Decimal

A decimal is a number that uses a dot (called a decimal point) to separate the whole number part from the fractional part. For example, 3.75 means 3 whole units and 75 hundredths.

A decimal is a number expressed in the base-10 (decimal) number system, where digits to the right of the decimal point represent fractional values based on powers of ten. The first digit after the decimal point represents tenths ( $\frac{1}{10}$ ), the second represents hundredths ( $\frac{1}{100}$ ), the third represents thousandths ( $\frac{1}{1000}$ ), and so on. Every decimal can be written as a fraction with a denominator that is a power of 10.

Worked Example

Problem: Write the decimal 4.365 as a fraction, and identify the place value of each digit after the decimal point.

Step 1: Identify the digits after the decimal point. The digits are 3, 6, and 5.

Step 2: Assign each digit its place value. The 3 is in the tenths place, the 6 is in the hundredths place, and the 5 is in the thousandths place.

3 \times \frac{1}{10} + 6 \times \frac{1}{100} + 5 \times \frac{1}{1000}

Step 3: Since the last digit is in the thousandths place, write the fractional part over 1000.

4.365 = 4\frac{365}{1000}

Step 4: Convert to an improper fraction if needed by multiplying the whole number by 1000 and adding the numerator.

4.365 = \frac{4 \times 1000 + 365}{1000} = \frac{4365}{1000}

Answer: 4.365 has a 3 in the tenths place, a 6 in the hundredths place, and a 5 in the thousandths place. As a fraction, it equals

\frac{4365}{1000}

Visualization

Why It Matters

Decimals appear everywhere in daily life — prices at a store ($4.99), measurements in science (3.14 cm), and batting averages in sports (0.325). Understanding decimals is essential for working with money, reading instruments like rulers and scales, and eventually learning about percentages and scientific notation.

Common Mistakes

Mistake: Thinking that more digits after the decimal point means a larger number (e.g., believing 0.125 is greater than 0.5).

Correction: Compare decimals by looking at each place value from left to right. Since 0.5 = 0.500, it is larger than 0.125. Adding zeros to the right can help you compare.

Mistake: Ignoring the decimal point and treating the number as a whole number when adding or subtracting (e.g., lining up 3.5 + 0.25 incorrectly).

Correction: Always line up the decimal points vertically before adding or subtracting. You can add trailing zeros so both numbers have the same number of decimal places: 3.50 + 0.25 = 3.75.

Related Terms

Fraction — Decimals and fractions represent the same values differently
Number Line — Decimals can be placed on a number line