Cambridge Lower Secondary CheckpointStage 8

Number (8Np)

Mathematics Stage 8 Chapter Notes

What this chapter covers

Number - Place value, ordering and rounding
ShareWhatsAppPost
Number (8Np) notes

Unable to load PDF

The notes viewer could not load. Please refresh the page.

Read online free. Download a watermarked copy with a free account.

Read the notes

The full Number (8Np) notes as text: skim, search, and jump between subtopics.

~9 min read

In this chapter

  1. 1Multiplying and Dividing Integers
  2. 2Squares, Cubes, and Roots
  3. 3Natural, Integer, and Rational Numbers
  4. 4Order of Operations (BIDMAS)

1. Multiplying and Dividing Integers

Integers are whole numbers, which can be positive, negative, or zero. When multiplying or dividing them, there are simple rules to follow based on their signs. Think of it this way: if the signs are the same, the answer is positive. If the signs are different, the answer is negative.

(+) × (+) = (+)

(-) × (-) = (+)

(+) × (-) = (-)

(-) × (+) = (-)

(+) ÷ (+) = (+)

(-) ÷ (-) = (+)

(+) ÷ (-) = (-)

(-) ÷ (+) = (-)

Key term

Integer: An integer is a whole number that can be positive, negative, or zero.

Common pitfall

A common mistake is forgetting that multiplying two negative numbers results in a positive number. For example, many students incorrectly write -4 × -5 = -20, when the correct answer is 20.

Fun fact

The plus (+) and minus (-) symbols we use today first appeared in print in a German arithmetic book in 1489. Before that, words like 'pui' (plus) and 'meno' (minus) were used.

Worked example 12 marks

Calculate the value of -9 × 6.

  1. 1

    Step 1: Identify the signs of the numbers. We have a negative number (-9) and a positive number (6).

  2. 2

    Step 2: Recall the rule: a negative multiplied by a positive gives a negative result.

  3. 3

    Step 3: Multiply the numerical values: 9 × 6 = 54.

  4. 4

    Step 4: Apply the negative sign to the result. So, -9 × 6 = -54.

Worked example 22 marks

Calculate the value of -72 ÷ -8.

  1. 1

    Step 1: Identify the signs. Both numbers are negative (-72 and -8).

  2. 2

    Step 2: Recall the rule: a negative divided by a negative gives a positive result.

  3. 3

    Step 3: Divide the numerical values: 72 ÷ 8 = 9.

  4. 4

    Step 4: The result is positive. So, -72 ÷ -8 = 9.

Recap

  • Multiplying or dividing two numbers with the same sign (both positive or both negative) gives a positive answer.
  • Multiplying or dividing two numbers with different signs (one positive, one negative) gives a negative answer.
  • Integers are the set of all whole numbers, including zero, and their negative counterparts.
  • Remember that any number multiplied by zero is zero.

Quick check

  1. What is -11 × -5?1 mark
  2. What is 60 ÷ -12?1 mark

2. Squares, Cubes, and Roots

Squaring a number means multiplying it by itself (e.g., 5² = 5 × 5 = 25). Cubing a number means multiplying it by itself twice (e.g., 2³ = 2 × 2 × 2 = 8). Roots are the opposite operation. A square root of 9 is 3 because 3² = 9. A cube root of -8 is -2 because (-2)³ = -8. The rules for positive and negative numbers are very important here.

a² = a × a

(-a)² = (-a) × (-a) = a²

a³ = a × a × a

(-a)³ = (-a) × (-a) × (-a) = -a³

√x (where x > 0) has two roots: one positive, one negative

∛x has one root

Key term

Square Root: A square root of a number is a value that, when multiplied by itself, gives the original number.

Examiner insight

Examiners expect you to know that a positive number has two square roots, but the √ symbol by itself refers only to the principal (positive) root. A question asking for 'the square roots' requires both, but a question asking to 'evaluate √25' requires only 5.

Fun fact

The symbol for a square root, √, is called a radical. It is thought to have originated from a handwritten lowercase 'r', for the Latin word 'radix' meaning 'root'.

Worked example 12 marks

Find the value of (-5)² and -5².

  1. 1

    For (-5)²: The brackets mean we square the entire number -5.

  2. 2

    (-5)² = -5 × -5 = 25. (A negative times a negative is a positive).

  3. 3

    For -5²: There are no brackets, so the squaring operation applies only to the 5, not the negative sign.

  4. 4

    -5² = -(5 × 5) = -25.

  5. 5

    The answers are 25 and -25 respectively.

Worked example 23 marks

Explain why -16 has no real square root, but -64 has a cube root.

  1. 1

    Step 1: Consider the square root of -16. A square root is a number multiplied by itself. If we multiply a positive number by itself, the result is positive (e.g., 4 × 4 = 16). If we multiply a negative number by itself, the result is also positive (e.g., -4 × -4 = 16).

  2. 2

    Step 2: Since there is no real number that gives -16 when multiplied by itself, -16 has no real square root.

  3. 3

    Step 3: Consider the cube root of -64. A cube root is a number multiplied by itself twice. If we try -4, we get (-4) × (-4) × (-4).

  4. 4

    Step 4: (-4) × (-4) = 16. Then 16 × (-4) = -64. So, the cube root of -64 is -4.

Recap

  • A positive number has two square roots, one positive and one negative (e.g., roots of 36 are 6 and -6).
  • A negative number has no real square roots.
  • The square of a negative number is always positive, e.g., (-7)² = 49.
  • Every number (positive or negative) has exactly one real cube root.
  • The cube root of a negative number is always negative, e.g., ∛-8 = -2.

Quick check

  1. What are the two square roots of 81?1 mark
  2. What is the value of ∛-1?1 mark

3. Natural, Integer, and Rational Numbers

Mathematicians classify numbers into different sets. Natural numbers are the positive 'counting' numbers (1, 2, 3...). Integers include all natural numbers, their negative opposites, and zero (...-2, -1, 0, 1, 2...). Rational numbers are any numbers that can be written as a fraction p/q, where p and q are integers and q is not zero. This includes all integers, all terminating decimals (like 0.5 = 1/2), and all recurring decimals (like 0.333... = 1/3).

Natural Numbers (ℕ): {1, 2, 3, ...}

Integers (ℤ): {..., -2, -1, 0, 1, 2, ...}

Rational Numbers (ℚ): {p/q | p, q are integers, q ≠ 0}

Key term

Rational Number: A rational number is any number that can be expressed as a fraction (or ratio) of two integers, where the denominator is not zero.

Common pitfall

Thinking that decimals are not rational. Any decimal that stops (e.g., 0.25) or repeats in a pattern (e.g., 0.141414...) can be written as a fraction and is therefore rational.

Worked example 13 marks

Classify each number as natural, integer, and/or rational.a) 8b) -4c) 0.75

  1. 1

    a) 8 is a counting number, so it is a Natural Number. Since all natural numbers are also integers and rational numbers, it is all three. (It can be written as 8/1).

  2. 2

    b) -4 is not a positive counting number, so it is not natural. It is a whole number, so it is an Integer. Since all integers are rational, it is also a Rational Number. (It can be written as -4/1).

  3. 3

    c) 0.75 is not a whole number, so it is not natural or an integer. It can be written as the fraction 3/4, so it is a Rational Number.

Worked example 22 marks

Is the statement 'All integers are natural numbers' true or false? Explain your answer.

  1. 1

    The statement is false.

  2. 2

    Explanation: Natural numbers are only the positive integers (1, 2, 3...).

  3. 3

    Integers include negative numbers (like -5) and zero (0), which are not natural numbers.

  4. 4

    Therefore, not all integers are natural numbers.

Recap

  • Natural numbers are for counting: 1, 2, 3...
  • Integers add zero and negatives: ..., -2, -1, 0, 1, 2...
  • Rational numbers add fractions and terminating/recurring decimals.
  • A number can belong to more than one set. For example, 5 is a natural number, an integer, and a rational number.
  • A number that cannot be written as a fraction, like π or √2, is called irrational.

Quick check

  1. Give an example of an integer that is not a natural number.1 mark
  2. Is -3/5 a rational number? Explain why or why not.2 marks

4. Order of Operations (BIDMAS)

When a calculation has multiple operations, we need a standard order to ensure everyone gets the same answer. This order is given by the acronym BIDMAS (or BODMAS). It stands for Brackets, Indices, Division, Multiplication, Addition, Subtraction. 'Indices' includes powers and roots. Division and Multiplication have equal priority, as do Addition and Subtraction; for these pairs, you simply work from left to right.

B: Brackets ()

I: Indices (e.g., x², √x, ∛x)

D: Division (left to right)

M: Multiplication (left to right)

A: Addition (left to right)

S: Subtraction (left to right)

Key term

BIDMAS: An acronym that specifies the correct sequence for evaluating a mathematical expression: Brackets, Indices, Division, Multiplication, Addition, Subtraction.

Examiner insight

Examiners often create questions where a simple left-to-right calculation gives an incorrect answer. By showing your calculations step-by-step according to BIDMAS, you demonstrate your understanding and are more likely to get method marks even if you make a final slip.

Fun fact

Different countries use different acronyms for the same rule! The UK uses BIDMAS or BODMAS, the USA uses PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction), and Canada uses BEMDAS.

Worked example 13 marks

Calculate 20 - 3 × (2 + 3)².

  1. 1

    Step 1 (Brackets): First, calculate the expression inside the brackets: 2 + 3 = 5. The expression becomes 20 - 3 × 5².

  2. 2

    Step 2 (Indices): Next, evaluate the index (power): 5² = 25. The expression becomes 20 - 3 × 25.

  3. 3

    Step 3 (Multiplication): Now, perform the multiplication: 3 × 25 = 75. The expression becomes 20 - 75.

  4. 4

    Step 4 (Subtraction): Finally, perform the subtraction: 20 - 75 = -55.

Worked example 23 marks

Calculate 12 + ∛-8 - 10 ÷ 2.

  1. 1

    Step 1 (Brackets): There are no brackets.

  2. 2

    Step 2 (Indices): Evaluate the root: ∛-8 = -2. The expression becomes 12 + (-2) - 10 ÷ 2.

  3. 3

    Step 3 (Division/Multiplication): Work from left to right. There is one division: 10 ÷ 2 = 5. The expression becomes 12 + (-2) - 5.

  4. 4

    Step 4 (Addition/Subtraction): Work from left to right. First, 12 + (-2) = 12 - 2 = 10. The expression becomes 10 - 5.

  5. 5

    Step 5: Finally, 10 - 5 = 5.

Recap

  • Always follow the BIDMAS order strictly.
  • Indices refers to both powers (like 4²) and roots (like √16).
  • Division and Multiplication are partners; do them in order from left to right.
  • Addition and Subtraction are partners; do them in order from left to right.
  • Show your working one step at a time to avoid errors and gain method marks.

Quick check

  1. Calculate 5 + 3 × 4.1 mark
  2. Calculate (10 - 6)² ÷ 8.2 marks

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. Calculate the value of (-8) × 3 + (-24) ÷ (-4).2 marks
  2. Write down the positive and negative square roots of 144.2 marks
  3. Evaluate 6² - ∛-125. Show your working.3 marks
  4. Here is a list of numbers: -7, 3.14, 1, -2/3, 0, √9, 50. From this list, write down: a) all the integers, b) all the natural numbers, c) all the rational numbers.4 marks
  5. Calculate (2 - 5)³ + 10 × 4. Show your working.3 marks
  6. Insert one pair of brackets into the following expression to make it correct: 50 - 30 ÷ 5 + 2 = 28.2 marks
  7. The cube root of a number is -4. What is the square of that number?4 marks
  8. Write a rational number that lies between 1/4 and 1/5. Give your answer as a fraction.3 marks
  9. Find the value of x if 2x + (-4)² = ∛-64.4 marks
  10. Is the statement 'The cube of a negative integer is always smaller than the integer itself' always true, sometimes true, or never true? Justify your answer with at least two examples.4 marks

Go deeper

Practise and revise with member-only material for this chapter.

Workbook

Exam-style practice questions with full worked solutions.

Unlock with a plan

Chapter at a Glance

A single-page summary of every key idea for fast revision.

Unlock with a plan

Free notes are just the start.

Unlock every Workbook and Chapter at a Glance, and generate your own worksheets and predicted papers.

Explore plans

Related chapters

Number (8Ni)Number (8Nf)Algebra (8Ae)Algebra (8As)Geometry and Measure (8Gg)Geometry and Measure (8Gp)