- #1
noboost4you
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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
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