Write # as a ratio of two integers..

noboost4you

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

Hurkyl

Actually, it is.


P.S. any particular reason you were grouping the nines in pairs?

noboost4you

technically, it is, but is that correct though? and no, there was no reason i paired them up.

master_coda

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.

HallsofIvy

Techically it's true but is it correct??? Is that what you are asking?

"True" is "true"- there is no "technically"! And if it's true, then it's correct.