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Number (8Ni)

Mathematics Stage 8 Chapter Notes

What this chapter covers

Number - Integers, powers and roots
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In this chapter

  1. 1The Number Hierarchy: Integers and Rationals
  2. 2Squares and Square Roots
  3. 3Cubes and Cube Roots
  4. 4Understanding and Using Indices
  5. 5Using Inequalities to Compare Numbers

1. The Number Hierarchy: Integers and Rationals

To understand numbers, we classify them into groups. The simplest are 'Natural Numbers' (1, 2, 3, ...), which we use for counting. If we include zero and all the negative whole numbers, we get 'Integers' (... -3, -2, -1, 0, 1, 2, 3, ...). A broader group is 'Rational Numbers'. These are any numbers that can be written as a fraction p/q, where p and q are integers and q is not zero. This group includes all integers (e.g., 5 can be written as 5/1), all terminating decimals (e.g., 0.75 = 3/4), and all recurring decimals (e.g., 0.333... = 1/3).

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 look for a clear understanding of the hierarchy; for example, knowing that an integer like 5 is also a rational number because it can be written as 5/1.

Common pitfall

Confusing the definitions. A common mistake is thinking that integers are not rational numbers. Remember that any integer 'n' can be written as the fraction 'n/1', so it is always rational.

Worked example 14 marks

Place the following numbers in the correct region of a Venn diagram with circles for 'Integers' and 'Rational Numbers': 7, -4, 0.5, -3/4, 0

  1. 1

    Step 1: Identify the properties of each number.

  2. 2

    7 is a positive whole number, so it is an integer.

  3. 3

    -4 is a negative whole number, so it is an integer.

  4. 4

    0 is a whole number, so it is an integer.

  5. 5

    0.5 is a terminating decimal (can be written as 1/2), so it is a rational number but not an integer.

  6. 6

    -3/4 is a fraction, so it is a rational number but not an integer.

  7. 7

    Step 2: Place the numbers. All integers are also rational numbers, so the 'Integers' circle is completely inside the 'Rational Numbers' circle.

  8. 8

    Inside the 'Integers' circle: 7, -4, 0.

  9. 9

    Inside the 'Rational Numbers' circle but outside the 'Integers' circle: 0.5, -3/4.

Worked example 22 marks

Write down a rational number that lies between 1/2 and 3/4.

  1. 1

    Step 1: To compare the fractions, find a common denominator. The lowest common multiple of 2 and 4 is 4.

  2. 2

    1/2 is equivalent to 2/4. So we need a number between 2/4 and 3/4.

  3. 3

    Step 2: To find a number between them, we can increase the denominator. Let's use a denominator of 8.

  4. 4

    2/4 is equivalent to 4/8.

  5. 5

    3/4 is equivalent to 6/8.

  6. 6

    Step 3: A rational number between 4/8 and 6/8 is 5/8. Other answers are possible (e.g., by choosing a larger denominator or converting to decimals like 0.6 which is 6/10).

Recap

  • Natural numbers are positive whole numbers used for counting (1, 2, 3...).
  • Integers include all whole numbers: positive, negative, and zero (...-2, -1, 0, 1, 2...).
  • Rational numbers can be written as a fraction, including integers, terminating decimals, and recurring decimals.
  • All integers are rational numbers, but not all rational numbers are integers.
  • To compare fractions, find a common denominator.

Quick check

  1. Is -15 a natural number, an integer, or a rational number? List all that apply.2 marks
  2. Is π (pi) a rational number? Explain why or why not.1 mark

2. Squares and Square Roots

Squaring a number means multiplying it by itself. For example, 5 squared is 5² = 5 × 5 = 25. An interesting point is that squaring a negative number also gives a positive result: (-5)² = (-5) × (-5) = 25. A square root is the reverse operation. It's the number you would square to get the original number. Because both 5² and (-5)² equal 25, the number 25 has two square roots: 5 and -5. However, the square root symbol (√) specifically means the 'principal' or positive square root. So, √25 = 5. If you need to show both roots, you use the ± symbol, as in ±√25 = ±5. You cannot find the square root of a negative number (like √-16) within the real number system, because multiplying any number (positive or negative) by itself will always result in a positive answer.

n² = n × n

(-n)² = (-n) × (-n) = n²

The square roots of a positive number 'a' are √a and -√a.

Key term

Square Root: A value that, when multiplied by itself, gives the original number.

Examiner insight

Marks are often awarded for remembering that a positive number has both a positive and a negative square root, unless the context (like length) implies only the positive root is needed.

Common pitfall

Forgetting the negative square root. When asked for 'the square roots of 25', the full answer is 5 and -5. If the question asks for '√25', the answer is just 5.

Fun fact

The square root symbol (√) is called a 'radix'. It is thought to have originated from a handwritten lowercase 'r', the first letter of the Latin word 'radix' which means 'root'.

Worked example 12 marks

Find the value of 10² - √64.

  1. 1

    Step 1: Deal with the powers and roots first, following the order of operations (BIDMAS/PEMDAS).

  2. 2

    Calculate the square: 10² = 10 × 10 = 100.

  3. 3

    Calculate the square root: √64 = 8 (remember the symbol asks for the positive root).

  4. 4

    Step 2: Perform the subtraction.

  5. 5

    100 - 8 = 92.

Worked example 22 marks

A square has an area of 49 cm². What are the possible lengths of its sides? What is the actual length of its side?

  1. 1

    Step 1: The area of a square is side length squared (A = L²). To find the length, we need to find the square root of the area.

  2. 2

    The area is 49. The square roots of 49 are 7 and -7, because 7² = 49 and (-7)² = 49.

  3. 3

    Step 2: Consider the context of the question. The question asks for a length.

  4. 4

    Length cannot be negative. Therefore, the only possible side length is 7 cm.

Recap

  • Squaring a number means multiplying it by itself.
  • Squaring a positive or a negative number results in a positive number.
  • A positive number has two square roots: one positive and one negative.
  • The symbol √ represents the positive (principal) square root.
  • You cannot find the square root of a negative number in the real number system.

Quick check

  1. What are the two square roots of 121?1 mark
  2. What is the value of √81?1 mark
  3. Why is it not possible to find √-25?1 mark

3. Cubes and Cube Roots

Cubing a number means multiplying it by itself twice. For example, the cube of 4 is 4³ = 4 × 4 × 4 = 64. Unlike squaring, cubing a negative number results in a negative answer. For example, the cube of -4 is (-4)³ = (-4) × (-4) × (-4) = -64. The cube root is the reverse operation. The cube root of a number is the value that, when cubed, gives the original number. The symbol for cube root is ∛. For example, ∛64 = 4 because 4³ = 64. Crucially, every number, whether positive or negative, has exactly one real cube root. So, you can find the cube root of a negative number: ∛-64 = -4 because (-4)³ = -64.

n³ = n × n × n

(-n)³ = (-n) × (-n) × (-n) = -n³

∛a gives the single cube root of a.

Key term

Cube Root: A value that, when multiplied by itself twice (cubed), gives the original number.

Examiner insight

Examiners will test your knowledge of the difference between square and cube roots, particularly regarding negative numbers. Be ready to explain why ∛-8 has a solution but √-4 does not.

Common pitfall

Applying the rules for square roots to cube roots. Students sometimes mistakenly think you cannot find the cube root of a negative number, but you can. For example, ∛-8 = -2.

Worked example 11 mark

Find the value of ∛-27.

  1. 1

    Step 1: We are looking for a number that, when multiplied by itself twice, gives -27.

  2. 2

    Let's try some negative integers: (-1)³ = -1, (-2)³ = -8, (-3)³ = -27.

  3. 3

    Step 2: The number is -3.

  4. 4

    So, ∛-27 = -3.

Worked example 22 marks

Calculate the value of 5³ - ∛1000.

  1. 1

    Step 1: Evaluate the cube and the cube root separately, following the order of operations.

  2. 2

    Calculate the cube: 5³ = 5 × 5 × 5 = 125.

  3. 3

    Calculate the cube root: ∛1000 = 10, because 10 × 10 × 10 = 1000.

  4. 4

    Step 2: Perform the subtraction.

  5. 5

    125 - 10 = 115.

Recap

  • Cubing a number means multiplying it by itself twice (e.g., x³ = x × x × x).
  • Cubing a positive number gives a positive result.
  • Cubing a negative number gives a negative result.
  • Every number (positive or negative) has exactly one real cube root.
  • The cube root of a positive number is positive, and the cube root of a negative number is negative.

Quick check

  1. What is the value of (-2)³?1 mark
  2. What is the value of ∛-125?1 mark

4. Understanding and Using Indices

Indices (or powers) are a shorthand way of writing repeated multiplication. In the term aⁿ, 'a' is the base and 'n' is the index. It means 'a' multiplied by itself 'n' times. There are several important rules, or 'laws', for working with indices.

  1. Multiplication Law: When you multiply terms with the same base, you add the indices. For example, 5³ × 5⁴ = 5³⁺⁴ = 5⁷.
  2. Division Law: When you divide terms with the same base, you subtract the indices. For example, 8⁶ ÷ 8² = 8⁶⁻² = 8⁴.
  3. Zero Index: Any non-zero number raised to the power of zero is equal to 1. For example, 9⁰ = 1.

aⁿ (a is the base, n is the index)

Multiplication Law: aᵐ × aⁿ = aᵐ⁺ⁿ

Division Law: aᵐ ÷ aⁿ = aᵐ⁻ⁿ

Zero Index: a⁰ = 1 (for a ≠ 0)

Key term

Index (plural: Indices): A number that shows how many times a base number is to be multiplied by itself.

Examiner insight

Examiners expect to see the laws of indices applied correctly step-by-step. Show your working by writing out the intermediate stages, like 2³ × 2⁴ = 2³⁺⁴ = 2⁷, to secure method marks.

Common pitfall

Incorrectly applying the laws. A common error in multiplication is multiplying the bases (e.g., 3² × 3⁴ = 9⁶) instead of keeping the base the same and adding the powers (correct answer is 3⁶).

Worked example 11 mark

Simplify 4⁵ × 4³. Give your answer as a single power of 4.

  1. 1

    Step 1: Identify that the bases are the same (both are 4) and the operation is multiplication.

  2. 2

    Step 2: Apply the multiplication law for indices: aᵐ × aⁿ = aᵐ⁺ⁿ.

  3. 3

    Add the indices: 5 + 3 = 8.

  4. 4

    Step 3: Write the final answer: 4⁸.

Worked example 23 marks

Calculate the value of (2⁸ × 2³) ÷ 2⁹.

  1. 1

    Step 1: Simplify the expression in the brackets first. Use the multiplication law: 2⁸ × 2³ = 2⁸⁺³ = 2¹¹.

  2. 2

    The expression is now 2¹¹ ÷ 2⁹.

  3. 3

    Step 2: Now apply the division law: aᵐ ÷ aⁿ = aᵐ⁻ⁿ.

  4. 4

    Subtract the indices: 11 - 9 = 2.

  5. 5

    The expression simplifies to 2².

  6. 6

    Step 3: Calculate the final value: 2² = 4.

Worked example 32 marks

Find the value of 5 + 12⁰.

  1. 1

    Step 1: Apply the zero index rule first: a⁰ = 1.

  2. 2

    So, 12⁰ = 1.

  3. 3

    Step 2: Substitute this value back into the expression.

  4. 4

    5 + 1 = 6.

Recap

  • When multiplying powers with the same base, add the indices.
  • When dividing powers with the same base, subtract the indices.
  • Any non-zero number to the power of zero is 1.
  • The index laws only work when the base numbers are the same.

Quick check

  1. Simplify 10⁹ ÷ 10⁶.1 mark
  2. What is the value of 7¹ × 7⁰?1 mark

5. Using Inequalities to Compare Numbers

Inequalities are mathematical sentences used to compare the size of two values. They are essential for describing ranges of numbers. There are four main symbols: < means 'is less than'. > means 'is greater than'. ≤ means 'is less than or equal to'. ≥ means 'is greater than or equal to'. For example, the statement 'x is a number greater than -2' can be written as x > -2. The statement '-3 < y ≤ 1' means that y is greater than -3 but less than or equal to 1. The integers that satisfy this would be -2, -1, 0, and 1. Notice that -3 is not included, but 1 is included.

< : less than

> : greater than

≤ : less than or equal to

≥ : greater than or equal to

Key term

Inequality: A mathematical statement that compares two values that are not equal, using symbols like <, >, ≤, or ≥.

Examiner insight

When asked to list integers that satisfy an inequality like -2 < x ≤ 1, examiners check that you correctly exclude -2 but include 1. Paying close attention to whether the endpoint is included is crucial for full marks.

Common pitfall

Mixing up the < and ≤ symbols. If an inequality is 'less than' (<), the boundary number is not included in the set of possible integers. If it is 'less than or equal to' (≤), it is included.

Worked example 12 marks

Write down all the integers that satisfy the inequality -4 ≤ x < 2.

  1. 1

    Step 1: Understand the inequality. 'x' is greater than or equal to -4, AND 'x' is less than 2.

  2. 2

    Step 2: 'Greater than or equal to -4' means we start at -4.

  3. 3

    Step 3: 'Less than 2' means we go up to, but do not include, 2.

  4. 4

    Step 4: List the integers in this range: -4, -3, -2, -1, 0, 1.

Worked example 22 marks

The temperature, T, in degrees Celsius, was greater than -5 but no more than 3. Write this as an inequality.

  1. 1

    Step 1: Break down the sentence. 'Greater than -5' means T > -5, or -5 < T.

  2. 2

    Step 2: 'No more than 3' means it can be 3, but not higher. This is 'less than or equal to 3', so T ≤ 3.

  3. 3

    Step 3: Combine these into a single inequality statement: -5 < T ≤ 3.

Recap

  • The 'pointy' end of the inequality symbol always points to the smaller number.
  • The symbols < and > mean the endpoint is not included.
  • The symbols ≤ and ≥ mean the endpoint is included.
  • When listing integers for an inequality, read the symbols carefully to see if the start and end numbers are part of the set.

Quick check

  1. Write the inequality for 'x is a number less than or equal to 10'.1 mark
  2. List the integers that satisfy -1 < n ≤ 3.2 marks

End-of-chapter exercise

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

  1. From the list of numbers -10, 0.8, 3/5, 9, -1.25, π, identify all the rational numbers.2 marks
  2. Find the value of √144 and ∛-1.2 marks
  3. List all the integers that satisfy the inequality -3 ≤ x < 2.2 marks
  4. Calculate the value of (-6)² - √121.3 marks
  5. Simplify the expression (3⁷ × 3²) ÷ 3⁶. Give your answer as a single power of 3.2 marks
  6. Calculate the value of 4³ + ∛-64.3 marks
  7. The cube root of a number is -5. What is the square of that number?3 marks
  8. State whether the following statements are true or false, and give a reason for your answer. a) All integers are natural numbers. b) All natural numbers are rational numbers.4 marks
  9. Evaluate (8⁴ × 8⁰) ÷ 8². Show your working.3 marks
  10. Find a rational number that lies exactly halfway between -1/4 and 1/2.3 marks

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