The recurring decimal

I can't even do high school maths, but this one always bothered me.

1/9 equals 0.11111111111111111 forever
2/9 equals 0.22222222222222222 forever
3/9 equals 0.33333333333333333 forever
...
9/9 must equal 0.9999999999999999 forever

But 9/9 equals 1 ! How is this so?

Originally posted by Kewpie
I can't even do high school maths, but this one always bothered me.

1/9 equals 0.11111111111111111 forever
2/9 equals 0.22222222222222222 forever
3/9 equals 0.33333333333333333 forever
...
9/9 must equal 0.9999999999999999 forever

But 9/9 equals 1 ! How is this so?

Originally posted by iamatiger
x = 0.999999999

10x = 9.99999999

10x - x = 9x = 9.9999999 - 0.999999999 = 9

so
9x = 9

x = 1

9/9 = 0.9999999999999 = 1

Originally posted by Mephisto2


10x - x = 9x = 9.9999999 - 0.999999999 = 8.99999991 which is not 9.

Or what?

Just to say that the infinite recurrence of the 9's is the crucial point.

Originally posted by forkedknight
Using what math does

9.999... - 0.999... = 8.99999999991
because in Iamatiger's answer, there were no '.....' after the 9's. Try it on your calculator.
?

I couldn't get a 9 with a dot over it, which is how I would usually write "recurring". I assumed everybody would know what I meant, and I suspect even the quibblers did know actually!

Originally posted by Kewpie
I can't even do high school maths, but this one always bothered me.

1/9 equals 0.11111111111111111 forever
2/9 equals 0.22222222222222222 forever
3/9 equals 0.33333333333333333 forever
...
9/9 must equal 0.9999999999999999 forever

But 9/9 equals 1 ! How is this so?

The way I understood it, any recurring digit approaches the next whole number but can never quite reach it, so 0.9 recurring can never quite equal 1. That makes the thing paradoxical to me. Perhaps the problem is at the other end: maybe 1/9 doesn't really equal 0.1 recurring, perhaps it's just "near enough". And that minuscule difference is always there but ignored by everybody.

Originally posted by Kewpie
The way I understood it, any recurring digit approaches the next whole number but can never quite reach it, so 0.9 recurring can never quite equal 1.

It seems to me that there's a circular argument here. I say 0.9 recurring tends to 1 but can never equal 1. You say 0.9 does equal one. Any proof using fractions doesn't make it so, because recurring numbers aren't able to be resolved this side of infinity.

We aren't the only ones having this discussion:

http://forum.boagworld.com/discussion/596/is-0.9-recurring-equal-to-one-the-big-debate/p1

There are (at least) two ways of representing 1/9
1/9 and 0.1111111111(1), these are exactly equal


Similarly for 2/9 = 0.22222222(2), 3/9, 4/9 etc

So 9.9 is not an oddity. if you say that 0.99999(9) <> 9/9 then you are also claiming that 0.11111111(1) /= 1/9

Another looking at it is the power of infinity, as we add more 9s to the end of 0.9 we are 1/10^n away from 9/9, and n increases with each 9 we add

In the limit, we are 1/n^(infinity) away from 9/9 and 1/infinity = 0

Originally posted by iamatiger

In the limit, we are 1/n^(infinity) away from 9/9 and 1/infinity = 0

One way to think of it, of course, is as a sum of 0.9 x 10^(-n) where n gets the values 1, 2, 3, ..., that is 0.9 + 0.09 + 0.009 + ...

Wonder if it is more convincing to ask how much 0.9999... differs from 1?

And that would be by 0.00000000.....

so.. not by very much.


While on the topic of sums... can you evaluate

1 - 1/2 + 1/3 - 1/4 + 1/5 - ..... + (-1)^n / n + ... ?

Originally posted by talzamir
One way to think of it, of course, is as a sum of 0.9 x 10^(-n) where n gets the values 1, 2, 3, ..., that is 0.9 + 0.09 + 0.009 + ...

Wonder if it is more convincing to ask how much 0.9999... differs from 1?

And that would be by 0.00000000.....

so.. not by very much.


While on the topic of sums... can you evaluate

1 - 1/2 + 1/3 - 1/4 + 1/5 - ..... + (-1)^n / n + ... ?

Originally posted by Shallow Blue
Then you understood it wrong. 0.9 recurring is, in fact, quite equal to 1.

Richard