be performed 1000 times per second, then in 10 years, it should be at 315,360,000,000.
In 10 years, I'll write a similiar program.
main(){
x = 0;
while(1){
x*=10;
}
}
If the same computers again performs that operation 1000 timers per second, how long will it take to exceed your number?
The correct answer is 12 milliseconds.
My point here is that the highest number "counted to" is a dumb idea. Assuming the number counts by 1, I can count by 2's, or 4's, or I can raise it to the power of 10...
But let's say you get smart...and you decided that you are always going to make yours raise by a power larger then mine.
I decided to make mine multiply by a power to the power of yours (which would be your power squared...)...
Well then, you decided that ...
And this is why we have infinity..(among other reasons)...
Quote: TriplellIf the operation can be performed 1000 times per second, then in 10 years, it should be at 315,360,000,000.
You're missing the point, Triplell. You have to do the math first. You can't just leap ahead. You get one computer, counting by 1's. Let's say we start the computer counting on this day April 5, 2012, In ten years, the limit for .999... would be ( by your calculations) 315,360,000,000. And, 1 second later, the limit would be 315,360,001,000. So, if you're doing a calculation it would be very important to know exactly what time you started the calculation, that way you would know what .999...'s limit is. If you want to be accurate you have to be diligent.
By the way, Triplell. You leaped ahead of the 0% question. Would I win the bet if I said you have a 0% chance of picking Pi between the numbers of 3 & 4?
.000... = 0 = No chance at all of picking Pi.
Quote: JyBrd0403You're missing the point, Triplell. You have to do the math first. You can't just leap ahead. You get one computer, counting by 1's. Let's say we start the computer counting on this day April 5, 2012, In ten years, the limit for .999... would be ( by your calculations) 315,360,000,000. And, 1 second later, the limit would be 315,360,001,000. So, if you're doing a calculation it would be very important to know exactly what time you started the calculation, that way you would know what .999...'s limit is. If you want to be accurate you have to be diligent.
By the way, Triplell. You leaped ahead of the 0% question. Would I win the bet if I said you have a 0% chance of picking Pi between the numbers of 3 & 4?
.000... = 0 = No chance at all of picking Pi.
Why does the computer count need to count by 1?
And if your bet is that you would pick Pi, then yes, you would lose....
Quote: TriplellWhy does the computer count need to count by 1?
And if you're bet is that you would pick Pi, then yes, you would lose....
You have to count by 1's because there are numbers that have not been identified and named yet.
Thank you, Triplell. That's why I wouldn't make that bet. :)
Ah, damn it all. I just leaped ahead. The bet is that you have a 0% chance of picking Pi.
.000...= 0 = No chance at all of picking Pi.
Quote: JyBrd0403You have to count by 1's because there are numbers that have not been identified and named yet.
Thank you, Triplell. That's why I wouldn't make that bet. :)
Ah, damn it all. I just leaped ahead. The bet is that you have a 0% chance of picking Pi.
.000...= 0 = No chance at all of picking Pi.
The Wizard's original questions was a dart thrown at a number line landing on "Pi", which would have a 0% probability, because Pi is not a rational number.
If I told you I am thinking of a number between 3 and 4, what is it? then the probability of it being pi is theoretically 1 out of infinity, which approaches 0, however it isn't impossible, as every number has the same possibility of being picked, which includes pi...
Quote: TriplellThe Wizard's original questions was a dart thrown at a number line landing on "Pi", which would have a 0% probability, because Pi is not a rational number.
If I told you I am thinking of a number between 3 and 4, what is it? then the probability of it being pi is theoretically 1 out of infinity, which approaches 0, however it isn't impossible, as every number has the same possibility of being picked, which includes pi...
The dart thrown at a dart board hitting Pi has a different problem. Pi is such a small number that it would be impossible to just hit Pi. The dart might hit Pi and 3.14 and 3.144 and 3.15 and 3.12 and 3.1234 etc. The point of the dart head would cover a ton of numbers ... in this "reality".
If you can imagine a dart being thrown at a dart board that has only one destination, that destination being Pi, what would have to happen for that dart to hit Pi? Somehow, the dartboard will have to become Infinitely big. It would have to grow to immense proportions for the dart to land on Pi. Probabilities of that happening are incredibly small, but (with no limits applied) there is a possibility of it happening. That is what would normally be called imagination.
The probability of the number picked being pi approaches zero, but is not zero. That's why I wouldn't make the bet. It means .000... != 0. .000... may have millions of values, but 0 has no value at all.
Quote: JyBrd0403So, if I bet 1 billion dollars that you have a 0% chance (0% being no chance at all) of picking pi from a number between 3 & 4, would I win the bet?
0.000... = 0 = No chance at all of picking pi. Would I win the bet?
Well, the chances of picking pi are 0%, but it could still happen. If you bet on "0% chance" you would win. If you bet on "no chance" then I would say you would lose, but it would depending on the exact definition of "no."
I know this all may sound ridiculous on the surface, but constraints in English betray explaining how something can have a 0% chance and still happen.
