.999... = 1

[natty dread]

Well I have exactly the same non finite string of zero's

You can't have an infinite string of zero's with a one at the end...

Trinary numbers really explain it the best IMO.

1/10 = 0.1 !

[nippersean]

lol 1/10 = 0.1 no shit sherlock

no it doesn't it equals 0.09999........

[jiminski]

this kind of mathematics is philosophy as there is no definitive answer, to wit, the infinitive go!

[Snorri1234]

lol 1/10 = 0.1 no shit sherlock

no it doesn't it equals 0.09999........

That is indeed also true.

[natty dread]

lol 1/10 = 0.1 no shit sherlock

no it doesn't it equals 0.09999........

What part about trinary numbers do you not understand?

Trinary 10 = decimal 3.

Trinary 0.1 = 1/3

[TheProwler]

this kind of mathematics is philosophy as there is no definitive answer, to wit, the infinitive go!

Exactly.

And whether or not modern mathematicians define it one way or the other, it doesn't change the fact that the "issue" is not resolved.

I am quite sure that there have been instances in the past where mathematicians agreed on a particular theory and that theory is eventually proved to be wrong.

This is really directed at snorri and the.killing.44. I ignore posts that threaten to entirely derail the thread.

[Titanic]

Great discussion, I used to be of the straight forward expression that .999 does not equal 1 as theres always that 0.000....1 let over at the end, but nowadays I'm with the .999=1 crowd.

If we're talking about philosophical answers to maths questions, answer me this, what is 0/0. I love that question...

[Snorri1234]

this kind of mathematics is philosophy as there is no definitive answer, to wit, the infinitive go!

Exactly.

And whether or not modern mathematicians define it one way or the other, it doesn't change the fact that the "issue" is not resolved.

I am quite sure that there have been instances in the past where mathematicians agreed on a particular theory and that theory is eventually proved to be wrong.

This is really directed at snorri and the.killing.44. I ignore posts that threaten to entirely derail the thread.

The issue is resolved from a mathematical standpoint. From the logical foundations that math is based on it follows that 0.999... is 1.

You can say that it isn't the case for the real world, but that simply implies that the math here doesn't apply to the real world. Which is not a problem because lots of math doesn't apply to the real world.

[Frigidus]

this kind of mathematics is philosophy as there is no definitive answer, to wit, the infinitive go!

Exactly.

And whether or not modern mathematicians define it one way or the other, it doesn't change the fact that the "issue" is not resolved.

I am quite sure that there have been instances in the past where mathematicians agreed on a particular theory and that theory is eventually proved to be wrong.

This is really directed at snorri and the.killing.44. I ignore posts that threaten to entirely derail the thread.

The issue is resolved from a mathematical standpoint. From the logical foundations that math is based on it follows that 0.999... is 1.

You can say that it isn't the case for the real world, but that simply implies that the math here doesn't apply to the real world. Which is not a problem because lots of math doesn't apply to the real world.

I mean, at this point proofs have been put forward. That's pretty much the nail in the coffin on this one.

[AAFitz]

could it be that mans mind cant quite grasp infinity?

could we be failible?

could there be a greater power that has estabilished all this

could it be that somethings we will never quite understand because we were not created with the ability nor purpose to understand.

If this really is such an easy and simple question as many have said then how come there is so much controversy....infinity means just that there is no closure no final equation. Man cannot grasp that something might not have an expliantion because we live in a a finite world at least on earth everything has a solution and final ending...but that is not so in eternity and also in the infinity of space. Hahaha for all our great accomplishments man really is a pathetic creature and very small in true knowledge

Oh yeah, tell that to the crocodile. Its been on this planet for millions of years. He still sleeps in the water, and we wear him as boots, belts and carry our money around in wallets made of him. We are discussing a creature that has existed for millions of years, on a planet that we completely dominated and took over. ((Please do not think I do not understand the fact that this same intelligence, may very well be our undoing. Furthermore, and technically speaking, such undoing only really matters to those around when said undoing unfolds. The rest of us still get to play with our toys and internet until it happens...))

The very fact that we can ask such questions, and ponder them shows your statement to be pathetic, and small in knowledge. The human imagination actually knows no bounds and no limits. It does take time, however. Further, you are assuming that people dont actually understand infinity, simply because everyone doesnt agree what infinity is... the fact is, some may know and understand it perfectly and be perfectly correct in their opinions. The fact that others disagree does not make them incorrect, only not fully accepted. Those that realized the earth orbited the moon, were thought of as madmen in their day. They of course were right, even while the rest of the population disagreed.

The great thing about being correct, is that it does not require acceptance of all, many or even one other person. If you are right, you are right, and it simply does not matter who disagrees, so, to say no one here understands infinity, or lacks the imagination to fully understand it, only shows that you do not understand the infinite possibilities of the human mind.

Rah rah rah!!!

Three cheers for us!!!!

C'mon AA...with all our "great intelligence", we still haven't scratched the surface. Yeah, we might "think" we have a lot of things nailed-down, and we might "think" that our imaginations are unlimited. But look at some of the facts. As an example: Space Travel. Great explorers that we are...we've gone....to our own moon!! Wow.

And we have no idea how powerful our imagination really is....or how accurate.

I agree with THORNHEART. No offense.

Well, I agree that some of us have failings, and one of them is without a doubt; pessimism... Infinite pessimism. Luckily, however, we do not all suffer the same.

More importantly, we could easily have traveled further than our moon. That has nothing to do with intelligence per-se. That has only been governed by decisions. We have not decided to do more. We have decided to do the things we have done, and nothing more, and nothing less. True, this does perhaps say something about our intelligence, but speaks nothing of our potential, and our imagination. We may have decided to not visit the moon as well, and it ironically could have been a better decision not to...but either way, it says nothing of our intelligence. We were capable of going to the moon, and we did. Had we not gone to the moon, we were still capable of it, as we are capable of a great number of things, that we simply have not decided to do.

What you and thornheart have done is confuse what we have done as some kind of definition of our intelligence, which lacks intelligence right there. Potential, and imagination are nearly disassociated with actual actions most of the time.

Feel free to try to elevate yourself, by taking some higher stance of looking down on the human race, in some egotistical effort to feel superior in your own way, but it is the human race which is superior, and has and will overcome challenges on a global scale. If you need to discount these achievements simply because you are not and will not be a part of them, then that is your right, but it will not make you right in your assessment of their limitations.

And it is ok, its a normal psychological tendency, so no offense. Not all can rise above such limitations, but luckilly, many can, and have, and will continue to do so, even while those who cant sit back and say it means nothing.

[jonesthecurl]

I'm still waiting for my exactly 1/3 of a pizza.
How can I be sure it's exactly 1/3? I don't wanna be ripped off by getting a mere 0.33... of a pizza.
What instruments should I use to convince myself I haven't been ripped off?
And when's the pizza gonna be delivered? If it doesn't turn up soon, it's free.

[Doc_Brown]

I see this thread is still buzzing. Maybe I didn't kill it after all! Prowler: You asked why I didn't address everyone else's mathematical flaws in this thread and whether I would focus in on only one student when I was teaching. It's a valid question. First of all, when I was teaching, I often got a lot of spurious questions, and if I tried to answer all of them I'd never get through the important material. That said, I'm not trying to be a math teacher in this thread. The central question is "Does .999... = 1?" not "Can Doc_Brown fix your flawed mathematical assumptions?" So it's perfectly reasonable for me to spend more time addressing valid challenges to the fact that .999... = 1, rather than correcting all the flaws in proofs leading to the fact (which are typically flawed more in terms of notation or completeness than they are outright wrong). As a last point, people that use flawed logic or poor notation to conclude that .999... = 1 pose no major problems to mathematics as a whole. Their notation and logic is correctable. However, those that would claim that .999... < 1 are attacking the very foundation of calculus and must be challenged. More on this in a minute.

I noticed that nippersean made the very mistake I addressed a few pages back. He claims that the difference between 1 and .999... is 0.000...1. When it was pointed out that there in an infinite string of 9s after the decimal, he responded that his infinitesimal is composed of an infinite string of 0s followed by a 1. The problem is that his "number" is a logical impossibility. His number has a "1" in the infinity+1 decimal digit, or, in other words, he is putting a 1 after the end of infinity. Expressed slightly differently, he is placing a 1 after the end of that which has no end. Now, I'm comfortable working with the concept of infinity, but I wouldn't dare try to go beyond the end of infinity!

Ok, back to the relationship between this question and the foundations of calculus. I think the central question really boils down to whether or not the sum of an infinite series has a specific well-defined value. To explain, 0.999... can be expressed as the sum of an infinite series by the following:
0.999... = sum[n=1 to infinity, 9*(10^-n)]
This can be simplified to:
9 * sum[n=1 to infinity, 10^-n]
The sum part of the expression is a geometric series of the form:
a + ar + ar^2 + ar^3 + ...
where a = 0.1 and r = 0.1.
If this was a finite series composed of N terms, the sum would be: 0.1 * (1 - 0.1^N) / (1 - 0.1) = (1/9) * (1 - 0.1^N).
Or, if you multiply by 9 to get the original 0.999... you get a finite sum given by: (1 - 0.1^N).

Now, I think that Prowler was willing to accept it if I stated that the limit of this expression approaches 1 as N goes to infinity. The argument is simply over whether the value of 0.999... can be expressed exactly without resorting to limits. I'm claiming that it can be, and even must be, while Prowler would argue that it is impossible to exactly sum an infinite series. Is that a fair statement?

I will quickly note that most math texts give the sum of an infinite geometric series as the exact value of a/(1-r), which would be (1/9) for the sum in the example above. Multiplying by 9 again, gives you 0.999... = 9/9 = 1. But let's set that aside for now and consider the implications to calculus if infinite sums are not exact.

Let me pose a question for you: What is the exact value of the derivative of y = x^3 at x=1?
Anyone that has completed a first semester calculus class knows that the derivative of x^3 is 3x^2, so plugging in x=1 gives the result dy/dx at x=1 is 3. All well and good. But where did the formula come from? Let's go back and derive it from the basic definition of the derivative:

df(x)/dx at x=x0 = lim(h goes to 0, [f(x0+h) - f(x0)]/h)
Plugging in, we get the limit as h goes to 0 of:
[(x0+h)^3 - x0^3] / h
= [x0^3 + 3(x0^2)(h) + 3(x0)(h^2) + h^3 - x0^3] / h
= 3(x0^2) + 3(x0)(h) + h^2
Now, if we use my claims, we can say that the derivative has an exact value: the h and h^2 terms are 0 when h goes to 0, and the derivative is exactly 3x0^2 = 3 (when x0=1). Prowler's claims about limits would force us to say that the derivative approaches 3 but is not quite exactly that value (it would either be slightly greater or slightly smaller than 3 depending on whether you perform the computation from the h>0 or h<0 side).

This will probably raise the objections that we can get to 0, just not infinity. Fine. Let's turn to the integrals instead. What is the integral of f(x)=3x^2 evaluated from 0 to 1? Easy, right? The integral of 3x^2 = (x^3). Evaluated from 0 to 1 gives a value of 1. Good. You remembered the formula. Where did it come from?

The answer is that the integral is the area under the curve. To compute the area, we break the region from x0 to x1 up into N narrow rectangles of width dx and whose height is f(x0 + n*dx) (where n is the rectangle number). Side note: You can express the integral using various partitions, all of which will produce the same answer.
The area of each rectangle is then: dx*f(x0 + n*dx)
The total area is given by:
sum[n=1 to N, dx*f(x0 + n*dx)]
Let's simplify this some. Let x0=0 as in the original question. N*dx=1 since we want to integrate between x=0 and x=1. That means that dx=(1/N). So, the discrete version of the integral becomes:
sum[n=1 to N, (1/N)*f(n/N)]
= sum[n=1 to N, (1/N) * 3(n/N)^2]
= sum[n=1 to N, (3/N^3) * (n^2)]
Now, a set of discrete rectangles aren't going to exactly cover the area under the curve. The only way to get this to be exact is if the rectangles have an infinitesimal width, or rather, if there is an infinite number of them. In other words, we want the exact value of that sum as N goes to infinity. This is the sum of an arithmetic series rather than a geometric one, but that shouldn't matter. First of all, the sum of (n^2) for n=1 to N is given by:
N(N+1)(2N+1)/6.
You can prove this using mathematical induction (I'll fill in the details in a later post if someone wants to see it).
So, the integral is given by:
limit[N goes to infinity, (3/N^3)*N(N+1)(2N+1)/6]
= limit[N goes to infinity, (N+1)(2N+1)/(2*N^2)]
= limit[N goes to infinity, (2N^2+3N+1)/(2*N^2)]
= limit[N goes to infinity, 1 + 3/(2N) + 1/(2*N^2)]
= 1

Here's the big question: Is that last "=" exact? If it is, you have to say that 0.999...=1 exactly because it is an equivalent expression. On the other hand, if 0.999... != 1, than the integral is not exact. It's only very close to 1. If you want to make this claim, I'd suggest you also tell me what the range of uncertainty on the value is. I'm sure NASA would like to know that information before they launch their next multi-billion $ satellite and expect it to go exactly where they intend. Likewise, there are a lot of nuclear engineers that might be interested in knowing that the equations they used to calculate how close their reactor designs come to critical have some uncertainty in them!

[TheProwler]

Feel free to try to elevate yourself, by taking some higher stance of looking down on the human race, in some egotistical effort to feel superior in your own way, but it is the human race which is superior, and has and will overcome challenges on a global scale. If you need to discount these achievements simply because you are not and will not be a part of them, then that is your right, but it will not make you right in your assessment of their limitations.

And it is ok, its a normal psychological tendency, so no offense. Not all can rise above such limitations, but luckilly, many can, and have, and will continue to do so, even while those who cant sit back and say it means nothing.

Hahaha!!

Holy shit dude, realizing one's limitations doesn't mean we are sitting around, complacent and feeling defeated.

Look, flap your arms all you want, you'll never be able to fly like an eagle. By realizing this limitation, man has gone ahead and invented airplanes. It isn't a bad thing to realize what you can and cannot do.

[TheProwler]

I'm still waiting for my exactly 1/3 of a pizza.
How can I be sure it's exactly 1/3? I don't wanna be ripped off by getting a mere 0.33... of a pizza.
What instruments should I use to convince myself I haven't been ripped off?
And when's the pizza gonna be delivered? If it doesn't turn up soon, it's free.

The pizza's gone. I ate it. At least 0.999... of it.

[TheProwler]

I see this thread is still buzzing. Maybe I didn't kill it after all! Prowler: You asked why I didn't address everyone else's mathematical flaws in this thread and whether I would focus in on only one student when I was teaching. It's a valid question. First of all, when I was teaching, I often got a lot of spurious questions, and if I tried to answer all of them I'd never get through the important material. That said, I'm not trying to be a math teacher in this thread. The central question is "Does .999... = 1?" not "Can Doc_Brown fix your flawed mathematical assumptions?" So it's perfectly reasonable for me to spend more time addressing valid challenges to the fact that .999... = 1, rather than correcting all the flaws in proofs leading to the fact (which are typically flawed more in terms of notation or completeness than they are outright wrong). As a last point, people that use flawed logic or poor notation to conclude that .999... = 1 pose no major problems to mathematics as a whole. Their notation and logic is correctable. However, those that would claim that .999... < 1 are attacking the very foundation of calculus and must be challenged. More on this in a minute.

"people that use flawed logic or poor notation to conclude that .999... = 1 pose no major problems to mathematics as a whole."

No, flawed logic. Nothing wrong with that.

I noticed that nippersean made the very mistake I addressed a few pages back. He claims that the difference between 1 and .999... is 0.000...1. When it was pointed out that there in an infinite string of 9s after the decimal, he responded that his infinitesimal is composed of an infinite string of 0s followed by a 1. The problem is that his "number" is a logical impossibility. His number has a "1" in the infinity+1 decimal digit, or, in other words, he is putting a 1 after the end of infinity. Expressed slightly differently, he is placing a 1 after the end of that which has no end. Now, I'm comfortable working with the concept of infinity, but I wouldn't dare try to go beyond the end of infinity!

This just goes to show how we don't really grasp the idea of infinity. We can talk about it all day, but we understand a beginning and an end. We understand finite.

Ok, back to the relationship between this question and the foundations of calculus. I think the central question really boils down to whether or not the sum of an infinite series has a specific well-defined value. To explain, 0.999... can be expressed as the sum of an infinite series by the following:
0.999... = sum[n=1 to infinity, 9*(10^-n)]
This can be simplified to:
9 * sum[n=1 to infinity, 10^-n]
The sum part of the expression is a geometric series of the form:
a + ar + ar^2 + ar^3 + ...
where a = 0.1 and r = 0.1.
If this was a finite series composed of N terms, the sum would be: 0.1 * (1 - 0.1^N) / (1 - 0.1) = (1/9) * (1 - 0.1^N).
Or, if you multiply by 9 to get the original 0.999... you get a finite sum given by: (1 - 0.1^N).

Now, I think that Prowler was willing to accept it if I stated that the limit of this expression approaches 1 as N goes to infinity. The argument is simply over whether the value of 0.999... can be expressed exactly without resorting to limits. I'm claiming that it can be, and even must be, while Prowler would argue that it is impossible to exactly sum an infinite series. Is that a fair statement?

"The argument is simply over whether the value of 0.999... can be expressed exactly without resorting to limits."

I'd add "or resorting to using the concept of infinity which is a concept we don't fully grasp or understand."

I will quickly note that most math texts give the sum of an infinite geometric series as the exact value of a/(1-r), which would be (1/9) for the sum in the example above. Multiplying by 9 again, gives you 0.999... = 9/9 = 1. But let's set that aside for now and consider the implications to calculus if infinite sums are not exact.

Let me pose a question for you: What is the exact value of the derivative of y = x^3 at x=1?
Anyone that has completed a first semester calculus class knows that the derivative of x^3 is 3x^2, so plugging in x=1 gives the result dy/dx at x=1 is 3. All well and good. But where did the formula come from? Let's go back and derive it from the basic definition of the derivative:

df(x)/dx at x=x0 = lim(h goes to 0, [f(x0+h) - f(x0)]/h)
Plugging in, we get the limit as h goes to 0 of:
[(x0+h)^3 - x0^3] / h
= [x0^3 + 3(x0^2)(h) + 3(x0)(h^2) + h^3 - x0^3] / h
= 3(x0^2) + 3(x0)(h) + h^2
Now, if we use my claims, we can say that the derivative has an exact value: the h and h^2 terms are 0 when h goes to 0, and the derivative is exactly 3x0^2 = 3 (when x0=1). Prowler's claims about limits would force us to say that the derivative approaches 3 but is not quite exactly that value (it would either be slightly greater or slightly smaller than 3 depending on whether you perform the computation from the h>0 or h<0 side).

This will probably raise the objections that we can get to 0, just not infinity. Fine. Let's turn to the integrals instead. What is the integral of f(x)=3x^2 evaluated from 0 to 1? Easy, right? The integral of 3x^2 = (x^3). Evaluated from 0 to 1 gives a value of 1. Good. You remembered the formula. Where did it come from?

The answer is that the integral is the area under the curve. To compute the area, we break the region from x0 to x1 up into N narrow rectangles of width dx and whose height is f(x0 + n*dx) (where n is the rectangle number). Side note: You can express the integral using various partitions, all of which will produce the same answer.
The area of each rectangle is then: dx*f(x0 + n*dx)
The total area is given by:
sum[n=1 to N, dx*f(x0 + n*dx)]
Let's simplify this some. Let x0=0 as in the original question. N*dx=1 since we want to integrate between x=0 and x=1. That means that dx=(1/N). So, the discrete version of the integral becomes:
sum[n=1 to N, (1/N)*f(n/N)]
= sum[n=1 to N, (1/N) * 3(n/N)^2]
= sum[n=1 to N, (3/N^3) * (n^2)]
Now, a set of discrete rectangles aren't going to exactly cover the area under the curve. The only way to get this to be exact is if the rectangles have an infinitesimal width, or rather, if there is an infinite number of them. In other words, we want the exact value of that sum as N goes to infinity. This is the sum of an arithmetic series rather than a geometric one, but that shouldn't matter. First of all, the sum of (n^2) for n=1 to N is given by:
N(N+1)(2N+1)/6.
You can prove this using mathematical induction (I'll fill in the details in a later post if someone wants to see it).
So, the integral is given by:
limit[N goes to infinity, (3/N^3)*N(N+1)(2N+1)/6]
= limit[N goes to infinity, (N+1)(2N+1)/(2*N^2)]
= limit[N goes to infinity, (2N^2+3N+1)/(2*N^2)]
= limit[N goes to infinity, 1 + 3/(2N) + 1/(2*N^2)]
= 1

Here's the big question: Is that last "=" exact? If it is, you have to say that 0.999...=1 exactly because it is an equivalent expression. On the other hand, if 0.999... != 1, than the integral is not exact. It's only very close to 1. If you want to make this claim, I'd suggest you also tell me what the range of uncertainty on the value is. I'm sure NASA would like to know that information before they launch their next multi-billion $ satellite and expect it to go exactly where they intend. Likewise, there are a lot of nuclear engineers that might be interested in knowing that the equations they used to calculate how close their reactor designs come to critical have some uncertainty in them!

"The argument is simply over whether the value of 0.999... can be expressed exactly without resorting to limits or resorting to using the concept of infinity which is a concept we don't fully grasp or understand."

You used a lot of limits and the concept of infinity in your discussion.

Anyways, limits are approached. They are never reached. Right?

When we say "the limit of A is equal to B" we are not saying "A is equal to B".

And if nuclear engineers somehow are dealing with 0.999... and not 1, it will not matter as they are so close that the difference is unmeasurable. That's why jones is so worried about his fair share of pizza.

[2dimes]

I'm here for the pizza.

[jonesthecurl]

Hey, Prowler: I have a small piece of pizza here which you dropped. You know how big it is? It's the bit left when three pieces, each .33... of the pizza were shared out. Those three pieces added up to .99... of the pizza.

So how much pizza do I have here?

[Phatscotty]

.999... does not equal 1

[2dimes]

Hey, Prowler: I have a small piece of pizza here which you dropped. You know how big it is? It's the bit left when three pieces, each .33... of the pizza were shared out. Those three pieces added up to .99... of the pizza.

So how much pizza do I have here?

I'm thinking not enough now that I'm here.

[natty dread]

.999... equals 1


Also: the trinary number 0.222... also equals the number 1!


Also: the binary number 0.111... equals the number 1!

Even though trinary number 2 does not equal decimal 9, and neither does binary number 1...


You see: the decimal number 0.333... equals the trinary number 0.1. The trinary number 0.1 times 3 equals 1 in both decimal and trinary. 1 is 1 in both systems. However... since 0.222.. in trinary also equals 1, if you divide it by 3 (which is 10 in trinary) you get 0.1 trinary! Even though it should, by all logic, be 0.022... since 1 divided by trinary 10 is 0.1 trinary....

(of course, the trick here being, that 0.022... also equals 0.1 in trinary)

Confusing numbers are confusing

[StephenB]

Hey, Prowler: I have a small piece of pizza here which you dropped. You know how big it is? It's the bit left when three pieces, each .33... of the pizza were shared out. Those three pieces added up to .99... of the pizza.

So how much pizza do I have here?

I'm thinking not enough now that I'm here.

Yeah, you better go buy another one, Jones. I suggest we play a game of Risk to see who gets the bigger piece and which two of us will have to settle for 33.333...% of the pizza.

[thegreekdog]

I'm still waiting for my exactly 1/3 of a pizza.
How can I be sure it's exactly 1/3? I don't wanna be ripped off by getting a mere 0.33... of a pizza.
What instruments should I use to convince myself I haven't been ripped off?
And when's the pizza gonna be delivered? If it doesn't turn up soon, it's free.

Further proof (as if it was needed), that pizza solves all problems... including whether .999 = 1.

[StephenB]

Further proof (as if it was needed), that pizza solves all problems... including whether .999 = 1.

No, you cannot have a piece of ours. Go get your own.

[thegreekdog]

Further proof (as if it was needed), that pizza solves all problems... including whether .999 = 1.

No, you cannot have a piece of ours. Go get your own.

Can I please just have a third of your third? That's only .11111111111

[2dimes]

I don't know if this discussion will improve once the beer starts flowing with them pizzas.