Does 0.9999 repeating 9 equal 1?

[Larry]

I've decided to start claiming that the slope of the function y = x^2, at x=0 isn't 0.

See, it's really, really, close to zero. But when you take the function at x=0, and take the function at x=0+(some really, really small number), y will never exactly equal zero, and therefore the slope will never exactly equal zero.

Congratulations! I've just proven that basic Calculus doesn't work! It's just an approximation. An approximation that will never be right! I'm smarter than Newton!

[PeterMP]

I'm fine with that. And I can see where you're going. Point taken. Thank-you :thumbsup:

If you are really fine with that, especially when taken to the extreme (two points that are not identical, but infinitely close (even 1/infinity close) have as many points between them as two points that are infinitely far apart (an infinite number)), then you are geometric savant and truly missed your calling, and I'd like to talk to you sometime about modeling the distance between two points that are not identical in a space with an infinite number of dimensions sometime.

[DjTj]

the limit as the number of integers of .333 approaches infinity is 1/3

the limit as the number of integers of .999 approaches infinity is 1

Yes, and .999 repeating is the limit of the series as it approaches infinity. That's what "repeating" means. It goes to infinity.

.999 repeating = the limit of the series as it approaches infinity = 1.

I don't know how you can understand limits and not understand that .999 repeating is a limit.

[PokerPacker]

Yes, and .999 repeating is the limit of the series as it approaches infinity. That's what "repeating" means. It goes to infinity.

.999 repeating = the limit of the series as it approaches infinity = 1.

I don't know how you can understand limits and not understand that .999 repeating is a limit.

I swore I was done after page 20, but I have to clear this up. I'm saying that limits are not the same as what something truly is. You can have a graph where the limit at point x is value y, but there is no actual point x therefore no value y at point x. there is only the limit as the graph approaches that point.

[PeterMP]

Yes, and .999 repeating is the limit of the series as it approaches infinity. That's what "repeating" means. It goes to infinity.

.999 repeating = the limit of the series as it approaches infinity = 1.

I don't know how you can understand limits and not understand that .999 repeating is a limit.

Because it isn't really that simple due to the nature of infinity.

Let me try again. From everything said here, I'd assume you agree that 1/9999... = 0

I have two points that aren't the same. The distance between them IS NOT 0. They are as close as you can get w/o them being 0. I have a ruler that measure infintismly small distances. However small the distance you want to measure, it can measure it. What is the distance between my two points?

PokerPacker, talk to your calculus professor. Maybe hearing it from him will make you happy. We aren't talking about the limit, but the existance at an infinite number of 9s, which makes the limit essentially irrelevant, which while practically is impossible isn't mathematically irrelevant.

[Larry]

I swore I was done after page 20, but I have to clear this up. I'm saying that limits are not the same as what something truly is.

They are when the thing is defined to be that way.

The words "0.9 repeating 9" specify an infinite number of 9's.

Not "well, pick a big number, and pretend it stops there".

Not "well, pretend it starts out short but it's getting longer".

An infinite number of 9's.

Just as the definition of "slope" is a limit (taken to infinity). That's it's definition.

The slope of y = x^2, at x=0, doesn't approach zero. It is zero. Because the definition of slope specifies the limit as delta-x approaches zero.

-----

I swore I was done after page 20,

I'm only on page 16, and I thought I had my preferences set for the smallest number of posts per page.

Obviously I'm sitting in a distortion of space-time.

[jnhay]

http://www.youtube.com/watch?v=kJJA1vvMc4I

[Mark The Homer]

If you are really fine with that, especially when taken to the extreme (two points that are not identical, but infinitely close (even 1/infinity close) have as many points between them as two points that are infinitely far apart (an infinite number)), then you are geometric savant and truly missed your calling, and I'd like to talk to you sometime about modeling the distance between two points that are not identical in a space with an infinite number of dimensions sometime.

Haha - I'm fine seeing the direction you're going and the point you're making - I'm not saying I necessarily understand it or even necessarily agree with it. But I see what you're saying and I'm acknowledging that you have a point and I think this is an area of math that needs to be studied to understand it. And I'm not that guy.

I swore I was done after page 20, but I have to clear this up. I'm saying that limits are not the same as what something truly is. You can have a graph where the limit at point x is value y, but there is no actual point x therefore no value y at point x. there is only the limit as the graph approaches that point.

The problem, PP, is you keep talking about limits, when nobody else is. This issue has nothing to do with limits.

"You can't see the forest for the trees" is a perfect analogy. You're making it way more complicated than it is.

The proofs above are indisputable.

[PokerPacker]

The slope of y = x^2, at x=0, doesn't approach zero. It is zero. Because the definition of slope specifies the limit as delta-x approaches zero.

slope = derivative. derivative of x^2 is 2x. 2x at x=0 is 0

[Thinking Skins]

slope = derivative. derivative of x^2 is 2x. 2x at x=0 is 0

but the definition of derivarive is based on limits. its the rate of change in f(x) divided by the rate of change in x.

lim (\delta x -> 0) (f(x + \delta x) - f(x))/ (\delta x)

In order to be able to say that the derivative we've got to be able to talk about what happens as \delta x APPROACHES 0. we can't talk about what happens when it reaches 0 because when it reaches 0, we're dividing by 0, which is not allowed.

The .999... example is the same way. We can't talk about what happens WHEN THE LIMIT REACHES infinity because it never reaches infinity. However, we can still calculate the limit AS IT APPROACHES infinity, and in this case the limit is 1.

So we say that the sum .9 + .09 + .009 + .0009 + ... converges to 1,

also written as .9999999... = 1,

also written as \sum_{k = 1 to infinity} 9 * 10^-k = 1

[velocet]

This proof:

x=.999...

10x = 9.999...

10x-x= 9.999... - .999...

9x=9

x= 1

.999...= 1

...can be found in the first part of this book:

...which might be a useful little guide for the mathematicians in this thread in case they ever want to show someone without benefit of mathematical maturity a thing or two about proofs.

The math majors might get a chuckle from this vignette:

I had just gotten my copy of this book:

...freshly unwrapped, it was sitting out, and a friend of mine stopped by.

He sees it sitting on the table starts and with the laughing.

"Algebra? hahahahaha! ALGEBRA?!?!!?! AHHHHAHAHAHAHA! Dude, I did that in like 9th grade. (more insults at my expense, more laughing, etc)

He was having such a good time busting on my apparently slow progress in math I just let slide.

velocet

[Larry]

slope = derivative. derivative of x^2 is 2x. 2x at x=0 is 0

And the reason the derivitive of x^2 is 2x, is because 2x is the slope.

Not approximates the slope. Not "the slope is really, really, close to 2x, but never quite gets there". It is the slope.

Because the definition of the slope is the limit of (f(x+dx)-f(x))/dx, as dx approaches zero. The definition of slope is the limit.

You cannot pick a dx, such that ((0+dx)^2-0^2)/dx equals zero. (Other than zero, and you can't use zero, because then the slope is dividing by zero).

But the slope does, in fact, equal zero. Absolutely and totally.

Well, the definition of the value of 0.9 repeating 9, is the limit, as the number of 9's becomes infinite.

0.9 repeating 9 doesn't approach 1, as it gets longer. Because it's already infinitely long. By definition. It doesn't approach 1. It is 1.

[DjTj]

I swore I was done after page 20, but I have to clear this up. I'm saying that limits are not the same as what something truly is. You can have a graph where the limit at point x is value y, but there is no actual point x therefore no value y at point x. there is only the limit as the graph approaches that point.

We aren't talking about an actual point x. 0.999... is not an actual point. It is a limit. The limit is 1.

Because it isn't really that simple due to the nature of infinity.

Let me try again. From everything said here, I'd assume you agree that 1/9999... = 0

I have two points that aren't the same. The distance between them IS NOT 0. They are as close as you can get w/o them being 0. I have a ruler that measure infintismly small distances. However small the distance you want to measure, it can measure it. What is the distance between my two points?

At any point that you can actually measure, the distance is 1/9999... (with however many 9's where you stopped). But we're not talking about actual points or actual measurements. 1/9999... is not an actual point in space that you can measure with a ruler. It is a limit as the number of 9's approaches infinity. And the limit equals 0.

slope = derivative. derivative of x^2 is 2x. 2x at x=0 is 0

As Thinking Skins points out, slope is not equal to derivative. The limit of the slope as delta approaches zero is the derivative. When you say slope = derivative, you are really talking about a limit.

There are no two adjacent points in the function x^2 where the slope is 0. Everything on the right side of zero has a positive slope, and everything on the left side has a negative slope. There are no two adjacent points where it IS zero.

...but as you say, it IS zero. Because the limit is zero. When we say the slope is zero, we mean that the limit is zero. When we say 0.9999 repeating IS 1, we mean that the limit is 1.

(it depends on what your definition of "is" is).

[Spec138]

This proof:

x=.999...

10x = 9.999...

10x-x= 9.999... - .999...

9x=9

x= 1

.999...= 1

Arg this proof has been posted like 1203021321 times.

What are ya'll arguing about now PP?

[Thinking Skins]

...but as you say, it is zero. Because the limit is zero. When we say the slope is zero, we mean that the limit is zero. When we say 0.9999 repeating is 1, we mean that the limit is 1.

(it depends on what your definition of "is" is).

ftw

[PeterMP]

We aren't talking about an actual point x. 0.999... is not an actual point. It is a limit. The limit is 1.

At any point that you can actually measure, the distance is 1/9999... (with however many 9's where you stopped). But we're not talking about actual points or actual measurements. 1/9999... is not an actual point in space that you can measure with a ruler. It is a limit as the number of 9's approaches infinity. And the limit equals 0.

My ruler will measure out to 1/999999999....

The distance IS NOT zero. What is the distance?

**EDIT***

Even better w/ respect to the question. The distance between my two points is not one (using whatever units you want), but is as close to one as you get. What is the distance?

[skinsfan07]

Any doubt that I had about your screen name was just shattered by this post right here.

lol agreed! Thinking Skins has the perfect username. haha

[Spec138]

My ruler will measure out to 1/999999999....

The distance IS NOT zero. What is the distance?

**EDIT***

Even better w/ respect to the question. The distance between my two points is not one (using whatever units you want), but is as close to one as you get. What is the distance?

If there is no number between the two they are, in fact, equal.

[PeterMP]

If there is no number between the two they are, in fact, equal.

I'm stipulating that they aren't equal (or they aren't 1 whatever units apart in the context of the discussion). They are as close to one another (or in the context of the discussion as close to being 1 whatever units apart) as possible.

My ruler will measure infintismal distances. How far are they apart?

[Mark The Homer]

I think I've just realized what the problem is.

If you'll notice, the ones in this thread that refuse to accept that .999 repeating equals one and refuse to accept that .333 repeating equals one-third - all these members are YOUNG. They're kids! Look at their ages! They're like nineteen. :doh:

These kids have done math on calculators all their lives! And because of that, they are limited in their thinking and in their minds. They are limited to eight place marks or sixteen place marks or however many their respective computer/calculator goes to. That's why they don't understand the concept of, for example, .333 repeating equals EXACTLY one-third. Their calculator only tells them it's close. But not exact. And they believe it!!! Their calculators have been giving them bad information for their whole lives in respect to infinitely repeating numbers. And these kids have never known anything different.

Those of use who grew up doing math with a pencil and a sheet of white paper can think outside 16 place marks. That's why we understand that .333 repeating is EXACTLY one-third.

[LaxBuddy21]

I think I've just realized what the problem is.

If you'll notice, the ones in this thread that refuse to accept that .999 repeating equals one and refuse to accept that .333 repeating equals one-third - all these members are YOUNG. They're kids! Look at their ages! They're like nineteen. :doh:

These kids have done math on calculators all their lives! And because of that, they are limited in their thinking and in their minds. They are limited to eight place marks or sixteen place marks or however many their respective computer/calculator goes to. That's why they don't understand the concept of, for example, .333 repeating equals EXACTLY one-third. Their calculator only tells them it's close. But not exact. And they believe it!!! Their calculators have been giving them bad information for their whole lives in respect to infinitely repeating numbers. And these kids have never known anything different.

Those of use who grew up doing math with a pencil and a sheet of white paper can think outside 16 place marks. That's why we understand that .333 repeating is EXACTLY one-third.

Or the older generation has not been exposed to the higher level of math actually required to understand the concept of limits and infinity. I will qualify myself here by stating I am currently getting my graduate degree in statistics at GW right now and have spent a lot of time with higher level math courses and especially limits. Infinity for example is not a tangible number. Infinity only exists conceptually. .9999999.... APPROACHES 1 as the number of digits reaches infinity. It never gets to 1. NEVER! Its an asymptote to 1 like many others have said. The reason for limits is to approximate values as they become extremely large to turn a continuous value into something discrete so it can be worked with. Its a manipulation in order for mathematicians to be able to work with functions. .99999.... is 1 for all practical purposes meaning if we were to use it in some way but only because we have manipulated it. The true value is and never will be 1.

[Chopper Dave]

Actually, I seem to recall mathematically proving, back in High School, that the answer is "yes".

Although I've always thought of myself as more of an engineer than a mathematician.

And to an engineer, "close enough" = "yes".

You are right, Larry. This wiki talks pretty extensively about the proof.

[PokerPacker]

I think I've just realized what the problem is.

If you'll notice, the ones in this thread that refuse to accept that .999 repeating equals one and refuse to accept that .333 repeating equals one-third - all these members are YOUNG. They're kids! Look at their ages! They're like nineteen. :doh:

These kids have done math on calculators all their lives! And because of that, they are limited in their thinking and in their minds. They are limited to eight place marks or sixteen place marks or however many their respective computer/calculator goes to. That's why they don't understand the concept of, for example, .333 repeating equals EXACTLY one-third. Their calculator only tells them it's close. But not exact. And they believe it!!! Their calculators have been giving them bad information for their whole lives in respect to infinitely repeating numbers. And these kids have never known anything different.

Those of use who grew up doing math with a pencil and a sheet of white paper can think outside 16 place marks. That's why we understand that .333 repeating is EXACTLY one-third.

In calculus, we do not use calculators. When I was in high school, I lost my calculator and went through the entire year of Algebra 2 without it and aced the class. Hell, I got 100% on the SOLs.

[Spec138]

Or the older generation has not been exposed to the higher level of math actually required to understand the concept of limits and infinity. I will qualify myself here by stating I am currently getting my graduate degree in statistics at GW right now and have spent a lot of time with higher level math courses and especially limits. Infinity for example is not a tangible number. Infinity only exists conceptually. .9999999.... APPROACHES 1 as the number of digits reaches infinity. It never gets to 1. NEVER! Its an asymptote to 1 like many others have said. The reason for limits is to approximate values as they become extremely large to turn a continuous value into something discrete so it can be worked with. Its a manipulation in order for mathematicians to be able to work with functions. .99999.... is 1 for all practical purposes meaning if we were to use it in some way but only because we have manipulated it. The true value is and never will be 1.

If you consider yourself well versed in mathematics, I would hope that you accept .999... = 1. The hole reason .999...=1 is that infinity is not tangible.

I'm still yet to see anyone disprove my proof.

[motorhead]

Those of use who grew up doing math with a pencil and a sheet of white paper can think outside 16 place marks. That's why we understand that .333 repeating is EXACTLY one-third.

I knew we were smarter than the young bucks.

We weren't even allows to use calculators.

I meant adding machines.(Maybe I'm the dumb one.)