Decimals

Place value to thousandths

The decimal point separates the whole-number part from the parts smaller than one. Each place is one tenth of the place to its left: ones, then tenths (1/10), hundredths (1/100) and thousandths (1/1000). In 3.472 the 7 means 7 hundredths, or 0.07.

Reading and writing decimals

Read the whole-number part, say “and” for the point, then read the digits after it as one number followed by the name of the last place. 3.472 is “three and four hundred seventy-two thousandths.” 0.06 is “six hundredths.” (In everyday talk people also say “three point four seven two.”)

Comparing and ordering

Line the numbers up by their decimal points and compare place by place from the left. A common trap: 0.45 has more digits than 0.5, but 0.5 is bigger. Add a zero so both have the same number of places: 0.50 vs 0.45, and 50 hundredths beats 45 hundredths. Adding zeros at the end of a decimal does not change its value.

Rounding

To round to a place, look at the digit just to its right. If it is 5 or more, round up; if it is less than 5, keep the digit, and drop everything after it. On a number line, you are choosing the nearer of the two neighbors.

Adding and subtracting

Line up the decimal points, fill empty places with zeros, then add or subtract as with whole numbers, putting the point straight down in the answer. Lining up the points lines up the places, so you add tenths to tenths and hundredths to hundredths, just as you add ones to ones.

Multiplying

Multiply as if there were no decimal points. Then count the decimal places in both factors together, and give the answer that many places. Why? 0.3 × 0.4 = 3/10 × 4/10 = 12/100 = 0.12. Tenths times tenths make hundredths, so 1 + 1 = 2 places.

Dividing

If the divisor is a whole number, divide and put the point in the answer directly above the point in the dividend. If the divisor is a decimal, make it whole: multiply both numbers by 10, 100 or 1000. That doesn’t change the answer, just as 6/3 = 60/30.

Converting between fractions and decimals

Decimal to fraction: read it. 0.36 is 36 hundredths, 36/100, which simplifies to 9/25 (see Fractions).

Fraction to decimal: divide the numerator by the denominator, or rename it with a denominator of 10, 100 or 1000. Some fractions give a terminating decimal that ends (3/8 = 0.375). Others give a repeating decimal: 1/3 = 0.333… and 2/11 = 0.181818…, written with a bar over the repeating part: 0.3 and 0.18. A fraction in lowest terms terminates exactly when its denominator has no prime factors other than 2 and 5.

Money and measurement

Money uses two decimal places: $3.07 is 3 dollars and 7 cents, because a cent is one hundredth of a dollar. When an answer comes out to 3.1, write it as $3.10. Metric units are built on tens too: 1.25 m is 1 meter and 25 centimeters, and 2.5 kg is 2500 g. For percents, 0.35 = 35/100 = 35% (see Percents).

How do we know?

Why line up the points? Each digit’s value depends on its place. Lining up points lines up places, so 4.68 + 13.7 adds 8 hundredths + 0 hundredths, 6 tenths + 7 tenths, and so on. Lining up the last digits instead would add hundredths to tenths.

Why count decimal places? Every decimal is a whole number divided by a power of ten: 2.4 = 24 ÷ 10 and 0.15 = 15 ÷ 100. So 2.4 × 0.15 = (24 × 15) ÷ (10 × 100) = 360 ÷ 1000 = 0.36. The places add because the powers of ten multiply: 10 × 100 = 1000 has 1 + 2 = 3 zeros.

Why is multiplying both numbers in a division allowed? A division is a fraction: 1.56 ÷ 0.4 = 1.56/0.4. Multiplying top and bottom by 10 gives an equivalent fraction, 15.6/4, with the same value.

History: Stevin’s “The Tenth”

In 1585 the Flemish mathematician Simon Stevin published a short booklet, usually known as De Thiende (“The Tenth”), giving an elementary and thorough account of decimal fractions. A French version, La Disme, appeared the same year. Stevin did not invent decimals: MacTutor notes that decimals were “used by the Arabs and the Chinese long before” him. But his booklet helped bring them into European mathematics, and he predicted that decimal coins, measures and weights would come into use everywhere. MacTutor reports that an English translation inspired Thomas Jefferson to propose a decimal currency for the United States. A tenth of a dollar is still called a dime.

Stevin didn’t use a decimal point; he marked each place with a small circled number. The point and the comma came into common use later; Wikipedia says John Napier’s logarithm tables of 1614 and 1619 used a period. Today the United States, the UK and many other countries write 3.5, while much of continental Europe and Latin America writes 3,5.

Practice problems

Try each one, then open it to compare.

1. What is the value of the 9 in 4.309?

It is in the thousandths place: 0.009 (9 thousandths).

2. Which is greater, 0.8 or 0.79?

0.80 vs 0.79: 0.8 is greater.

3. Round 12.645 to the nearest hundredth.

The digit after the hundredths place is 5, so round up: 12.65.

4. Find 7.5 + 2.86.

7.50 + 2.86 = 10.36.

5. Find 3.2 × 0.06.

32 × 6 = 192; 1 + 2 = 3 places: 0.192.

6. Find 4.2 ÷ 0.6.

Multiply both by 10: 42 ÷ 6 = 7.

7. Write 7/20 as a decimal.

Multiply top and bottom by 5: 35/100 = 0.35.

8. Write 2/3 as a decimal with bar notation.

2 ÷ 3 = 0.666…, so 0.6.

Talk About It

• A friend says 0.45 > 0.5 “because 45 is bigger than 5.” How would you convince them?
• Why does multiplying by 0.5 make a number smaller?
• Why does 1/4 terminate but 1/3 repeat?
• Where have you seen a decimal comma, like 3,5? How might it cause confusion?

Sources

• MacTutor History of Mathematics: Simon Stevin (the 1585 booklet, earlier Arab and Chinese decimals, and Jefferson)
• Wikipedia: De Thiende (the Dutch and French versions)
• Wikipedia: Decimal separator (Napier’s period and today’s point and comma conventions)
• Khan Academy: Arithmetic with decimals (more free practice)

Credits: All diagrams on this page are original SVG drawings made for Jugami. Prices in the examples are made up for practice.