Cambridge Lower Secondary CheckpointStage 9

Number (9Np)

Mathematics Stage 9 Chapter Notes

What this chapter covers

Number - Place value, ordering and rounding
ShareWhatsAppPost
Number (9Np) 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 (9Np) notes as text: skim, search, and jump between subtopics.

~8 min read

In this chapter

  1. 1Rational and Irrational Numbers
  2. 2Understanding Indices
  3. 3Applying the Index Laws
  4. 4Estimating Roots and Surds

1. Rational and Irrational Numbers

Numbers can be sorted into different groups. Natural numbers are the positive whole numbers we use for counting (1, 2, 3, ...). Integers include all whole numbers, both positive, negative, and zero (...-2, -1, 0, 1, 2...). 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 includes all integers (e.g., 5 can be written as 5/1), all terminating decimals (e.g., 0.25 = 1/4), and all recurring decimals (e.g., 0.333... = 1/3). An irrational number cannot be written as a simple fraction. Their decimal representations go on forever without repeating. Famous examples include Pi (π) and surds, which are the roots of numbers that are not perfect squares or cubes (like √2 or ∛10).

Key term

Rational Number: A number that can be expressed as a fraction p/q, where p and q are integers and q is not zero.

Examiner insight

Examiners frequently test classification of numbers. Be prepared to justify your choice by defining 'rational' or 'irrational' in your answer.

Common pitfall

Confusing long decimals with irrational numbers. A recurring decimal like 0.8333... is rational because it can be written as a fraction (5/6), whereas an irrational decimal never repeats.

Fun fact

The ancient Greeks, who believed all numbers were rational, were shocked by the discovery of √2. Legend has it that the mathematician who discovered it was drowned at sea for revealing this unsettling truth!

Worked example 14 marks

Sort the following numbers into two groups: Rational and Irrational. The numbers are: -12, √100, 5/8, √3, π, 0.71, ∛64.

  1. 1

    Step 1: Analyse each number individually.

  2. 2

    -12 is an integer, so it is rational (-12/1).

  3. 3

    √100 = 10. This is an integer, so it is rational.

  4. 4

    5/8 is already in fraction form, so it is rational.

  5. 5

    √3: 3 is not a perfect square, so its square root is a non-terminating, non-repeating decimal. It is irrational.

  6. 6

    π is a well-known irrational number.

  7. 7

    0.71 is a terminating decimal, so it is rational (71/100).

  8. 8

    ∛64 = 4. This is an integer, so it is rational.

  9. 9

    Step 2: Group the numbers based on the analysis.

  10. 10

    Rational: -12, √100, 5/8, 0.71, ∛64

  11. 11

    Irrational: √3, π

Recap

  • Rational numbers can be written as a fraction.
  • Integers, terminating decimals, and recurring decimals are all rational.
  • Irrational numbers cannot be written as a simple fraction.
  • The decimal form of an irrational number goes on forever without repeating.
  • Pi (π) and the square roots of non-square integers are common irrational numbers.

Quick check

  1. Is the number 0.121212... rational or irrational? Explain why.2 marks
  2. Is √81 a surd? Explain your answer.2 marks

2. Understanding Indices

An index (also called a power or exponent) tells you how many times to multiply a number by itself. For example, in 5³, the '5' is the base and the '3' is the index. It means 5 × 5 × 5. There are three key rules for individual indices. A positive index 'n' means multiply the base by itself 'n' times. The zero index rule states that any non-zero number raised to the power of zero is 1 (e.g., 99⁰ = 1). A negative index indicates a reciprocal; it means '1 divided by the number with a positive index'. For example, a⁻ⁿ = 1/aⁿ.

a⁰ = 1 (for a ≠ 0)

a⁻ⁿ = 1 / aⁿ

Key term

Index: A number indicating how many times a base number should be multiplied by itself.

Examiner insight

Marks are consistently awarded for correctly converting a negative power into a fraction. Show this step clearly in your working.

Common pitfall

Thinking that a negative index gives a negative answer (e.g., 4⁻² = -16). It does not; it creates a fraction.

Worked example 12 marks

Calculate the value of 4⁻².

  1. 1

    Step 1: Identify the negative index. The rule is a⁻ⁿ = 1/aⁿ.

  2. 2

    Step 2: Apply the rule to 4⁻². Here, a=4 and n=2.

  3. 3

    4⁻² = 1 / 4²

  4. 4

    Step 3: Calculate the value of the denominator. 4² = 4 × 4 = 16.

  5. 5

    Step 4: Write the final answer. 4⁻² = 1/16.

Worked example 22 marks

Evaluate 7¹ + 7⁰.

  1. 1

    Step 1: Evaluate each term separately.

  2. 2

    7¹ is just 7.

  3. 3

    7⁰ follows the rule a⁰ = 1, so 7⁰ = 1.

  4. 4

    Step 2: Add the results together.

  5. 5

    7 + 1 = 8.

Recap

  • An index shows the number of times a base is multiplied by itself.
  • Any non-zero number to the power of zero equals 1.
  • A negative index means you take the reciprocal of the base raised to the positive index.
  • For example, 5⁻³ = 1/5³ = 1/125.

Quick check

  1. Write 10⁻³ as a decimal.1 mark
  2. What is the value of (3 + 4)⁰?1 mark

3. Applying the Index Laws

When you need to combine terms with indices, you can use a set of rules called index laws, as long as the base numbers are the same.

  1. Multiplication Law: To multiply powers with the same base, you add the indices. For example, aᵐ × aⁿ = aᵐ⁺ⁿ.
  2. Division Law: To divide powers with the same base, you subtract the indices. For example, aᵐ ÷ aⁿ = aᵐ⁻ⁿ.
  3. Power of a Power Law: To raise a power to another power, you multiply the indices. For example, (aᵐ)ⁿ = aᵐⁿ. These laws work for positive, negative, and zero indices.

aᵐ × aⁿ = aᵐ⁺ⁿ

aᵐ ÷ aⁿ = aᵐ⁻ⁿ

(aᵐ)ⁿ = aᵐⁿ

Key term

Base: The number that is being repeatedly multiplied in an exponential expression (e.g., the 'a' in aⁿ).

Examiner insight

Examiners look for the correct application of the laws step-by-step. For a question like (p⁷)³ ÷ p¹², show the simplification of the bracket first, then the division.

Common pitfall

Applying the wrong operation to the indices, such as multiplying them when adding is required (e.g., writing 5³ × 5² as 5⁶ instead of 5⁵).

Worked example 12 marks

Simplify 3⁵ × 3⁻².

  1. 1

    Step 1: The bases are the same (3), and the terms are being multiplied, so we use the multiplication law: aᵐ × aⁿ = aᵐ⁺ⁿ.

  2. 2

    Step 2: Add the indices: 5 + (-2) = 3.

  3. 3

    Step 3: Write the result with the original base: 3³.

  4. 4

    Final Answer: 3³ (or 27 if asked to evaluate).

Worked example 23 marks

Simplify (p⁷)³ ÷ p¹².

  1. 1

    Step 1: First, simplify the bracketed term using the power of a power law: (aᵐ)ⁿ = aᵐⁿ.

  2. 2

    (p⁷)³ = p⁷ˣ³ = p²¹.

  3. 3

    Step 2: The expression is now p²¹ ÷ p¹².

  4. 4

    Step 3: The bases are the same, and we are dividing, so use the division law: aᵐ ÷ aⁿ = aᵐ⁻ⁿ.

  5. 5

    Subtract the indices: 21 - 12 = 9.

  6. 6

    Step 4: Write the final answer: p⁹.

Recap

  • To multiply powers with the same base, add the indices.
  • To divide powers with the same base, subtract the indices.
  • To raise a power to another power, multiply the indices.
  • The index laws only work when the base numbers are the same.

Quick check

  1. Write 11⁻² ÷ 11⁻⁵ as a single power of 11.2 marks
  2. Simplify (2k³)⁵.2 marks

4. Estimating Roots and Surds

While a calculator can find the value of a root instantly, you need to be able to estimate it without one. The method is to 'box in' the number with perfect squares or perfect cubes. To estimate √55, you find the two perfect squares it lies between. We know 7² = 49 and 8² = 64. Since 49 < 55 < 64, it follows that √49 < √55 < √64, which means 7 < √55 < 8. Because 55 is closer to 49 than to 64, we can estimate that √55 is about 7.4. The same logic applies to cube roots, using perfect cubes (1, 8, 27, 64, 125...).

Key term

Perfect Square: An integer that is the square of another integer (e.g., 9 is a perfect square because 3² = 9).

Examiner insight

Full marks require showing the bounding inequality (e.g., 9 < √90 < 10). Without this working, even a correct final answer may not receive full credit.

Common pitfall

Mixing up perfect squares and perfect cubes when estimating, or simply guessing a value without showing the bounding integers.

Worked example 13 marks

Estimate the value of √90. State which integer it is closest to.

  1. 1

    Step 1: Find the two perfect squares that 90 lies between.

  2. 2

    9² = 81 and 10² = 100.

  3. 3

    So, 81 < 90 < 100.

  4. 4

    Step 2: Take the square root of the entire inequality.

  5. 5

    √81 < √90 < √100.

  6. 6

    This means 9 < √90 < 10.

  7. 7

    Step 3: Determine which integer it is closer to. Find the difference between 90 and the two square numbers.

  8. 8

    90 - 81 = 9.

  9. 9

    100 - 90 = 10.

  10. 10

    Since 90 is slightly closer to 81 than to 100, its square root will be slightly closer to 9 than to 10.

  11. 11

    Final Answer: √90 is between 9 and 10, and it is closest to 9.

Worked example 22 marks

Between which two consecutive integers does ∛110 lie?

  1. 1

    Step 1: Find the two perfect cubes that 110 lies between.

  2. 2

    We know 4³ = 64 and 5³ = 125.

  3. 3

    So, 64 < 110 < 125.

  4. 4

    Step 2: Take the cube root of the entire inequality.

  5. 5

    ∛64 < ∛110 < ∛125.

  6. 6

    Step 3: This simplifies to 4 < ∛110 < 5.

  7. 7

    Final Answer: ∛110 lies between 4 and 5.

Recap

  • To estimate a square root, find the two perfect squares it lies between.
  • To estimate a cube root, find the two perfect cubes it lies between.
  • This gives you the two integers the root lies between.
  • Check which perfect square/cube the number is closer to for a more accurate estimate.

Quick check

  1. Find two consecutive integers such that ... < √19 < ...1 mark
  2. Estimate √40 to one decimal place.2 marks

End-of-chapter exercise

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

  1. Evaluate 15⁰ + 5⁻².2 marks
  2. Classify each number as rational or irrational: √144, -3.5, √15, π - π, 4/9.3 marks
  3. Simplify fully: (2x⁵)³ ÷ 4x¹⁰.4 marks
  4. Estimate the value of √115. State which integer it is closest to and justify your answer.3 marks
  5. Write 7⁻⁸ ÷ 7⁻³ as a single power of 7.2 marks
  6. A square garden has an area of 80 m². a) Find the exact length of one side of the garden. b) Estimate the perimeter of the garden to the nearest whole metre.4 marks
  7. Find the value of n such that 2ⁿ = 1/32.3 marks
  8. Place the following values in ascending order, showing your working: 3², √75, ∛512, 8.54 marks
  9. Write down a rational number and an irrational number that lie between 4 and 5.2 marks
  10. A cube has a volume of 300 cm³. The side length, L, is given by L = ∛300. a) Find two consecutive integers, a and b, such that a < L < b. b) Is the side length L rational or irrational? Explain your reasoning.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 (9Ni)Number (9Nf)Algebra (9Ae)Algebra (9As)Geometry and Measure (9Gg)Geometry and Measure (9Gp)