Integers and rational numbers form the bedrock of secondary mathematics, underpinning algebra, geometry and every topic across the O-Level syllabus.
This topic covers integers on the number line, the sign rules for operations, the order of operations (BODMAS) and rational numbers. It also covers converting between fractions and decimals and putting rational numbers in order.
What these notes cover
- •Integers
- •Ordering integers on the number line
- •Adding and subtracting integers
- •Sign rules for × and ÷
- •Order of operations (BODMAS)
- •Rational numbers
- •Converting fractions to decimals
- •Converting decimals to fractions
- •Recurring decimal to fraction
- •Ordering rational numbers
- •Common mistakes
- •Summary
Key definitions
- Integer
- Any whole number: positive, negative or zero. Integers do not include fractions or decimals: …, −3, −2, −1, 0, 1, 2, 3, …
- Ordering on the number line
- Integers increase to the right and decrease to the left, with zero as the boundary. The integer further to the right is the greater value, so −1 > −3 because −1 lies to the right of −3.
- BODMAS
- The order of operations. Work out Brackets first, then Orders (powers and roots), then Division and Multiplication from left to right, and finally Addition and Subtraction from left to right.
- Rational number
- Any number that can be expressed in the form p ÷ q, where p and q are integers and q ≠ 0. This includes all integers, fractions, terminating decimals and recurring decimals.
Key formulas
Sign rules for × and ÷
(+) × (+) = + (−) × (−) = + (+) × (−) = − (−) × (+) = −
Same signs give a positive result and different signs give a negative result. The same rules apply to division.
Subtracting a negative
a − (−b) = a + b
Subtracting a negative number is the same as adding. For example, 3 − (−5) = 3 + 5 = 8.
Form of a rational number
ℤ is the set of integers. Every integer is rational, for example −7 = −7 ÷ 1.
Fraction to decimal
Divide the numerator by the denominator. Some results terminate, such as 0.375. Others recur, such as 1 ÷ 6 = 0.1666…
Terminating decimal to fraction
Write the decimal as a fraction (here over 100), then simplify it to its simplest form.
Recurring decimals are rational
A recurring decimal can be written as a fraction, so it is a rational number. Use the algebraic method to convert it.
Worked example
Evaluate (−2)³ − 3 × (4 − 7).
- 1Brackets first: 4 − 7 = −3.
- 2Orders next: (−2)³ = −8.
- 3Multiply: 3 × (−3) = −9.
- 4Subtract: −8 − (−9) = −8 + 9 = 1.
- 5Answer: 1.
Watch out: Subtracting a negative becomes addition, so −8 − (−9) must be rewritten as −8 + 9.
Common mistakes
(−3)² vs −3²
Wrong: −3² = 9
Right: −3² = −(3²) = −9, but (−3)² = 9. The bracket changes what the exponent applies to.
Two negatives in × or ÷
Wrong: (−4) × (−3) = −12
Right: (−4) × (−3) = 12. Same signs always give a positive.
Subtracting a negative
Wrong: 5 − (−2) = 3
Right: 5 − (−2) = 5 + 2 = 7. Subtracting a negative becomes addition.
Questions students ask
What is the difference between an integer and a rational number?
An integer is a whole number (positive, negative or zero) and does not include fractions or decimals. A rational number is any number that can be written as p ÷ q, where p and q are integers and q ≠ 0. Every integer is a rational number, for example −7 = −7 ÷ 1.
Is a recurring decimal a rational number?
Yes. Recurring decimals are rational numbers, for example 0.333… = 1 ÷ 3. Rational numbers include all integers, fractions, terminating decimals and recurring decimals.
Why is −1 greater than −3?
On the number line, the integer further to the right is the greater value. −1 lies to the right of −3, so −1 > −3, even though 3 > 1. If you are unsure, plot both values on a number line.
How do you arrange fractions and decimals in ascending order?
Convert all the numbers to decimals, or to fractions with a common denominator. Then read them off from smallest to largest. For example, −3 ÷ 4 = −0.75 and 2 ÷ 3 ≈ 0.667, which makes them easy to compare with other decimals.
In the full PDF
- •Worked example 2: expressing a recurring decimal as a fraction in its simplest form using the algebraic method
- •Worked example 3: arranging a mix of fractions and decimals in ascending order
- •Number-line method for adding and subtracting integers, with examples
- •Two further common mistakes: adding fractions without a common denominator, and zero denominators
- •A five-point summary of the whole topic