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(intimidated newbie here, please be gentle) I got curious about adding repeating decimals while working on a "convert degrees/min/sec to degrees" problem.

To convert 5°13'11'' to degrees, I can add $$5 + \frac{13}{60}+\frac{11}{3600}$$ to get $$5+\frac{791}{3600} = 5+0.2197\overline{2}$$ just fine, but just to see what would happen I tried converting the fractions to decimals before adding, and I broke Arithmetic.

$$\frac{13}{60} = .21\overline{6}$$ $$\frac{11}{3600} = .0030\overline{5}$$ $$0.2166666+0.0030555 = .2197221$$

Now obviously this is * * almost * * right, but there must be a way to add repeating decimals so they come out right. It doesn't look like rounding helps; the last digit might be right, or wrong, depending on the repeat. Do you just agree to ignore the final digit because you know it's gonna be funky? Or is there some way to account for it arithmetically? I don't even want to think about what happens when you get into decimals that repeat differently (e.g., 3 digits vs 7 digits) until I understand this simple version.

sloopie72
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  • Nice question. By extending the pre-periodic part, and going to the lcm of the periods, it can be done. – André Nicolas Jul 26 '14 at 16:57
  • There is no final digit. The numbers repeat forever. – Mike Jul 26 '14 at 16:59
  • A left-to-right algorithm is possible. For instance in adding 36 + 57, we can add the tens first getting 8 tens, and then 13 ones. We then carry back to the tens getting 9 tens and 3 ones. The same idea can be used for repeating decimals. – paw88789 Jul 26 '14 at 17:00

1 Answers1

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\begin{align} 0 & .2166\bar{6} \\ +\ 0 & .0030\bar 5 \\ \hline 0 &.2197\bar 2 \end{align}

The $\bar 6$ plus the $\bar 5$ will be one more than $6+5$ because the previous pair will have a carry of $1$.

Broseph
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