Decimal notation and ordering

Learning outcome

Read decimal place values, recognise equal decimal forms, and arrange positive and negative decimals without treating their digits as separate whole numbers.

Understanding decimal notation

A decimal extends place value beyond the ones column. Moving one place right divides the value of a position by 10. The first three positions after the decimal point are tenths, hundredths and thousandths. A digit’s contribution depends on its position, not merely its face value.

Zeros can hold essential positions. A zero between other decimal digits cannot simply be removed. However, zeros added at the right-hand end of the fractional part do not change the value: 0.6 = 0.60. This is exact equality, not rounding.

For positive decimals, compare whole-number parts first. If these match, compare fractional digits from left to right. You may append zeros to create equally long fractional parts. The first unequal place decides the order; having more decimal digits does not automatically make a number larger.

For negative decimals, compare their distances from zero and reverse that order. The number farther below zero is smaller. On a number line, values increase towards the right.

Worked examples

Example 1: Read 47.308.

Its expansion is 40 + 7 + 3/10 + 0/100 + 8/1000. Thus, 3 contributes 0.3 and 8 contributes 0.008. The zero preserves the hundredths position; removing it would change the number.

Example 2: Order 0.56, 0.506, 0.560 and 0.605.

Write 0.56 as 0.560. Among numbers beginning with five tenths, the hundredths digit separates 0.506 from 0.560. Six tenths is larger. Therefore, 0.506 < 0.56 = 0.560 < 0.605.

Example 3: Order -2.08, -2.008 and -1.98.

Write the first number as -2.080. It is farther below zero than -2.008. Both are below -1.98. The increasing order is -2.08 < -2.008 < -1.98.

Common mistakes

Do not compare decimal tails as whole numbers. Do not delete internal zeros or assume that a longer decimal is larger. With negatives, a larger distance from zero means a smaller value, not a larger one.

Practice questions

1. What is the place value of 7 in 5.072?
2. Arrange 3.405, 3.45 and 3.045 in increasing order.
3. Insert <, > or =: -0.7 _ -0.65; 6.040 _ 6.04.
4. Give one decimal strictly between 1.23 and 1.24.

Worked answers

1. The 7 is in the hundredths position, so its place value is 7/100 = 0.07.
2. Compare 3.405, 3.450 and 3.045 place by place. The order is 3.045 < 3.405 < 3.45.
3. Since -0.70 is farther below zero, -0.7 < -0.65. The final zero changes no value, so 6.040 = 6.04.
4. One answer is 1.235. Appending zeros gives 1.230 < 1.235 < 1.240, which verifies that it lies strictly between the endpoints.