Decimal Numbers: Read, Compare and Operate with Confidence

A decimal number is one that has a fractional part separated by a point (in English-speaking countries) or a comma (in many others). To the left of the decimal point are whole units; to the right are parts smaller than 1. For example, 3.75 means 3 wholes and 75 hundredths.

Each position to the right of the decimal point has a specific name: the first place is tenths (1/10), the second is hundredths (1/100), the third is thousandths (1/1000), and so on. So 0.3 is 'three tenths', 0.07 is 'seven hundredths' and 0.005 is 'five thousandths'. Knowing these names lets you read any decimal number aloud without confusion.

There is an important distinction between 0.3 and 0.30 in terms of notation, but no difference in value: both represent three tenths. Adding trailing zeros after the last decimal digit does not change the value — you just need to be careful not to confuse it with 0.03, which is three hundredths, a value ten times smaller.

Comparing decimals is straightforward once you remember that each digit's position defines its weight. To compare 0.4 and 0.38, it is not enough to notice that 38 is greater than 4 among whole numbers: 0.4 is equivalent to 0.40, which has 40 hundredths, while 0.38 has only 38 hundredths. Therefore 0.4 > 0.38.

The safest method is to align the decimal points and pad with trailing zeros until both numbers have the same number of decimal places. Then compare from left to right, digit by digit, as if they were whole numbers. The first differing digit you encounter determines which is greater.

Another efficient strategy is to convert the decimals to fractions with the same denominator. So 0.4 = 4/10 = 40/100 and 0.38 = 38/100. Since 40 > 38 in the numerator, 0.4 > 0.38. This conversion is also useful for verifying answers on tests that mix fractions and decimals.

Converting a fraction to a decimal is simple: divide the numerator by the denominator. The fraction 3/4 becomes 3 ÷ 4 = 0.75. The fraction 1/3 becomes 1 ÷ 3 = 0.333…, a repeating decimal. The fraction 5/8 becomes 5 ÷ 8 = 0.625. Any calculator can do this, but it is worth understanding why: the fraction a/b literally means 'a divided by b'.

To convert a decimal to a fraction, use place value. The decimal 0.7 is 7/10. The decimal 0.35 is 35/100, which simplifies to 7/20. The decimal 1.25 is 125/100 = 5/4 or 1 and 1/4. After forming the fraction, always check whether it can be simplified using the GCD of the numerator and denominator.

Some decimals are terminating (they end at some point), such as 0.25 or 0.125. Others are infinitely repeating, such as 0.333… (= 1/3) or 0.142857142857… (= 1/7). Fractions whose denominators contain only the prime factors 2 and 5 always produce terminating decimals. Any other prime factor in the denominator will generate a repeating decimal.

Addition and subtraction of decimals have one golden rule: align the decimal points. When adding 3.7 + 12.45, write 3.70 below 12.45 with the decimal points in the same column. Add as if they were whole numbers: 370 + 1245 = 1615. Place the decimal point in the correct position: 16.15. This rule ensures you are adding tenths to tenths, hundredths to hundredths.

Multiplication of decimals is done by ignoring the decimal points during the calculation and replacing them at the end. To multiply 2.3 × 1.4: calculate 23 × 14 = 322. Count the total decimal places in both factors: 1 + 1 = 2. Place the decimal point 2 positions from the right: 3.22. This method works because you temporarily converted the decimals to whole numbers by multiplying by powers of 10, then undid that transformation.

Division of decimals becomes easier when you eliminate the decimal point from the divisor. To calculate 4.8 ÷ 0.6, multiply both by 10: 48 ÷ 6 = 8. If the divisor has 2 decimal places, multiply by 100. This transformation does not change the result because it multiplies both numerator and denominator of a fraction by the same value.

Money is the most everyday example of decimals. $12.75 has 12 dollars (whole part) and 75 cents (decimal part). When adding $4.90 + $7.35, align the decimal points: 4.90 + 7.35 = 12.25. The most common mistake is forgetting the alignment and mixing cents with dollars.

Measurements of length, weight and volume constantly use decimals. A baby weighing 3.850 kg has 3 kilograms and 850 grams. A piece of fabric measuring 1.75 m has 1 metre and 75 centimetres. In pharmacies, medications are dosed in milligrams: 0.5 mg is half a milligram. Precision with decimals is critical in this context.

Fuel efficiency is expressed in km/l or miles per gallon with decimal places — 12.4 km/l and 13.1 km/l look similar, but over 1,000 km that represents a difference of 5.6 litres. In athletics, a race can be decided by a margin of 0.01 second. These examples show why knowing how to work with decimals correctly matters.