.999... = 1

[InkL0sed]

This is similar to 0.999recurring equaling 1. It is almost there. Almost. Another trillion 9's and we're almost there. Almost. You will never get there.

Some students regard 0.999… as having a fixed value which is less than 1 by an infinitesimal but non-zero amount.

These ideas are mistaken in the context of the standard real numbers, although some may be valid in other number systems, either invented for their general mathematical utility or as instructive counterexamples to better understand 0.999…

Dude, enough with the wiki. It doesn't prove a thing. It proves that someone agrees with you.

On this topic, like many others, a very small percentage of people will be able to really understand it. It doesn't come down to a vote count.

That small percentage being (you / population of the world) x 100%

[TheProwler]

Then I saw Prowler's post, above, and he is right. 3.333 recurring + 3.333 recurring + 3.333 recurring does not equal 9, as Crocodile Dumdum believes.

I have never claimed that 3.333... +3.333... + 3.333 = 9 I clearly claimed that 3+3+3=9. 3=/= 3.333... I'm not sure where this confusion comes from.

You should look at the example I provided until you understand it as KLOBBER did.

Xeno's paradox? You should really learn basic calculus.

Define Xeno's paradox.

I'm pretty sure you still don't understand the example I provided to help you get your head around where you are making your algebraic mistake.

[TheProwler]

For Prowler and Klobber: How much smaller is 0.99999recurring than 1.0? You both seem to agree that it is smaller. So by how much?

And infinitely small, non-zero, number. You should read the thread.

That number is impossible. Infinitely small = 0

Hey, you are correct about something. That number is impossible.

So when something is impossible, it equals zero? Haha! Good job!

[Timminz]

The problem is, you're trying to define the end of an infinite string.

[TheProwler]

On this topic, like many others, a very small percentage of people will be able to really understand it. It doesn't come down to a vote count.

That small percentage being (you / population of the world) x 100%

Haha! You crack me up! Good thing you multiplied that number by 100%. Next time, just for kicks, multiple it by 0.999recurring. Hahaha!

I appreciate your praise. But I'm quite sure there are at least a few hundred others out there who would understand.

[owheelj]

Then I saw Prowler's post, above, and he is right. 3.333 recurring + 3.333 recurring + 3.333 recurring does not equal 9, as Crocodile Dumdum believes.

I have never claimed that 3.333... +3.333... + 3.333 = 9 I clearly claimed that 3+3+3=9. 3=/= 3.333... I'm not sure where this confusion comes from.

You should look at the example I provided until you understand it as KLOBBER did.

Xeno's paradox? You should really learn basic calculus.

Define Xeno's paradox.

I'm pretty sure you still don't understand the example I provided to help you get your head around where you are making your algebraic mistake.

In the paradox of Achilles and the Tortoise, Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 feet. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run 100 feet, bringing him to the tortoise's starting point. During this time, the tortoise has run a much shorter distance, say, 10 feet. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, because there are an infinite number of points Achilles must reach where the tortoise has already been, he can never overtake the tortoise. Of course, simple experience tells us that Achilles will be able to overtake the tortoise, which is why this is a paradox.[4][5]

[Timminz]

In the paradox of Achilles and the Tortoise, Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 feet. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run 100 feet, bringing him to the tortoise's starting point. During this time, the tortoise has run a much shorter distance, say, 10 feet. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, because there are an infinite number of points Achilles must reach where the tortoise has already been, he can never overtake the tortoise. Of course, simple experience tells us that Achilles will be able to overtake the tortoise, which is why this is a paradox.[4][5]

I've never seen this one before. That's awesome.

[TheProwler]

The problem is, you're trying to define the end of an infinite string.

Not really...the problem is people are trying to apply basic algebraic rules on a set of numbers that will cause the rules to fail.

3 * 0.333recurring does not equal 0.999recurring.

That's why I've tried to simplify it by showing that as far as you want to go, the highest degree of precision ends with 3 1/3, not 3...

0.3333333...until 3/1000000000...people are trying to multiple with by 3 and getting 0.9999999...until 9/1000000000. But it is really 0.3333333...until 3 1/3 /1000000000....which results in 1.0 when multiplied by 3. I realize I am showing a finite amount of precision. But it is in an attempt to get you to understand something that is difficult to grasp.

Here...check out the pattern

3 * (3/10) = 0.9
3 * (3/10 + 3/100) = 0.99
3 * (3/10 + 3/100 + 3/1000) = 0.999
3 * (3/10 + 3/100 + 3/1000 + 3/10000) = 0.9999
...
3 * (3/10 + 3/100 + 3/1000 + 3/10000 + 3/100000 + 3/1000000 + 3/10000000 + 3/100000000 + 3/1000000000) = 0.999999999

Now check out this pattern.

3 * (3 1/3/10) = 1.0
3 * (3/10 + 3 1/3/100) = 1.0
3 * (3/10 + 3/100 + 3 1/3/1000) = 1.0
3 * (3/10 + 3/100 + 3/1000 + 3 1/3/10000) = 1.0
...
3 * (3/10 + 3/100 + 3/1000 + 3/10000 + 3/100000 + 3/1000000 + 3/10000000 + 3/100000000 + 3 1/3/1000000000) = 1.0

The first pattern seems to follow 3 * 0.33recurring. But it isn't exact.

The second pattern exactly follows 3 * 0.333recurring.

And it all equals 1, baby!


This is just an illustration to help you understand. Did it help?

[Timminz]

Did you read the proof I posted earlier? There is absolutely no reference to 1/3, or 0.333...

[KLOBBER]

3 * (3/10) = 0.9
3 * (3/10 + 3/100) = 0.99
3 * (3/10 + 3/100 + 3/1000) = 0.999
3 * (3/10 + 3/100 + 3/1000 + 3/10000) = 0.9999
...
3 * (3/10 + 3/100 + 3/1000 + 3/10000 + 3/100000 + 3/1000000 + 3/10000000 + 3/100000000 + 3/1000000000) = 0.999999999

Now check out this pattern.

3 * (3 1/3/10) = 1.0
3 * (3/10 + 3 1/3/100) = 1.0
3 * (3/10 + 3/100 + 3 1/3/1000) = 1.0
3 * (3/10 + 3/100 + 3/1000 + 3 1/3/10000) = 1.0
...
3 * (3/10 + 3/100 + 3/1000 + 3/10000 + 3/100000 + 3/1000000 + 3/10000000 + 3/100000000 + 3 1/3/1000000000) = 1.0

The first pattern seems to follow 3 * 0.33recurring. But it isn't exact.

The second pattern exactly follows 3 * 0.333recurring.

And it all equals 1, baby!


This is just an illustration to help you understand. Did it help?

Prowler, you are AWESOME! I already understood, and now I understand even better. Your example is final, and it is conclusive.

[john9blue]

Nonbelievers: do you agree that .333... = 1/3?

If so then:

.333... = 1/3
(1/3)*3 = 3/3 = 1
Therefore, .333...*3 = 1

.333... = sum of all 3/(10^n) as n goes from 1 to infinity.
(3/(10^n))*3 = 9/(10^n)
sum of all 9/(10^n) as n goes from 1 to infinity = .999...
Therefore, .333...*3 = .999...

If .333... = 1 and .333... = .999..., then .999... = 1. QED,MF.

[TheProwler]

In the paradox of Achilles and the Tortoise, Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 feet. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run 100 feet, bringing him to the tortoise's starting point. During this time, the tortoise has run a much shorter distance, say, 10 feet. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, because there are an infinite number of points Achilles must reach where the tortoise has already been, he can never overtake the tortoise. Of course, simple experience tells us that Achilles will be able to overtake the tortoise, which is why this is a paradox.[4][5]

This has very little to do with what we are discussing. This can be displayed with a linear graph (or simple algebra). Where the points intersect, is where dude catches turtle.

We are not talking about a linear relationship.

This was, to be honest, disappointing. It makes me think you have less understanding that I thought you did.

[TheProwler]

Did you read the proof I posted earlier? There is absolutely no reference to 1/3, or 0.333...

The "proof" I posted on the first page of this thread? It is another example of the limitations of standard algebra when dealing with....

Wait!! Do you read what I post?

[TheProwler]

Nonbelievers: do you agree that .333... = 1/3?

If so then:

.333... = 1/3
(1/3)*3 = 3/3 = 1
Therefore, .333...*3 = 1

.333... = sum of all 3/(10^n) as n goes from 1 to infinity.
(3/(10^n))*3 = 9/(10^n)
sum of all 9/(10^n) as n goes from 1 to infinity = .999...
Therefore, .333...*3 = .999...

If .333... = 1 and .333... = .999..., then .999... = 1. QED,MF.

I'll ignore the mistake...I know what you meant.

Nah, we'll never know until you write it out in long form.

Again, you've only exposed a limitation of common methods when dealing with infinitely long numbers.

[owheelj]

Ok here is a really really really easy question.

What is 0.999... / 3

[TheProwler]

Prowler, you are AWESOME!

This would have been such a nice way to end the thread.

[Timminz]

So, you are honestly trying to suggest that the vast majority of mathematicians are wrong?

Although, you have managed to confound me in respect to one thing. I can't tell which is greater, your ignorance, or your arrogance. Does anyone have a proof for that?

[TheProwler]

Ok here is a really really really easy question.

What is 0.999... / 3

Wow, how tricky!

Again, you a exposing a limitation....

I'll play along for a second.

0.333recurring

But, that is not equal to the 0.333recurring that we get when attempting to display 1/3 without using fractions.

Your 0.333recurring is slightly smaller than my 0.333recurring. I know this is hard to grasp.

It is similar to two lines on a graph that represent equations with a certain limit. Call that limit Z. Both equations approach Z. But one equation approaches it more quickly. It does not make the two lines equal and it doesn't make the two equations equal. You mentioned calculus before....you should understand this.

[TheProwler]

So, you are honestly trying to suggest that the vast majority of mathematicians are wrong?

Although, you have managed to confound me in respect to one thing. I can't tell which is greater, your ignorance, or your arrogance. Does anyone have a proof for that?

Does your head hurt?

[owheelj]

so what you're saying is 0.333... =/= 0.333...

You are right, that is hard to grasp.

[TheProwler]

so what you're saying is 0.333... =/= 0.333...

You are right, that is hard to grasp.

Use the limit example and it might help you to understand.

[owheelj]

To what degree are those two numbers different? How many digits to I have to go through those numbers before they diverge, and which number changes from a 3 and what does it change to?

Or are you saying that at particular point along those infinite lines of 3s, the 3 on one line will be worth less than the three on the other?

[owheelj]

Let's say that the smallest possible difference between two numbers is 0.0... 1; zero followed by an infinite amount of zeros and then a one.

Therefore if there is a difference between 1/3 and 0.999.../3 then it needs to be at least 0.0... 1 difference. Therefore at the end of the decimal of 1/3 or 0.999.../3 there needs to be a digit that is 3 +/- 1 (so 4 or 2), otherwise they actually are the same numbers. I would really like to know whether it is a 2 or a 4.

[pimpdave]

BEWARE THE CATS IN SPACE

[Snorri1234]

Cat, what are you doing in space?