A question of numbers (one for the maths fans)

x=0.999999...
10x=9.999999...10x-x = 9.999999... - 0.999999...
9x = 9Therefore x=1Edited by Modulous, : No reason given.

Oh yes you can! Oh no it doesn't! Surely you mean 9? You have to subtract more than 1 to get 8.999991. If we have 9.999999... then you need to propose the number 8.9999999...1 (an infinite number of 9s followed by a 1!), which would assume infinity has an end, where a 1 is located. That's crazy talk!It's quite a provocative piece of maths, it feels very naughty don't you think? More here.

rR has a point in that he stated it can't be done. I have the same strength of point in that I said it can. That's as advanced as the debate is right now

That would imply that 0.999999999... + 0.11111111111.... is 1. Actually that bit of maths gets you 1.1111111111111.....Doing it in fractions:9x¹⁄₉ - 1 = 0Your maths '0.999999999..... - 1 = 0.11111111111111111111111111111111 ........' would look like this:9x¹⁄₉ - 1 = ¹⁄₉That leads to the unusual concept of ⁹⁄₉ + ¹⁄₉ = ⁹⁄₉

If 0.99999... doesn't exist, then isn't the statement 0.9999....=1 vacuously true? Maybe I'm just thinking about it too much?

What is 6 - πCan it be done?

That wasn't n that was πIt might not be clear but that is pi.

Yet it is a real ratio which real calculations can be performed on, without the need to approximate. It seems like you are rejecting a great swathe of mathematics.e^Πi+1=0 springs to mind. We cannot say this is true by your definitions. Likewise we know that Π = ^c⁄_d, but now you are proposing that^(Πd)⁄_c ≠ 1It's fine if you want to reject any number you cannot write down - but you need to appreciate just how that might affect the rest of maths.Incidentally, if it suits you to think in terms of approximation, then you know that if you made any approximation of 0.9999.... it becomes 1. If you don't like the decimal representation of an infinite geometric series, you can think of it in terms of fractions - it's just more difficult to write and less likely that people will understand the number being represented. It is strange that every decimal number has different ways of being represented - even though we accept it as common for fractions.

I hope not, but I can't be sure. It would seem a great angle for the fine-tuners to try and come from. Unfortunately, it's not like its an easy thing to search for! Come one, we've moved at least a little bit! If you fancy trying to mathematically prove something to somebody in this medium, go for it. I'm sure it'll be as entertaining as usual. I considered starting calculus 101 in light of rRs infinity issues.

But you don't lose a 9, you have exactly the same number of 9s - the only thing that has changed is the position of the decimal point. This happens basically by definition when you multiply by 10 in base 10. More generally when you multiply by n in base n. So we have the same number of digits (infinite), the decimal place has just shifted.Edited by Modulous, : No reason given.Edited by Modulous, : No reason given.

As I said, its a question of defintions and number systems. In base ten, if you multiply by 10 the decimal point moves. Actually, in any base if multpily by 10 the decimal point moves, in base ten 10 means ten. It's an integral part of what the number 10 is.In 0.999 we don't lose the 9/1000. All that happens is that the 1000ths column becomes 0 because the digits are shifted up one column (in base 10). when there are an infinite number of 9s we don't put a zero at the end, but the essential operation (shifting the digits up a column) can still happen. This is part and parcel of the number system. There aren't two proofs, there are two ways of multiplying normal numbers by 10. One is to add a zero at the end of it and shift it up. What is actually happening is that we are shifting the numbers up a column which has the result (in the case of finitely long strings of numbers) of adding a zero at the end. It has been dealt with by others, but I will do it again if you like.x=0.333...
10x= 3.333...10x-x = 9x9x = 3x = 3/9Therefore x =0.333... That doesn't make sense to me at all, the only way to do it would be to flip the equations around. If flipping the equations couldn't prove the converse concept, then there would be a contradiction and the maths would be wrong somewhere.

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.

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.

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.