Lesson 1.4: Decimals

New Concept

This lesson explores decimals, a key way to represent fractions with denominators that are powers of ten. You'll master rounding, performing arithmetic operations, and converting between decimals, fractions, and percents, seeing how they fit into the real number system.

What’s next

Next, you'll tackle interactive examples covering each decimal operation. Then, you'll test your skills with a series of practice cards and challenge problems.

Property

Decimals are another way of writing fractions whose denominators are powers of ten. When we work with decimals, it is often necessary to round the number. To round a decimal:

1. Locate the given place value and mark it with an arrow.
2. Underline the digit to the right of the place value.
3. If the underlined digit is greater than or equal to 5, add 1 to the digit in the given place value. If not, do not change the digit.
4. Rewrite the number, deleting all digits to the right of the rounding digit.

Examples

• Round 34.586 to the nearest hundredth. The digit to the right of the hundredths place (8) is 6. Since $6 \ge 5$, we add 1 to 8, which gives 9. The rounded number is 34.59.

• Round 12.934 to the nearest tenth. The digit to the right of the tenths place (9) is 3. Since $3 < 5$, we do not change the 9. The rounded number is 12.9.

Property

To add or subtract decimals, we line up the decimal points. By lining up the decimal points this way, we can add or subtract the corresponding place values. The steps are:

1. Determine the sign of the sum or difference.
2. Write the numbers so the decimal points line up vertically.
3. Use zeros as placeholders, as needed.
4. Add or subtract the numbers as if they were whole numbers. Then place the decimal point in the answer under the decimal points.

Examples

• Add $19.45 + 7.8$. We line up the decimal points: $19.45 + 7.80 = 27.25$.

• Subtract $8.15 - 12$. The result will be negative. We subtract the smaller value from the larger: $12.00 - 8.15 = 3.85$. So, the answer is $-3.85$.

Property

To multiply decimals, first determine the sign. Multiply the numbers as if they were whole numbers, ignoring the decimal points. The number of decimal places in the product is the sum of the decimal places in the factors. To divide decimals, first determine the sign. Make the divisor a whole number by moving its decimal point all the way to the right. Move the decimal point in the dividend the same number of places to the right. Then divide as usual.

Examples

• Multiply $(-4.5)(2.1)$. The product is negative. We multiply $45 \times 21 = 945$. There are two decimal places in total (one in 4.5 and one in 2.1), so the answer is $-9.45$.

• Divide $-42.75 \div (-0.05)$. The quotient is positive. Move the decimal two places in both numbers to get $4275 \div 5$. The result is $855$.

Property

To convert a decimal to a fraction, identify the place value of the last digit and use that as the denominator. A percent is a ratio whose denominator is 100.
To convert a percent to a decimal, move the decimal point two places to the left.
To convert a decimal to a percent, move the decimal point two places to the right and add the percent sign.

Examples

• Write $0.45$ as a fraction. The last digit, 5, is in the hundredths place, so the fraction is $\frac{45}{100}$, which simplifies to $\frac{9}{20}$.

• Convert $72\%$ to a decimal. We move the decimal point two places to the left, which gives $0.72$.

Property

If $n^2 = m$, then $m$ is the square of $n$. If $n^2 = m$, then $n$ is a square root of $m$. Every positive number has two square roots—one positive and one negative. The radical sign, $\sqrt{m}$, denotes the positive square root, also called the principal square root. \\ $\sqrt{m}$ is read 'the square root of m'. If $m = n^2$, then $\sqrt{m} = n$, for $n \ge 0$.

Examples

• Simplify $\sqrt{36}$. Since $6^2 = 36$, the principal square root of 36 is $6$.

• Simplify $\sqrt{169}$. Since $13^2 = 169$, the principal square root of 169 is $13$.

Property

A rational number is a number of the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. Its decimal form stops or repeats. An irrational number is a number that cannot be written as the ratio of two integers. Its decimal form does not stop and does not repeat. A real number is a number that is either rational or irrational.

Examples

• Given the set $\{-9, \frac{12}{5}, \sqrt{13}, 6.1, \sqrt{81}\}$, the integers are $-9$ and $\sqrt{81}$ (since $\sqrt{81}=9$).

• From the set $\{-9, \frac{12}{5}, \sqrt{13}, 6.1, \sqrt{81}\}$, the rational numbers are $-9$, $\frac{12}{5}$, $6.1$, and $\sqrt{81}$.