Lesson 2.1: Decimal Numbers

New Concept

Decimal numbers are fractions whose denominators are powers of 10, like $\frac{1}{10}$ or $\frac{1}{100}$. Understanding their place value is the key to comparing, rounding, and converting them, connecting them back to the fractions we already know.

What’s next

Next, you'll tackle interactive examples and practice cards to master comparing decimals, converting fractions, and rounding with confidence.

Property

The place-value of the last digit tells us the denominator of the fraction. Each decimal place is one tenth of the preceding place. For example, the places after the decimal point are tenths, hundredths, thousandths, and so on.

Examples

• The decimal $0.6$ is read as six tenths and is written as the fraction $\frac{6}{10}$.
• The decimal $0.09$ is read as nine hundredths because the last digit, 9, is in the hundredths place. It is written as $\frac{9}{100}$.
• The decimal $0.125$ is read as one hundred twenty-five thousandths and is written as $\frac{125}{1000}$.

Explanation

Think of place value as a digit's address. The further right it is from the decimal point, the smaller its value. Each spot is ten times smaller than the one to its left, just like a penny is worth less than a dime.

Property

To compare decimal fractions, compare the digits in each place value from left to right, starting with the tenths place. An extra zero at the end of a decimal fraction does not change its value. For example, $0.3 = 0.30 = 0.300$.

Examples

• To compare $0.6$ and $0.475$, we look at the tenths place. Since $6 > 4$, we know that $0.6 > 0.475$.
• To compare $0.82$ and $0.85$, the tenths are the same. We look at the hundredths place. Since $2 < 5$, we know that $0.82 < 0.85$.
• The decimals $0.9$ and $0.90$ are equal. Writing $0.9$ as $0.90$ shows they both have 9 tenths, so $0.9 = \frac{9}{10}$ and $0.90 = \frac{90}{100}$, which also reduces to $\frac{9}{10}$.

Explanation

To find the bigger decimal, check the tenths first. If they're tied, check the hundredths. Don't be fooled by length; a shorter decimal can be larger if its early digits are bigger!

Property

To convert a common fraction to a decimal, divide the numerator by the denominator. Some decimals terminate (end), while others have a repeating pattern of digits. We use a bar over the repeating digits, for example, $\frac{1}{3} = 0.333... = 0.\overline{3}$.

Examples

• To convert $\frac{3}{4}$ to a decimal, we calculate $3 \div 4$, which gives $0.75$. This is a terminating decimal.
• To convert $\frac{4}{9}$ to a decimal, we calculate $4 \div 9$, which gives $0.444...$. We write this repeating decimal as $0.\overline{4}$.
• The fraction $\frac{5}{12}$ is converted by calculating $5 \div 12 = 0.41666...$. We write this as $0.41\overline{6}$, with the bar only over the repeating digit.

Explanation

A fraction is just a division problem in disguise! When you perform the division, the answer is its decimal form. It either stops neatly (terminates) or repeats a pattern forever.

Property

When we round to any place value, we look at the digit to the right of that place. If that digit is less than 5, we round down. If that digit is 5 or greater, we round up. Rounding gives an approximation, not an exact value.

Examples

• To round $3.852$ to the nearest tenth, we look at the hundredths digit, which is $5$. Since it's $5$ or greater, we round the tenths digit up. So, $3.852 \approx 3.9$.
• To round $0.1239$ to the nearest hundredth, we look at the thousandths digit, which is $3$. Since it's less than $5$, we keep the hundredths digit. So, $0.1239 \approx 0.12$.
• Rounding $7.98$ to the nearest tenth means looking at the $8$. We round the $9$ up to $10$. This carries over, making the answer $8.0$.

Explanation

Rounding simplifies a number to its nearest, cleaner value. Look at the digit to the right of your target place: if it is 5 or more, round up; if it is 4 or less, the target digit stays the same.