Hi, I'm in grade eight and I have a maths problem which I just don't understand.

http://img683.imageshack.us/i/66153173.jpg/

I don't understand how you read the circled figure. Does the ... mean 111 repeating over and over again?

Or if not that, what does it mean?

Thanks!

2. Originally Posted by JesseElFantasma
Hi, I'm in grade eight and I have a maths problem which I just don't understand.

ImageShack&#174; - Online Photo and Video Hosting

I don't understand how you read the circled figure. Does the ... mean 111 repeating over and over again?

Or if not that, what does it mean?

Thanks!

Yes, apparently that's exactly what it means.

Be aware in that case it can be understood as a summation of geometric progression, hence:

$\displaystyle $2,...111... = 2,(1) = 2\frac{1}{9} = \frac{{19}}{9}$$

3. 2,111... (in the U.S. we would write 2.111...) where the "111" keeps repeating can be made into a fraction in two ways:
1) If x= 2,111.... then 10x= 21,111.... (notice that we never "run out" of "1"s on the right so the decimal part is exactly the same). Subtracting:
10x- x= 9x= 21,111...- 2,111= 19 so that $x= \frac{19}{9}$ as Pranas said.

That's a little "hand waving" since it uses arithmetic on infinite decimals that really should be proven first.

Better is to think of this, as Pranas says, as a geometric sum.

2,111...= 2+ .1+ .01+ .001+ ...= 1+ (1+ .01+ .01+ 0.001+ ...)= 1+ ((.1)^0+ (.1)^1+ (.1)^2+ (.1)^3+ ...)

The quantity in parentheses is a "geometric series" and the formula for the sum of such is well known:
$\sum_{n=0}^\infty r^n= 1+ r+ r^2+ r^3+ ...= \frac{1}{1- r}$
for |r|< 1, otherwise, it does not converge.

Here, r= .1, which is between -1 and 1, so the sum is $\frac{1}{1- .1}= \frac{1}{.9}= \frac{1}{\frac{9}{10}}= \frac{10}{9}$

Then, of course, the given number is $1+ \frac{10}{9}= \frac{19}{9}$.

(Oops! In the last part I have reverted to "." instead of ",". Hope you still understand.)