Overview
What you’ll learn
- Add, subtract, multiply and divide positive and negative numbers, and evaluate powers of negatives.
- Compare and order integers, fractions and decimals on a number line.
- Add, subtract, multiply and divide fractions, including mixed numbers.
- Convert between fractions and decimals, including recurring decimals.
- Classify numbers as integers, rational, irrational or real, and justify the classification.
In 30 seconds
- 1Same signs → positive answer; different signs → negative answer. This works for × and ÷ alike.
- 2Subtracting a bigger number goes below zero: 15 − 22 = −7.
- 3To add or subtract fractions, build a common denominator first; to divide, multiply by the reciprocal.
- 4A rational number can be written as a fraction of whole numbers; its decimal terminates or repeats.
- 5Irrational numbers (like √2) have decimals that never end and never repeat.
- 6Every integer is rational (n = n/1), every rational is real — but not every real is rational.
Notes · 1 of 5
Negative numbers and the four operations
The number line extends below zero, and the sign rules keep the operations consistent.
An integer is any whole number, positive, negative or zero: …, −3, −2, −1, 0, 1, 2, 3, … On the number line, numbers increase to the right, so −5 < −2 < 0 < 3. A common slip is to think −5 is bigger because 5 is — the negative sign flips the order.
| Operation | Same signs | Different signs |
|---|---|---|
| (+ ) × (+ ) or (− ) × (− ) | Positive | — |
| (+ ) × (− ) or (− ) × (+ ) | — | Negative |
| Division follows the same pattern | Positive answer | Negative answer |
- Adding a negative moves left: 4 + (−9) = −5. Adding a negative is the same as subtracting.
- Subtracting a negative moves right: 4 − (−9) = 4 + 9 = 13.
- Powers of negatives depend on how many times the base repeats: (−2)² = 4 but (−2)³ = −8, because an odd number of negative factors stays negative.
Notes · 2 of 5
Fractions and decimals
Fractions, mixed numbers and decimals are different costumes for the same quantities.
| Operation | What to do | Example |
|---|---|---|
| Add / subtract | Common denominator first | 2/5 + 1/3 = 6/15 + 5/15 = 11/15 |
| Multiply | Top × top, bottom × bottom | 7/8 × 4/21 = 1/6 after cancelling |
| Divide | Multiply by the flipped fraction | 2½ ÷ 5/8 = 5/2 × 8/5 = 4 |
| Mixed number | Convert to an improper fraction first | 2½ = 5/2 |
To convert a fraction to a decimal, divide top by bottom: 5/8 = 0.625. Some fractions terminate (3/4 = 0.75); others recur forever in a repeating block (2/3 = 0.666…). To convert a terminating decimal back, read off the place value: 0.35 = 35/100 = 7/20.
Notes · 3 of 5
Comparing and ordering numbers
Put every number in the same costume — decimal form — before you line them up.
- 1Convert
Write fractions as decimals: 3/4 = 0.75.
- 2Line up the signs
Negatives first (most negative is smallest), then zero, then positives.
- 3Compare place values
For positives, compare tenths, then hundredths: 0.6 < 0.75 < 0.8.
- 4Check against the number line
Left is smaller. −2.5 sits far left of −1.
The symbols < (less than) and > (greater than) always open towards the bigger number: −7 < −3, and 0.6 > 0.58. Exam answers often ask for ascending order (smallest first) — underline the direction before you write.
Notes · 4 of 5
Rational numbers
A number is rational when it can be written as one whole number over another.
A rational number is any number that can be written as a fraction a/b, where a and b are integers and b is not zero. This family is enormous: it contains every integer (5 = 5/1), every terminating decimal (0.7 = 7/10) and every recurring decimal (0.333… = 1/3).
Why does recurring count as rational? Because a repeating block can always be captured by a fraction: 0.363636… = 36/99 = 4/11. Terminating decimals are the same idea with denominators of 10, 100, 1000, …
- Fractions like −3/4 are rational (integers are allowed on top, just not on the bottom).
- Every integer is rational, because n = n/1.
- Every terminating or recurring decimal is rational.
Notes · 5 of 5
Irrational numbers and the real number line
The numbers that escape fractions — and where everything fits together.
An irrational number cannot be written as a fraction of whole numbers. Its decimal expansion goes on forever without terminating and without any repeating block. The classic examples are √2 (about 1.4142135…) and π (about 3.14159…). These decimals never settle into a cycle.
- Integers ℤ
- …, −2, −1, 0, 1, 2, …
- Rational numbers ℚ
- All fractions of integers — includes every integer and every terminating or recurring decimal.
- Irrational numbers
- √2, π, and decimals that never repeat — not rational.
- Real numbers ℝ
- All of them together: every point on the number line.
Together, the rational and irrational numbers fill the number line completely — every point is a real number. Each family nests inside the next: integers ⊂ rational ⊂ real, and irrational numbers sit inside the reals but outside the rationals.
Revise
Key terms
The words examiners expect you to use precisely.
- Integer
- Any whole number, positive, negative or zero: …, −2, −1, 0, 1, 2, …
- Rational number
- A number that can be written as a fraction a/b of two integers, with b not zero.
- Irrational number
- A number that cannot be written as a fraction of integers, such as √2; its decimal never terminates or repeats.
- Real number
- Any rational or irrational number — every point on the number line.
- Recurring decimal
- A decimal with a block of digits that repeats forever, e.g. 0.333… = 1/3.
- Terminating decimal
- A decimal that stops, e.g. 0.625; always rational.
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- Reciprocal
- The flipped fraction: the reciprocal of 5/8 is 8/5. Dividing by a fraction means multiplying by its reciprocal.
- Improper fraction
- A fraction whose top is bigger than its bottom, e.g. 7/4; equal to the mixed number 1¾.
- Mixed number
- A whole number with a fraction, e.g. 2½.
- Common denominator
- A shared bottom line for two fractions, needed before adding or subtracting them.
- Ascending order
- Arranged from smallest to largest.
Revise
Common mistakes
Where students most often lose marks in this chapter — and what to do instead.
Mistake: Saying −5 is greater than −2 “because 5 is bigger than 2”.
The negative sign reverses the comparison: −5 sits further left on the number line, where values are smaller.
Instead: For negatives, the number closer to zero is the larger one: −2 > −5.
Mistake: Multiplying (-3) × (-6) and writing −18.
Two negatives cancel when multiplied: the product of same signs is positive.
Instead: Same signs → positive; different signs → negative.
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Mistake: Adding fractions straight across: 2/5 + 1/3 = 3/8.
Fractions must refer to the same-sized parts before they can be counted together.
Instead: Build a common denominator first: 6/15 + 5/15 = 11/15.
Mistake: Dividing by flipping the wrong fraction, e.g. computing 2 ÷ 1/3 as 1/3 × 1/2.
You flip (take the reciprocal of) the fraction you divide **by**, not the first number.
Instead: 2 ÷ 1/3 = 2 × 3/1 = 6.
Mistake: Claiming 0.666… is irrational because “it goes on forever”.
Irrational means never ending **and never repeating**. A repeating block is exactly what makes it rational.
Instead: Every recurring decimal is rational: 0.666… = 2/3.
Mistake: Treating 22/7 as an irrational number because “it is used for π”.
22/7 is a fraction of two integers, so it is rational; it is merely a close approximation of π.
Instead: π itself is irrational; approximations of it need not be.
Mistake: Computing (-4)³ as +64.
Three negative factors multiply to a negative: (−4)(−4)(−4) = −64.
Instead: Odd powers of negatives stay negative; even powers turn positive.
Mistake: Calling the square root of 16 irrational “because it has a root sign”.
The root of a perfect square is a whole number: √16 = 4, which is an integer and therefore rational.
Instead: Check whether the number under the root is a perfect square before deciding.
Mistake: Subtracting fractions straight across, e.g. 5/6 − 1/2 = 4/4 = 1.
Tops and bottoms cannot be subtracted separately — the parts must be the same size first.
Instead: Use a common denominator: 5/6 − 3/6 = 2/6 = 1/3.
Mistake: Ignoring the sign when dividing, e.g. 6 ÷ (−2) = 3.
Dividing numbers with different signs gives a negative answer: 6 ÷ (−2) = −3.
Instead: Same signs → positive; different signs → negative, for both × and ÷.
Check yourself
Quick quiz
18 questions with instant explanations. About 5 minutes.
Question 1 of 3
- Question 1: current question
- Question 2: not answered
- Question 3: not answered
Add and subtract negative numbers
Evaluate (−6) + (−4).
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Practise
Topical questions
Exam-style practice in three tiers, from recall to data questions.
- 6Foundation
- 5Standard
- 4Challenging
15 questions worth 46 marks, each with a model answer, mark scheme and examiner tips.
Practise topical questionsExam practice
Practice paper
A 30-mark paper set out like a school paper, with a separate mark scheme. Give yourself 40 min.
Question paper
Chapter 2 Practice Paper
30 marks · 40 min · 5 pages
10 multiple-choice questions and structured questions with answer lines. Chapter 2.
Mark scheme
PROChapter 2 Practice Paper — mark scheme
Answers · Marking points · 3 pages
Model answers with a mark for each creditworthy point — for marking at home or in class.
Where this chapter is tested
Each school sets its own scope — check your assessment outline.
- Term 1Coming soonWeighted Assessment 1 (WA1)
- Term 4Coming soonEnd-of-Year Examination (EOY)
Questions
Frequently asked questions
What is Sec 1 Math Chapter 2 about?
Chapter 2 covers the number families of secondary maths: working with negative numbers and all four operations, adding, subtracting, multiplying and dividing fractions and decimals, and classifying integers, rational, irrational and real numbers.
How do I tell rational and irrational numbers apart?
Ask whether the number can be written as a fraction of two integers. Fractions, terminating decimals and recurring decimals are all rational. Numbers like √2 whose decimals never end and never repeat are irrational.
What are the rules for multiplying negative numbers?
Same signs give a positive answer; different signs give a negative answer. The same rule works for division. Powers of negatives are positive for even powers and negative for odd powers.
How do I divide by a fraction?
Multiply by its reciprocal: flip the fraction you are dividing by, then multiply. For example, 2½ ÷ 5/8 = 5/2 × 8/5 = 4. Convert mixed numbers to improper fractions first.
When is this chapter tested?
This material usually appears in Weighted Assessment 1 (Term 1) and the end-of-year exam. Every school sets its own scope, so check your school’s assessment outline.
Up next
Chapter 3: Approximation and Estimation