OK, I see where you're going with the point about an illegal operation, but that still leaves us with a strange problem.

.999~ x 9 = 8.9~
.999~ x 10 = 9.999~
9.999~ - .999~ = 9

.999~ x 10 - .999~ does not equal .999~ x 9

I think this goes back to my original opinion on this equation- when you ask what happens when you do the impossible, you get a wierd answer. Would I be correct in saying that it is not possible to perform operations (not perfectly anyway) with a repeating decimal?

Then we both concur that 0 is nothing.

Yes symbolically we can represent the square root of -1 as i persea, but that is no different that assigning it to any variable. However finding it's real value simply cannot be done. Therefore, again, I concur with using i as a place holder for that unknown quanity in order to manipulate the rest of the system.

Addititionally it has properties that can allow it to be re-usable into a real system.

SQR(-1) * SQR(-1) = -1
SQR(-1) ^ 4 = 1

Is there a Doctor in the house,my head really really really hurts.

And 1/3=.333
.333 x 3= .999

You can't do the inverse of the operation due to round off error. When the original calculation was made ( 1 ÷ 3), the decision was made to drop off part of the value. The all powerful computer (No matter which one you choose) does not have enough memory to represent the digit 3 carrying on to infinity. This brings up a lot of issues such as the black hole forming near the infinite amount of mass used to store the infinite amount of memory. That and the fact that most wimpy manufacturers only give you 3-4 banks to put memory cards in, ect.

So for lack of memory, we make the decision to only keep the first million digits or so. And this is all fine since the result is close enough to give out the correct change or to put a shuttle in space. Now when you try to use the result in a calculation, you just blew it. You're now compounding the problem since you're starting with a flawed number to begin with.

Say you need a 6 foot ladder to reach a light bulb, and you have one that's 5 1/2 feet tall. Close enough right? Sure, you can still reach the light bulb with no problem. But now if you try to add two of your almost 6 feet ladders to make a 12 ft. ladder you're off by a whole foot. If you tried to combine your incomplete ladders to reach the top of a 20 story building, you would be way, way, way, way, off. How much round off we allow depends on how important the calculation is. For a one time calculation, we don't care that much, but........

An egregious example of roundoff error is provided by a short-lived index devised at the Vancouver stock exchange (McCullough and Vinod 1999). At its inception in 1982, the index was given a value of 1000.000. After 22 months of recomputing the index and truncating to three decimal places at each change in market value, the index stood at 524.881, despite the fact that its "true" value should have been 1009.811.

In short, you've just stepped out of the protective world of integers, and ran into "real world math" or real numbers. It can be a bit scary at times. Also, try not to reuse numbers that have been rounded.

.9~ and 1 are equivalent. You have the gist of it in the very first post here:

x=.9~ (1)
=> 10x = 9.9~ (2)

subtract (1) from (2) =>

10x - x = 9.9~ - .9~ =>
9x = 9 =>
x=1.0

A good way to think about it conceptually is to consider what it means for two numbers to be different. Here is one definition: Two numbers are different if and only if you can find a third number which lies between them. For example, .95 and 1.0 are different because I know that .97 lies between them.

Try to come up with a number greater than .9~ and less than 1.0. Hint: it is impossible.

Think about this:

1 - 0.9~ = an infinitesimally small number, 1 / infinity. We can't grasp 1/inf., but if we could then we'd be able to distinguish between 1 and 0.9~.

1/inf is basically 0, because we can't understand infinitely close to 0.

Okay... some mathematics doctoring: It depends on what you are doing this math for.

If you are doing math for physical engineering or for real world experiments, there's a certain "precision point" which is as good as you can cut/construct/whatever and "almost 1" becomes 1 in this system. You can't cut something to an infinite number of nines because we have no way of producing this.

Now... in dealing with theoretical math, point-9 to infinity is not the same thing as 1.

In theoretical math, there's all sorts of interesting and bizarre things that happen (in theoretical math there are systems where you CAN add infinity to infinity and come up with an answer other than "infinity.") These actually have real-world applications, only I am not well enough educated in math to go much beyond this point.

Now... in dealing with theoretical math, point-9 to infinity is not the same thing as 1.

No, they are exactly equivalent.

In theoretical math, there's all sorts of interesting and bizarre things that happen (in theoretical math there are systems where you CAN add infinity to infinity and come up with an answer other than "infinity.")

You are probably thinking about set theory, which involves some unintuitive notions about different sizes of infinity. For example, there are the same number of positive even numbers (2, 4, 6, ...) as natural numbers (1, 2, 3, ...).

Originally posted by utrex
You are probably thinking about set theory, which involves some unintuitive notions about different sizes of infinity.

Following that line of thought 10X - X = 8.999 because you have only subtracted the infinite remainder one time.
(-0.000... repeating ...1)


[edit on 24-3-2005 by Seth76]

Following that line of thought 10X - X = 8.999 because you have only subtracted the infinite remainder one time.

0.9 is, fortunately, equal to itself:

10X - X = 9.9 - 0.9 =>
9X = 9 + 0.9 - 0.9 =>
9X = 9

(-0.000... repeating ...1)

-0.000... repeating ...1 = 0

Yes, I really can believe that 1/3 * 3 = 1. I would hate to think that by splitting a candy bar with two friends someone was loosing out.

However, understanding these concepts is what separates us from computers. The original equation wasn't stated as a fractional division. "Where X = 0.999" and 10 * X - X = 8.999 with a remainder of -9 * 0.000...1

Originally posted by utrex

Now... in dealing with theoretical math, point-9 to infinity is not the same thing as 1.

No, they are exactly equivalent.

That's not what the mathemeticial I married told me... now, he could have been giving me a hard time, but he does make a distinction with these things. .9-to-infinity APPROACHES 1 but is never exactly equal to 1.

You are probably thinking about set theory, which involves some unintuitive notions about different sizes of infinity. For example, there are the same number of positive even numbers (2, 4, 6, ...) as natural numbers (1, 2, 3, ...).

Partly, yes, but I was also thinking of (I believe they're called) the Lee(Lie?) Algebras and others.

From what I remember of my mathematics classes .9 repeating is indeed equal to 1, not just approaching it in a calculus sense.

In fact, there really is no such thing as .9 repeating...all rational numbers should be expressable as a fraction, .3 repeating is 1/3, .6 repeating is 2/3. But how would you express .9 repeating as a fraction? You really can't...it's 9/9 which equals 1. It's more of an anomaly or quirk in the way we express numbers that we can even say .9 repeating as something different than 1, it is not.


[edit on 3/24/2005 by djohnsto77]

Talking about infinities is infinitely long...

What I remember of infinities, is that there is an infinite number of infinities, some of them are countable (sorry if it is not the right word in English for that, I did my maths in French), some are not.

The set of infinities is a set called Aleph (no such character on my keyboard, sorry), which is the hebrew 'a'. Aleph0 (Aleph-zero) is the smallest countable infinite set, and the number of natural numbers equals the number of elements in aleph0.

Now, talking about 0... Even in real numbers, 0 is bizarre. You need the get to an infinite precision if you want no break between 0 and the next positive/negative number. Indeed, the first (let's say) positive number after 0 has to be 0.000...1, with an infinite number of 0's between the decimal point and the 1. So, basically, even real numbers are not real, they're just theoretical.

Originally posted by djohnsto77
But how would you express .9 repeating as a fraction?

I believe .9 recurring could be expressed as a fraction of infinity.

I really don't understand the point anyway. In the physical universe there's a point in all measurements where .9 recurring and 1 become the same.

It's the same with 0. You can't express 0 unless you have something to compare it against. You can't define 0 nothings. There's no pattern or structure and therefore mathematics has no solution.

In your proof that .333...*3=.999... you forgot to include the fact that .3333... is NOT 1/3. 1/3 if not a number that can be turned into a decimal in any way. I thought someone might like to know this fact.



PS:Slipknot Rocks

.999999 is equivalent to 1 because there's nothing you can do to it to distinguish it from 1. Infinity isn't a thing you can grasp, it's a concept you accept and work with to achieve certain results. Infinity isn't really anything, and so you get weird results (infinity * 0 can be equivalent to an integer). There's an infinite amount of integers, and an infinite amount of real numbers. There are, however, more real numbers and integers. And that isn't even saying anything about what happens when you count all the integers (omega, 2omega, omegaSquared, etc.).

Also, don't forget 1=1.000000. This is why the system of significant digits was invented, to be practical.

Man there's a lot of math misinformation in this thread

It's real simple. Given an amount of precision, .999999 is equivalent to 1 or it can be just .999999. It all depends upon how many decimal places you're willing to go before you round. In applied math, like engineering and the like, you work with precision.

In real math, theoretical math, all your numbers have infinite precision and therefore, you're working with infinite precision. Because of this, it's clear that
.999999 is NOT equivalent to 1.

[note: this is exactly what Byrd has written, but everyone seems to be ignoring it.]