# Rational Representation of a Repeating Decimal

1. Jul 10, 2010

1. The problem statement, all variables and given/known data
Find the rational number representation of the repeating decimal.

1.$$0.\overline{36}$$

2. Relevant equations

3. The attempt at a solution

I know it has something to do with infinite geometric sequences but I'm not sure what.

what would your ratio be for a repeating decimal, I've thought about it and cant seam to reason it out, however I know the answer is $$\frac{4}{11}$$ from my calculator.

2. Jul 10, 2010

### HallsofIvy

Staff Emeritus
$$x= 0.\overline{36}= 0.36363636...$$

$$100x= 36.36363636....$$
The point is that since the "repeating" portion never stops, those two numbers have exactly the same decimal part and subtracting cancels them:
100x- x= 99x= 36. Write that as a fraction and reduce it.

Another, more "rigorous" method (so you don't have to argue about "canceling" an infinite string of digits) is to use the fact that this is a geometric series:
$$0.\overline{36}= .36+ .0036+ .000036+ \cdot\cdot\cdot= .36(1+ .36+ .0036+ \cdot\cdot\cdot)$$
a geometric series with common ratio, r, equal to .36. Its sum, by the usual formula, is
$$\frac{.36}{1- .36}$$

3. Jul 10, 2010

simplifying $$\frac{0.36}{1-0.36}$$ does not come out to a fraction that equals $$0.\overline{36}$$

I have found the infinite geometric sequence summation to be:

$$\sum_{i=1}^\infty0.36(10)^{2-2n}$$

which would yield $$0.\overline{36}$$
I understand the formula for the sum of an infinite geometric sequence, however it doesn't seam to work in this case.

If r is really 10 then $$10^\infty$$ dosen't tend to 0 so the formula isen't valid. However, you have the exponents, I'm, not sure how you'd deal with thoes.

Last edited: Jul 10, 2010
4. Jul 10, 2010

### Dick

The common ratio r isn't 0.36 (Halls misspoke) and it isn't 10 either. What is it? You've got the geometric series correct. So what's the ratio between two successive terms like a_2/a_1?

5. Jul 10, 2010

### Staff: Mentor

The last part should be .36(1 + 10-2 + 10-4 + ... ).
The common ratio r is 1/100, so the sum is .36(1 - 1/100) = 36/99.

6. Jul 10, 2010

Ahh... the common ratio would be 10-2 or 1/100 which is the same thing.

so

$$\frac{36/100}{1-1/100}=\frac{3,600}{9,900}=\frac{36}{99}=\frac{4}{11}=0.\overline{36}$$

Thanks!

7. Jul 10, 2010

### hunt_mat

HallsofIvy's first method was perfectly rigourous and very neat. The second method is as follows:
$$0.363636... & = & 36/10^2+36/10^4+36/10^6+... = 36(10^{-2}+10^{-4}+10^{-6}+...)$$
The series in brackets is a geometric progression with first term 10^{-2} and common ratio 10^{-2}, the sum for this is series is 10^{-2}/(1-10^{-2})=1/99. So the rational form is 36/99 as HallsofIvy already calculated.

Mat