Write # as a ratio of two integers..

Problem: Write the number 3.1415999999999... as a ratio of two integers.

In my book, they have a similar example, but using 2.3171717... And this is how they solved that problem.

2.3171717... = 2.3 + (17/10^3) + (17/10^5) + (17/10^7) + ...

After the first term we have a geometric series with a = (17/10^3) and r = (1/10^2). Therefore:

2.3171717... = 2.3 + [(17/10^3) / (1 - (1/10^2))] = 2.3 + [(17/1000)/(99/100)] = (23/10) + (17/990) = 1147/495 == 2.3171717...

Thinking I could follow the similar steps with a different number, I thought it would work, but it really isn't.

This is what I did:

3.1415999999999... = 3.1415 + (99/10^6) + (99/10^8) + (99/10^10)

a = (99/10^6) and r = (1/10^2)

3.1415 + [(99/10^6) / (1 - (1/10^2))] = 3.1415 + [(99/1000000)/(99/100) = (31415/10000) + (1/10000) = (31416/10000) = 3.1416 which isn't 3.1415999999999...

What am I doing wrong?

Thanks
 PhysOrg.com science news on PhysOrg.com >> Galaxies fed by funnels of fuel>> The better to see you with: Scientists build record-setting metamaterial flat lens>> Google eyes emerging markets networks

Recognitions:
Gold Member
Staff Emeritus
 3.1416 which isn't 3.1415999999999...
Actually, it is.

P.S. any particular reason you were grouping the nines in pairs?
 technically, it is, but is that correct though? and no, there was no reason i paired them up.

Write # as a ratio of two integers..

 Quote by noboost4you technically, it is, but is that correct though? and no, there was no reason i paired them up.
3.1416=3.141599999999... is very true. So any fractional representation of one is a representation of the other. In fact, that's how I would have solved this problem; I wouldn't have bothered with an infinite geometric series in this case.

Recognitions:
Gold Member