Chapter 8.6, Problem 1P

Use the standard division algorithm to determine the decimal representation of $\frac{1}{37}$ .Determine whether the decimal representation repeats or terminates; if it repeats, describe the string of repeating digits.

To determine

To find: The decimal representation of $\frac{1}{37}$ using standard division algorithm and to determine whether the decimal representation repeats or terminates. Also to describe the string of repeating digits if it repeats.

Answer to Problem 1P

The decimal representation of $\frac{1}{37}$ is $0.027027\dots$ and the obtained decimal representation repeats itself. The string of digits that repeat are $027$ .

Explanation of Solution

Given information:

The fraction is $\frac{1}{37}$ .

Calculation:

For the fraction $\frac{1}{37}$ , the divisor is 37 and the dividend is 1.

By long division method,

  $37\begin{array}{c}\hfill 0.027027\\ \hfill \overline{)\begin{array}{l}1.000000\\ -\underset{\_}{74}\\ 260\\ -\underset{\_}{259}\\ 100\\ -\underset{\_}{74}\\ 260\\ -\underset{\_}{250}\\ 1\end{array}}\end{array}$

In this division the remainder 1 repeats hence the decimal representation does not terminates it repeats with the string of digits $\overline{027}$ .

Conclusion:

Thus, the decimal representation of $\frac{1}{37}$ is $0.027027\dots$ that repeats with the string of digits $027$ .