Decimals

[OBonio]

Is there such a number as 0.9 recurring.  If so, then how can decimals truly represent 1/3.  Is this not a huge failure of the system?

[QmunkE]

The decimal representation of 1/3 is 0.3333 recurring, and can never completely represent the fraction except by using a dot above the threes to mean "reccurring".

The decimal system is indeed flawed - most people believe it only exists because we have ten fingers (or "digits"). In binary, 1/10 is impossible to measure accurately. All number systems have their shortcomings I guess.

[Loosehead]

The recurring suffix simply means there is no exact resolution to the relevant division sum. eg 10/3 does not produce an exact figure, it is simply a way of representing this. It is not unique to base 10. There are lots of numbers that cannot be represented exactly, PI, root 2, e, etc

[Peter Pedant]

Yes indeedy there is a number .9 recurring, and its one!

( take .9 from one, then take .99 from one, then take .999 from one.....you get a sequence of numbers that get smaller and smaller, (by a factor of ten) and clearly have a limit, as you take it to infinity.

That limit is zero, and so one is driven to conclude .9 recurring is one.

[OBonio]

My point is that the limit isn't 0.  The limit converges towards 0, but will never get there.  Hence, 0.9 recurring will never be 1.  And 0.3 recurring will never be 1/3.  I'm going to throw away my maths qualifications now ;o).

[Tweed]

You're right OBonio, decimals cannot accurately represent some numbers, like 1/3, 1/9, Pi etc., but it can be accurate to n decimal places, which is good enough in most practical applications.

[Tweed]

Just to throw a spanner in the works

x = 0.999...

10x = 9.999...

9x = 9.999... - x

but x = 0.999... so

9x = 9.999... - 0.999...

9x = 9

x = 1

This is supposed to prove that 0.999... = 1. It is in fact the way to convert recurring decimals to fractions. I think it falls over in the step 10x = 9.999... because you can't multiply an infinite series on one side of the equation and expect the same on the other. I.e.you're saying that infinity = infinity, or more to the point, infinity/infinity = 1 which, as far as I know is undefined (like n/0).

[OBonio]

Thanks for that Tweed.  That's really boggled my brain now.

[Tweed]

I know what you mean. Now I'm not so sure myself

http://mathforum.org/library/drmath/view/55748.html

[OBonio]

Thinking of it from that other perspective :

If you start at 1, and try to move away from 1 and toward 0.99999..., how far do you have to go to get to 0.99999... ?  Any step you try to take will be too far, so you can't really move at all -  which means that to move from 1 to 0.99999..., you have to stay at 1.