Rational Representation of a Repeating Decimal

[themadhatter1]

Homework Statement

Find the rational number representation of the repeating decimal.

1.[tex]0.\overline{36}[/tex]

Homework Equations

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 can't seam to reason it out, however I know the answer is [tex]\frac{4}{11}[/tex] from my calculator.

 

[HallsofIvy]

[tex]x= 0.\overline{36}= 0.36363636...[/tex]

[tex]100x= 36.36363636...[/tex]
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:
[tex]0.\overline{36}= .36+ .0036+ .000036+ \cdot\cdot\cdot= .36(1+ .36+ .0036+ \cdot\cdot\cdot)[/tex]
a geometric series with common ratio, r, equal to .36. Its sum, by the usual formula, is
[tex]\frac{.36}{1- .36}[/tex]

 

[themadhatter1]

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

I have found the infinite geometric sequence summation to be:

[tex]\sum_{i=1}^\infty0.36(10)^{2-2n}[/tex]

which would yield [tex]0.\overline{36}[/tex]
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 [tex]10^\infty[/tex] 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.

 

[Dick]

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

I have found the infinite geometric sequence summation to be:

[tex]\sum_{i=1}^\infty0.36(10)^{2-2n}[/tex]

which would yield [tex]0.\overline{36}[/tex]
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 [tex]10^\infty[/tex] 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.

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?

 

[Mark44]

[tex]x= 0.\overline{36}= 0.36363636...[/tex]

[tex]100x= 36.36363636...[/tex]
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:
[tex]0.\overline{36}= .36+ .0036+ .000036+ \cdot\cdot\cdot= .36(1+ .36+ .0036+ \cdot\cdot\cdot)[/tex]

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.

a geometric series with common ratio, r, equal to .36. Its sum, by the usual formula, is
[tex]\frac{.36}{1- .36}[/tex]

 

[themadhatter1]

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?

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

so

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

Thanks!

 

[hunt_mat]

HallsofIvy's first method was perfectly rigourous and very neat. The second method is as follows:
[tex]
0.363636... & = & 36/10^2+36/10^4+36/10^6+... = 36(10^{-2}+10^{-4}+10^{-6}+...)
[/tex]
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