Does 0.9999 repeating 9 equal 1?

[Spec138]

lets say you start off at point 0. and every step you take you get half way closer to point 1. YOU WILL NEVER REACH POINT ONE!!!! EVER. But the difference between the two will be so infinitesimal that we just say that you arrive at point one.

Here is what you would get if you were to run a program that would calculate this.

Well isn't the concept of infinity is that it's unattainable?

Basically you could say that this series:

.9x10^0 + .9x10^1 + .9x10^2... is equal to .999... correct?

So each time you add another part of the above series, you are reducing the error between the series and 1 by a factor of 10.

Since you are reducing the error each time, as you approach infinity your error would be 1/(infinity) of what it originally was.

1 - 1/infinity = 1.

[PeterMP]

lets say you start off at point 0. and every step you take you get half way closer to point 1. YOU WILL NEVER REACH POINT ONE!!!! EVER. But the difference between the two will be so infinitesimal that we just say that you arrive at point one.

Here is what you would get if you were to run a program that would calculate this.

You're right, but that isn't really the question. People in this thread seem to be talking past one another.

Let me give you another example. Let's say that we have an infinitely long board and somebody starts writing 0.99999...... Now as they add nines the number gets bigger, and we could say that as time (the person writing has more time) approaches infinity the number approaches 1, but it never reaches one.

However, that isn't the question. The question is GIVEN the number 0.999999 out to infinity (the number has an infinite number of nines), is it equal to 1?

The answer is yes, it is one as indicated in the wiki link somebody provided before.

All of the people that are saying it isn't one are thinking with respect to approaching it, but that isn't the case or answer here. You don't have the person at the board writing the numbers. You have to be able to imagine the full blown number coming into existance already with an infinite number 9s.

[boofMcboof]

http://www.freshminds.com/animation/alan_watts_prickles.html

[Slacky McSlackAss]

Well isn't the concept of infinity is that it's unattainable?

Basically you could say that this series:

.9x10^0 + .9x10^1 + .9x10^2... is equal to .999... correct?

So each time you add another part of the above series, you are reducing the error between the series and 1 by a factor of 10.

Since you are reducing the error each time, as you approach infinity your error would be 1/(infinity) of what it originally was.

1 - 1/infinity = 1.

You cant divide by something non existant.

[PeterMP]

If. there. is. no. number. between. them. comma. then. they. are. the. same. number. period.

Well actually the number in between 0.99999 (an infinite number of 9s) and 1 is 0.99999 (an infinite number 9s + 1 more 9).

But they are still the same.

[Larry]

300 places is infinitely short of what you need.

Look at the proof. You can't refuse to accept the truth when it's staring you in the face.

OK. Maybe it is like a political thread.

[Slacky McSlackAss]

300 places is infinitely short of what you need.

Look at the proof. You can't refuse to accept the truth when it's staring you in the face.

Alright Mark. I've done my research and looked at the proof. I've come to the conclusion that they are essentially the same thing. Looks like PP is left fighting for the not equal side.

[Mark The Homer]

You're right, but that isn't really the question. People in this thread seem to be talking past one another.

Let me give you another example. Let's say that we have an infinitely long board and somebody starts writing 0.99999...... Now as they add nines the number gets bigger, and we could say that as time (the person writing has more time) approaches infinity the number approaches 1, but it never reaches one.

However, that isn't the question. The question is GIVEN the number 0.999999 out to infinity (the number has an infinite number of nines), is it equal to 1?

The answer is yes, it is one as indicated in the wiki link somebody provided before.

All of the people that are saying it isn't one are thinking with respect to approaching it, but that isn't the case or answer here. You don't have the person at the board writing the numbers. You have to be able to imagine the full blown number coming into existance already with an infinite number 9s.

Well said.

The concept of infiniti as it relates to an infinite number of repeating 9's is difficult to understand, because in our minds, it's always a work in progress. But in math, infiniti is not a work in progress. It's a finished product.

[Spec138]

You cant divide by something non existant.

Ok let's try it this way then.

Our error is reduced by a factor of ten as I sad before each time we add a number to the series I posted.

This would mean the error between 1 and (.9x10^0 + .9x10^1 + .9x10^2...) is equal to 1/(10^x), where x is equal to how many numbers we have added (i.e.: For .9*10^1, x = 1; for .9*10^2, x = 2 and so on.)

So our series is equal to one minus our error or:

.9x10^0 + .9x10^1 + .9x10^2... = .999... = 1 - error

As 1/(10^x) approaches infinity it equals 0.

Therefore, .999... = 1 - 0 = 1

[Mark The Homer]

Alright Mark. I've done my research and looked at the proof. I've come to the conclusion that they are essentially the same thing. Looks like PP is left fighting for the not equal side.

You're a wise man.

[Mark The Homer]

OK. Maybe it is like a political thread.

:rotflmao:

[Slacky McSlackAss]

Ok let's try it this way then.

Our error is reduced by a factor of ten as I sad before each time we add a number to the series I posted.

This would mean the error between 1 and (.9x10^0 + .9x10^1 + .9x10^2...) is equal to 1/(10^x), where x is equal to how many numbers we have added (i.e.: For .9*10^1, x = 1; for .9*10^2, x = 2 and so on.)

So our series is equal to one minus our error or:

.9x10^0 + .9x10^1 + .9x10^2... = .999... = 1 - error

As 1/(10^x) approaches infinity it equals 0.

Therefore, .999... = 1 - 0 = 1

I didn't think smart cowboys fans existed.

[Spec138]

I didn't think smart cowboys fans existed.

Few and far between. Trust me I know.

[Springfield]

I find this to be a very interesting thread.

When I first opened it up, my thought was that .999 repeating does NOT equal 1.

Then I got to thinking... If .999 (repeating) does not equal 1, what amount would you have to subtract from 1 to get .999 (repeating). How much would you have to add to .999 (repeating) to get to 1?

As far as my math knowledge goes, there is no .000 (repeating) 1.

[Spec138]

I find this to be a very interesting thread.

When I first opened it up, my thought was that .999 repeating does NOT equal 1.

Then I got to thinking... If .999 (repeating) does not equal 1, what amount would you have to subtract from 1 to get .999 (repeating). How much would you have to add to .999 (repeating) to get to 1?

As far as my math knowledge goes, there is no .000 (repeating) 1.

Exactly that's another proof I've seen.

1-.999...=.000...

[Slacky McSlackAss]

I think the argument lies in the fact that they are not the same number when looking at them on paper. But for the sake of human mathematics they are essentially the exact same thing. People, like myself, are (or were) over-thinking it, but I have seen enough proofs to be convinced.

[Slacky McSlackAss]

I find this to be a very interesting thread.

When I first opened it up, my thought was that .999 repeating does NOT equal 1.

Then I got to thinking... If .999 (repeating) does not equal 1, what amount would you have to subtract from 1 to get .999 (repeating). How much would you have to add to .999 (repeating) to get to 1?

As far as my math knowledge goes, there is no .000 (repeating) 1.

Why cant there be, its a number on the number line?

[Spec138]

Why cant there be, its a number on the number line?

An infinite amount of zeros can't end, much less in a 1.

[Springfield]

Why cant there be, its a number on the number line?

Sure it is but there isn't any way to express it... is there?

[PeterMP]

Well said.

The concept of infiniti is difficult to understand, because in our minds, it's always a work in progress. But in math, infiniti is not a work in progress. It's a finished product.

Well that's the fun with infinity. It is in fact both simultaneously, which let's you do some crazy things. For example, in the spirit of this thread. Most have agreed that

1= 0.9999(infinite number of 9s)

However using the same logic

0.9999(infinte number of 9s)= 0.999999(infinite number of 9s with one 8 at the end)

And:

0.999999(infinite number of 9s with one 8 at the end)=0.999999(infinite number 9s with one 7 at the end)

And:

0.999999(infinite number of 9s with one 7 at the end)=0.999999(infinite number 9s with one 6 at the end)

Therefore:

0.999999(infinite number 9s with one 6 at the end) = 1

I could keep going and end up with the conclusion that 0.5 (with an infinite number of some number trailing it) = 1, which everybody would automatically reject.

[Spec138]

Well that's the fun with infinity. It is in fact both simultaneously, which let's you do some crazy things. For example, in the spirit of this thread. Most have agreed that

1= 0.9999(infinite number of 9s)

However using the same logic

0.9999(infinte number of 9s)= 0.999999(infinite number of 9s with one 8 at the end)

And:

0.999999(infinite number of 9s with one 8 at the end)=0.999999(infinite number 9s with one 7 at the end)

And:

0.999999(infinite number of 9s with one 7 at the end)=0.999999(infinite number 9s with one 6 at the end)

Therefore:

0.999999(infinite number 9s with one 6 at the end) = 1

I could keep going and end up with the conclusion that 0.5 (with an infinite number of some number trailing it) = 1, which everybody would automatically reject.

An infinite amount of anything can't have anything "on the end". It doesn't end.

[PeterMP]

An infinite amount of anything can't have anything "on the end". It doesn't end.

Sure it can. Take an unending string of 9s and add something to the end.

[Spec138]

Sure it can. Take an unending string of 9s and add something to the end.

There is no end to an unending string. You're thinking finitely.

[Mark The Homer]

An infinite amount of anything can't have anything "on the end". It doesn't end.

I agree.

[jnhay]

According to your philosophy, 1/4, 4/16, .25, 0.25, and .250 are all distinct and unique numbers.

Those number equal the same amount. .25 isn't equal to .255. 0 is obviously not going to change a number. Just because it's close doesn't mean it's the same. The number approaches 1 forever. In practical uses, you might as well call it 1, but number literally mean that it can never be one.

EDIT: if .9 repeating equals 1, then .9 repeating doesn't exist. It's really just an imaginary number. People shouldn't use it if they're just going to say that it equals one.