A question of numbers (one for the maths fans)

It's a little wierd, I'll admit that.It's a matter of precision--and I like precision. Besides, that you you avoid making the mistake of writing .499 instead of .499 . . ..Technically speaking, .499 . . . does not equal .999 . . ., because
.499 . . . is actually .5. And you'll never get .999 . . . to be .499 . . . when dividing by 2. It will always be .5.Just a matter of precision

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All a man's knowledge comes from his experiences

Others have answered this but I'm not sure they got the point across..9 is 1/10 different from 1, or 0.1
.99 is 1/100 different or 0.01
.999 is 1/1000 different or 0.001
.999999 is 1/1000000 different or 0.0000001
Notice as I add 9's on the end the difference is getting very very small.With unending 9's the difference is zero..3 is 0.1 different from .4
.33 is 0.07 different from .4
.333 is 0.067 different from point 4
.3333 is 0.0667 different
.33333 is 0.06667 differentNotice that the difference is NOT getting very small.ABE the difference is not getting closer to zero it is getting closer and closer to 0.06666...Edited by NosyNed, : a little more explanation

I am dumb! I still don't get it.I define precision to be the number of digits I have.In addition you meant I presume, Technically ... does not equal .999... divided by 2 is what you meant.The precision of all the numbers you give is infinite in that they are all repeating .Thus 0.4999... is exactly equal to 0.5. So if half of .99999... is .5 it is also equal to 0.4999...Let me try the division:
2 into .9 is 0.45
2 into .99 is 0.495
2 into .999 is 0.4995and so on giving us
2 into .999 is 0.4999...Can you explain where I am confused?

It's partially my fault.If you divide .999... by 2, you will get .5 or .499....
But since .499... = .5, it's easier to write .5 than .499....
And my calculator will never give me .499... as the answer to x = .999.../2. Just .5.I like precision, but if .5 is = .499..., then why not write .5?
Besides, it helps prevent you from screwing up and writing .499 instead of .499...--and that can be a major difference.So technically, you are right. It's just better to write .5 in my opinion.

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All a man's knowledge comes from his experiences

Nobody is treating it as a number as far as I can see. The reason the zero doesn't get put there is because there is nowhere to put the number. The zero merely says 'no more', which doesn't make sense with the sum of an infinite series.The multiplication simply adds a 9/1 to the start of the series so it becomes9/1 + 9/10 + 9/100 9/1000 ... Why? The zero doesn't exist! It only gets mentioned when you shift a finite number of digits, and not always then (since if it comes after the decimal point it doesn't mean anything); but it was there already!0.9990 * 10 = 9.990If you can show us where the zero is at the end of 0.999... The result is different! The maths shows us clearly that 0.999... = 1 and that 0.333... ≠ 4I agree it looks crazy, but unfortunately for your sensibility it is true and the maths basically proves it.

It isn't helpful here. It is important not to confuse things any more than they are.But of course we would use 0.5 almost all the time.

Ok, that makes sense. But I still don't see why with unending 9's we can't have an unending difference. It's just like admitting infinity doesn't exist.

No, because if we put it into fractions, it makes perfect sense. Its just that decimal representations are a little funny. When I was studying maths, I always preferred to give the answer in fractions where possible - they are more beautiful As such 0.999... is easier to represented by 9/9 or just 1. So I'd write2 * 9/9 = 18/9 = 2And the jobs a good 'en. Time doesn't enter into infinte geometric series's. There are well defined and well proven methods for manipulating them in consistent ways. If that wasn't the case, Newton wouldn't have come up with calculus, and we wouldn't have any modern technology. Sorry - I didn't see this before my previous response. The concept is mind blowing and in maths it is easy to get lost. This stuff is OK with me now, but I steadfastly refused to accept that in calculus it is possible to divide (edit: the area under) a curve into infinitely small areas and then add the infinite number of infinitely small areas together. Eventually, I got it, but it was kicking and screaming. So, your position is one I can understand.Edited by Modulous, : No reason given.

Message 161Kind of shows how important it is to have that number at the end.In all of kuresu's explanations, he always makes that leap from .999... to 1, or .4999... to .5If we are going to start changing numbers before proving they are that number, then I have aproblem with it.Even kuresu admits that it is not precise. So I don't see how what I am saying is so off-beat..4999... * 2 just does not equal .999... Unless you have a 5 at the end of .4999... or you convert it to .5Just because his calculator says so, doesn't make it fact.It seems to me, there is no valid way of using recurring decimals as part of an equation other being an answer. It's like we are making rules to fit the problem, similar to creationists trying to make the evidence fit their theory.Also, every time we use a finite symbol to represent infinity, an infinite string, we lose what infinite is actually supposed to be.

riVeRrat You are close RR but you need to take that last step. Remember that the number .999... as an infinite series is simply a repetion of the same number over and over. Each 9 in the series has a corresponding 9 being subtracted. Since both are infinite they are the same size and thus we are simply subtracting a number {though infinite} from exactly the same number.
Since numbers are simply symbols representing an agreed upon value we can easily introduce the symbol ∞ for infinity,set it as equivalent to .999..., and proceed with subtraction and we get ∞ - ∞ = 0. Edited by sidelined, : Edit for clarification

That's what I said way back in the beginning, that the problem lies within our number system, and decimals being able to express thirds of 10.Of course we can divide 10/3, but how we express it is important. Thanks for your time in explaining all this to me.That's a whole lot better than kindergarden explanations of infinity

I understand that thought, but that very thought, seems to go against the very concept of infinity, or an infinite set of numbers. Exactly, I can understand that. I am fine with that, but the possibility remains that this also could be wrong. Like I showed before:∞ - ∞ +1 = 1
∞ +1 - ∞ = 0So it would seem to matter very much where infinity gets placed in an equation. Where as whole numbers it does not. That means one of several things.
Infinity is way beyond our grasp of understanding it.
Infinity cannot be used in regular equations.
Infinity is actually another demension (lol)
etc.

It shows we should treat the numbers in a correct manner. 0.4999... does equal exactly 0.5
0.9999... does equal exactly 1You can't have a '5' at the end of an infite series! However, one can prove that 0.499... = 0.5, and the proof follows from that. kuresu has clarified this point - it is precise, but the notation 0.4999... is crap - just use 0.5. 'Seems to' won't cut it, of course. Creationists use 'seems to'. Mathematics really can't afford 'seems to'. It is entirely valid to use recurring decimals (or any other irrational number like π ), if we know how to manipulate them correctly. It's difficult to understand it, but it really can be done. As I said - if we can't add an infinite amount of things together to form a definite solution, we can't do calculus and we're cream crackered...as far as any technology goes.

No, you missed what I said.When I was evaluating one of your equations, you used .499... at one point, and .499 at another. The difference is important. Because if the number is .499... (or .5), then it is half of .999...But if the number is .499, then it is NOT half of .999....Somewhere along the line, you wondered if .499 was half of .999. And I said no--.4995 is half of .999. The important fact here is that these decimals are nonrecurring. They stop. Which changes things completely.The reason I make the jump from .999... to 1 is because of Modulous's equation that proves that .999... = 1. The same equation can be used to prove that .499... =.5.Keep in mind, my calculator, a Ti-86, is programmed by professional mathematicians--and if they can't do the actual programming, then they explain the operations to the programmer. Either way, my caculator is right, because it follows the very same rules that Modulous used to prove that .999... IS 1.

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All a man's knowledge comes from his experiences

I suspect you are conflating physics and math. The universe maybe finite. Infinity doesn't exist in the real world. Neither do numbers. Things exist and we count them. But if you go looking for 1 or 2 they don't exist except as math concepts.I find an apple, I find another apple. If I can count I say I had no apples, found 1 then found another so I have two apples. So we count 1,2,3,... What does the [...] mean? It means in my mind I can go on adding 1 without ever stopping. Now if we could count all the apples in the world we would reach an end. They are finite.But in math the set of natural integers is infinite. Is this true of the universe? It is not neccessary that we know that. In math what we know is that we can add 1 to n, n+1 without limit. That is what we mean by infinity. It's not real, it doesn't exist, it's not a number, it's a property of some sets that they aren't limited. They are limitless. You can go on counting them forever and never reach an end.Now take the series of halving a distance. 1/2 then half of that is 1/4 then half of that is 1/8 and on and on. You can do this with out end because 1/2n for the natural numbers means n is not limited. It can get huge and 1/2n gets very very small. If you sum this infinite series up it will equal 1.This is an intuitive solution to Zeno's paradox. Zeno saw that you would have to go half way across a room, then half way across the remaining distance, then half of that, and the series was infinite so he stated you can't ever cross the room. But you can. Why? because intuitively the sum of 1/2+1/4+1/8... to infinity equals 1.Now the calculus does this rigorously. Decades ago I took a course for a year where the professor proved the entire calculus. Just wrote proof after proof on the board. I can't recall all that and I can't begin to reproduce it here. You would need to take a college course on calculus. But intuitively you should be able to take this and see what is happening.The other thing is you sometimes are confusing a notation with an operation. Notation in math is variable as you have seen 1/2=.5=.4999... or even 500/1000 or 1x/2x etc. etc. Don't confuse notation with what is notated. Although 1x/2x doesn't look like .5 that is what it is.lfen