This chapter extends decimal work to multiplication and division, treating decimals as the fractions they represent. To multiply two decimals, multiply the numbers as if there were no decimal points, then place the point so the number of decimal places in the product equals the sum of the decimal places in the factors — for example \(5.96 \times 24.8 = 147.808\). To divide by 10, 100, 1000, move the decimal point left by the number of zeros. Division uses the long-division (place-value) method: when ones are regrouped into tenths a decimal point is placed in the quotient, and regrouping continues into hundredths and thousandths. Some divisions never end, such as \(10 \div 3 = 3.333\dots\), revealing repeating decimals like \(1 \div 7 = 0.\overline{142857}\). When the divisor is itself a decimal, multiply both dividend and divisor by a power of 10 to make the divisor a whole number. The chapter ends with a fascinating real application: how leap-year rules (the \(0.2422\) extra days each year) keep our calendar aligned with the Earth's motion.
Recap of decimals; dividing by powers of 10Decimal multiplication and placing the pointDecimal division by long divisionRepeating decimals and decimal divisorsApplication: leap years and the calendar
Key terms
Decimal multiplication
Multiplying the numbers without the point, then placing the point by adding decimal places.
Decimal division
Dividing using long division and placing the point when ones become tenths.
Dividing by powers of 10
Moving the decimal point left by as many places as there are zeros in the divisor.
Repeating decimal
A decimal whose digits repeat without ending, e.g. \(1 \div 7 = 0.\overline{142857}\).
Decimal divisor
A divisor that is a decimal; make it a whole number by multiplying both numbers by a power of 10.
Leap year
A year with 366 days, added to keep the calendar aligned with the Earth's revolution.
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When multiplying two decimals, the number of decimal places in the product equals:
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