Convert rational numbers to terminating or repeating decimals

Learning goal: Convert a fraction to a decimal through long division, track the remainder and explain why a rational decimal terminates or eventually repeats. Distinguish an exact value from a rounded approximation.

A rational number can be written as a quotient of integers with a nonzero denominator. Its decimal representation comes from dividing numerator by denominator. The sign can be handled separately: first divide the magnitudes, then apply the sign of the quotient. For example, −3/8 is negative because exactly one of the numerator and denominator is negative.

To convert 3/8, divide 3.000 by eight. Eight does not fit into three whole units, so the whole-number digit is zero. Record the decimal point in the quotient above the decimal point in the dividend. Regroup the remaining three whole units as thirty tenths. Eight fits into thirty three times, representing three tenths in the quotient, with a remainder of six tenths after subtracting twenty-four tenths.

Regroup six tenths as sixty hundredths. Eight fits seven times, accounting for fifty-six hundredths and leaving four hundredths. Regroup those as forty thousandths. Eight fits five times with remainder zero. The quotient is 0.375. Each new place comes from regrouping the remaining amount into ten smaller units, not from inserting arbitrary digits.

When the remainder becomes zero, additional decimal digits would be zeros. Thus 3/8 = 0.375 = 0.3750 and so on. We call the decimal terminating because a finite number of nonzero places gives the exact value. Multiplication checks it: 0.375 × 8 = 3. Applying the sign gives −3/8 = −0.375.

Now divide two by eleven. Two whole units are less than eleven, so start with 0 and a decimal point. Twenty tenths divided by eleven gives one tenth, leaving nine tenths. Ninety hundredths divided by eleven gives eight hundredths, leaving two hundredths. The remainder's digit pattern has returned to two at a smaller place. Continuing creates the same sequence of division steps: twenty, one, remainder nine; ninety, eight, remainder two.

Therefore 2/11 = 0.181818… with the block 18 repeating forever. The dots indicate continuation of the stated repeating pattern. The finite decimal 0.18 is not exactly equal to 2/11: multiplying 0.18 by eleven gives 1.98 rather than two. A calculator may display only a limited number of digits, so a screen's last visible digit does not prove the decimal ends there.

A decimal can have a nonrepeating beginning before its repeating part. For 1/6, ten tenths divided by six gives one tenth and a remainder of four tenths. Each later step divides forty smaller units by six, producing six with a remainder of four again. Thus 1/6 = 0.1666…, where only the six repeats after the first tenth. Mark the repeating block accurately rather than repeating every digit already written.

Why must a rational decimal eventually terminate or repeat? With a positive whole-number divisor d, the whole-number remainder at each long-division step is one of 0, 1, 2, up to d − 1 after appropriate regrouping. If zero occurs, the process terminates. If not, only finitely many nonzero remainders are available, so a remainder must recur. The same remainder leads to the same next digit and next remainder, causing a repeating cycle.

The divisor condition is crucial: this reasoning applies to integer division with a nonzero denominator, the definition of a rational number. It does not claim that every nonterminating decimal is rational. The feature is eventual repetition, not simply having many digits. For this lesson, identify repetition through the division process rather than guessing from a short display.

Rounding produces an approximation for a purpose. To round 2/11 to the nearest hundredth, inspect the thousandths digit in 0.1818…; it is one, so the hundredths value remains 0.18. To the nearest thousandth, the next digit is eight, so 0.182 is appropriate. Use approximately equal when reporting the rounded value, and retain the exact fraction when an exact result is required.

Worked terminating conversion: 7/20 equals 35/100, so it is 0.35. Long division confirms this: seventy tenths divided by twenty gives three tenths with remainder ten tenths; one hundred hundredths divided by twenty gives five hundredths with no remainder. Equivalent fractions and the standard algorithm agree.

Worked repeating conversion: For 5/12, fifty divided by twelve gives four with remainder two; twenty divided by twelve gives one with remainder eight; eighty divided by twelve gives six with remainder eight. That last remainder recurs, so 5/12 = 0.41666… . Only the six repeats after the initial 41.

Worked negative rounding: −1/6 = −0.1666… . To the nearest hundredth it is −0.17, since that hundredth lies closer to the exact value than −0.16. Rounding the magnitude and restoring the negative sign works here; it does not mean every operation on a negative number can ignore direction.

Division record: For 3/8 and 2/11, show each partial dividend, quotient digit and remainder. Connect the digits with their place values. Circle the zero remainder in the first case and the repeated remainder in the second. Verify the terminating decimal by multiplication and explain why a short truncation fails an exact multiplication check for the repeating case.

Independent performance: Convert 7/20, −3/8, 1/6 and 5/12 using long division. Identify which terminate and state the exact repeating block for the others. Round each to the nearest hundredth where necessary, clearly separating exact and approximate forms. Explain to a partner why the possible remainders provide a reason for eventual repetition, rather than claiming the pattern only because several visible digits happen to repeat.