Cambridge Lower Secondary CheckpointStage 9

Number (9Nf)

Mathematics Stage 9 Chapter Notes

What this chapter covers

Number - Fractions, decimals, percentages, ratio and proportion
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Number (9Nf) notes

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In this chapter

  1. 1Understanding Rational and Irrational Numbers
  2. 2Working with Surds
  3. 3Mastering Directed Numbers
  4. 4Calculating with Fractions
  5. 5Accuracy: Upper and Lower Bounds

1. Understanding Rational and Irrational Numbers

In mathematics, we classify numbers into different sets. A 'rational' number is any number that can be written as a fraction, in the form a/b, where 'a' and 'b' are integers and 'b' is not zero. This includes all integers (e.g., 5 = 5/1), all terminating decimals (e.g., 0.5 = 1/2), and all recurring decimals (e.g., 0.333... = 1/3). In contrast, an 'irrational' number cannot be written as a simple fraction. Its decimal representation goes on forever without repeating. Famous examples include Pi (π) and the square roots of numbers that are not perfect squares, like √2 or √10. These specific irrational numbers are called surds.

Key term

Irrational Number: A number that cannot be expressed as a simple fraction p/q, where p and q are integers and q is not zero; its decimal representation is non-terminating and non-recurring.

Common pitfall

Assuming that all numbers involving a square root are irrational. Remember that the square root of a perfect square (e.g., √25 = 5) is a rational number.

Fun fact

The number π (pi) has been calculated to over 100 trillion decimal places by Google in 2022, and no repeating pattern has ever been found, confirming its irrational nature.

Worked example 14 marks

Sort the following numbers into two groups: Rational and Irrational. The numbers are: -7, √49, √15, 3/8, 0.57, π, 0.121212...

  1. 1

    Step 1: Analyse each number individually.

  2. 2

    -7 can be written as -7/1, so it is rational.

  3. 3

    √49 = 7, which can be written as 7/1, so it is rational.

  4. 4

    √15 is the square root of a non-perfect square, so it is irrational.

  5. 5

    3/8 is already in fraction form, so it is rational.

  6. 6

    0.57 is a terminating decimal, which can be written as 57/100, so it is rational.

  7. 7

    π is a well-known irrational number.

  8. 8

    0.121212... is a recurring decimal, so it is rational.

  9. 9

    Step 2: Group the numbers.

  10. 10

    Rational: -7, √49, 3/8, 0.57, 0.121212...

  11. 11

    Irrational: √15, π

Recap

  • Rational numbers can be written as a fraction a/b.
  • 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 is infinite and non-repeating.
  • Surds (like √2) and special numbers like π are irrational.

Quick check

  1. Is the number 0.123123123... rational or irrational? Explain why.2 marks
  2. Give an example of an irrational number between 4 and 5.1 mark

2. Working with Surds

A surd is an irrational number expressed as the root of an integer that cannot be simplified to a rational number. For example, √3 and ³√10 are surds, but √9 is not, because it equals 3. We can simplify surds by using the rule √ab = √a × √b. The goal is to find the largest perfect square that is a factor of the number under the root. For example, to simplify √50, we can write it as √25 × √2 = 5√2. We can also estimate the value of a surd. To estimate √45, we know it lies between √36 (which is 6) and √49 (which is 7). Since 45 is closer to 49, the value of √45 will be slightly less than 7.

√ab = √a × √b

√(a/b) = √a / √b

Key term

Surd: An irrational number that is the root of an integer, which cannot be simplified to a rational number (e.g., √2, ³√5).

Examiner insight

Examiners award marks for showing clear simplification steps, such as breaking down the number under the root into its factors to find the largest square factor.

Fun fact

The spiral of Theodorus, also known as the square root spiral, is a spiral composed of right triangles, where the hypotenuse of each triangle has a length equal to the square root of consecutive integers (√2, √3, √4, etc.).

Worked example 12 marks

Simplify the surd √48.

  1. 1

    Step 1: Find the largest square number that is a factor of 48. The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. The largest square factor is 16.

  2. 2

    Step 2: Rewrite √48 using this factor. √48 = √16 × 3

  3. 3

    Step 3: Apply the rule √ab = √a × √b. √16 × 3 = √16 × √3

  4. 4

    Step 4: Simplify the square root of the perfect square. √16 = 4.

  5. 5

    Step 5: Write the final simplified answer. 4√3

Worked example 23 marks

Estimate the value of √69 to one decimal place.

  1. 1

    Step 1: Find the two perfect squares that 69 lies between. 8² = 64 and 9² = 81. So, √69 is between 8 and 9.

  2. 2

    Step 2: Determine which square 69 is closer to. The difference between 69 and 64 is 5. The difference between 81 and 69 is 12. So, 69 is closer to 64.

  3. 3

    Step 3: This means √69 will be closer to 8 than to 9. Let's try a value like 8.3.

  4. 4

    Step 4: Check the estimate by squaring it. 8.3² = 8.3 × 8.3 = 68.89. This is very close to 69.

  5. 5

    Step 5: A slightly higher estimate, 8.4², would be 70.56, which is further away. So, 8.3 is a good estimate to one decimal place.

Recap

  • A surd is an irrational root, like √2 or √3.
  • To simplify a surd, find the largest square factor.
  • Use the rule √ab = √a × √b to simplify.
  • To estimate a surd, find the two closest perfect squares.
  • Remember that you cannot add or subtract surds like you do with numbers (e.g., √2 + √3 ≠ √5).

Quick check

  1. Simplify √72.2 marks
  2. Between which two whole numbers does √110 lie?1 mark

3. Mastering Directed Numbers

Directed numbers are numbers that have a positive (+) or negative (-) sign, indicating their position relative to zero. Mastering operations with them is crucial. For addition and subtraction, think of a number line. Adding a positive number moves you right. Adding a negative number (e.g., 5 + (-3)) is the same as subtraction, moving you left (5 - 3 = 2). Subtracting a negative number (e.g., 5 - (-3)) is the same as addition, moving you right (5 + 3 = 8). For multiplication and division, the rules are simple: if the signs are the same, the result is positive. If the signs are different, the result is negative. Always remember to apply the order of operations (BIDMAS/BODMAS).

(+) × (+) = (+)

(-) × (-) = (+)

(+) × (-) = (-)

(-) ÷ (+) = (-)

Key term

Directed Number: A number that has both magnitude (size) and direction (positive or negative).

Examiner insight

Examiners often use directed numbers in multi-step calculations to test both procedural fluency and understanding of the order of operations (BIDMAS/BODMAS).

Common pitfall

Mistakes with subtraction of a negative number are very common. For example, calculating 5 - (-2) as 3 instead of the correct answer, 7.

Worked example 12 marks

Calculate the value of -8 + (-12) ÷ 3.

  1. 1

    Step 1: Identify the operations. There is an addition and a division.

  2. 2

    Step 2: Apply the order of operations (BIDMAS/BODMAS). Division comes before addition.

  3. 3

    Step 3: Perform the division: (-12) ÷ 3. A negative divided by a positive is a negative. 12 ÷ 3 = 4. So, (-12) ÷ 3 = -4.

  4. 4

    Step 4: Rewrite the expression: -8 + (-4).

  5. 5

    Step 5: Perform the addition. Adding a negative is the same as subtracting: -8 - 4 = -12.

Worked example 23 marks

The temperature at midnight was -5°C. By 6 am, it had dropped by 4°C. By midday, it had risen by 11°C. What was the temperature at midday?

  1. 1

    Step 1: Start with the initial temperature: -5°C.

  2. 2

    Step 2: Calculate the temperature at 6 am. It dropped by 4°C, so we subtract 4. -5 - 4 = -9°C.

  3. 3

    Step 3: Calculate the temperature at midday. It rose by 11°C from the 6 am temperature, so we add 11. -9 + 11 = 2°C.

  4. 4

    Step 4: The final temperature at midday was 2°C.

Recap

  • Adding a negative number is the same as subtracting a positive number.
  • Subtracting a negative number is the same as adding a positive number.
  • For multiplication and division, if the signs are the same, the answer is positive.
  • For multiplication and division, if the signs are different, the answer is negative.
  • Always follow the order of operations (BIDMAS/BODMAS).

Quick check

  1. Calculate: (-7) × (-2) - 202 marks

4. Calculating with Fractions

Fractions are a way of representing parts of a whole. To add or subtract fractions, you must have a common denominator. Find the lowest common multiple (LCM) of the denominators, convert the fractions to their equivalents, and then add or subtract the numerators. Multiplication is more straightforward: multiply the numerators together and the denominators together. It's often easier to cancel common factors before multiplying. Division has one extra step: to divide by a fraction, you multiply by its reciprocal (you 'flip' the second fraction and multiply). This is often remembered by the phrase 'Keep, Change, Flip'.

a/b + c/d = (ad + bc) / bd

a/b × c/d = ac / bd

a/b ÷ c/d = a/b × d/c = ad / bc

Key term

Reciprocal: The reciprocal of a number is 1 divided by that number; for a fraction a/b, its reciprocal is b/a.

Examiner insight

Examiners look for clear working. Show the conversion to a common denominator for addition/subtraction, and the 'invert and multiply' step for division, to secure method marks even if you make a calculation error.

Common pitfall

Forgetting to find a common denominator when adding or subtracting fractions, and instead just adding the numerators and denominators together.

Worked example 13 marks

Calculate 3¾ ÷ 1¼. Give your answer as a mixed number.

  1. 1

    Step 1: Convert the mixed numbers into improper fractions.

  2. 2

    3¾ = (3×4 + 3)/4 = 15/4

  3. 3

    1¼ = (1×4 + 1)/4 = 5/4

  4. 4

    Step 2: Rewrite the division problem with the improper fractions: 15/4 ÷ 5/4.

  5. 5

    Step 3: Apply the 'Keep, Change, Flip' rule. Keep 15/4, change ÷ to ×, and flip 5/4 to 4/5. The problem becomes 15/4 × 4/5.

  6. 6

    Step 4: Multiply the fractions. You can cancel the 4s and also divide 15 and 5 by 5. (15/4) × (4/5) = (3×5)/(1×4) × (1×4)/(1×5). This simplifies to 3/1 × 1/1 = 3.

  7. 7

    Step 5: The answer is 3. As a mixed number, it is just 3.

Worked example 22 marks

A recipe requires 2/3 of a cup of flour. You only want to make half of the recipe. How much flour do you need?

  1. 1

    Step 1: Understand the problem. You need to find half 'of' 2/3 of a cup. 'Of' means multiply.

  2. 2

    Step 2: Set up the calculation: ½ × 2/3.

  3. 3

    Step 3: Multiply the numerators and the denominators: (1 × 2) / (2 × 3) = 2/6.

  4. 4

    Step 4: Simplify the resulting fraction. 2/6 can be simplified by dividing the numerator and denominator by 2. 2/6 = 1/3.

  5. 5

    Step 5: You need 1/3 of a cup of flour.

Recap

  • To add or subtract fractions, find a common denominator.
  • To multiply fractions, multiply the numerators and multiply the denominators.
  • To divide by a fraction, multiply by its reciprocal (Keep, Change, Flip).
  • Always convert mixed numbers to improper fractions before multiplying or dividing.
  • Simplify your final answer where possible.

Quick check

  1. Calculate 5/6 - 1/4.2 marks
  2. What is the reciprocal of 7?1 mark

5. Accuracy: Upper and Lower Bounds

When a measurement is rounded to a given unit, the actual value lies within a range. This range is defined by its 'lower bound' and 'upper bound'. To find these bounds, first identify the degree of accuracy (e.g., nearest 10, nearest cm, 1 decimal place). Then, take that unit of accuracy, halve it, and subtract this half-unit from the measurement to find the lower bound, and add it to find the upper bound. For example, if a length is 12 cm to the nearest cm, the unit of accuracy is 1 cm. Half of this is 0.5 cm. So, the lower bound is 12 - 0.5 = 11.5 cm and the upper bound is 12 + 0.5 = 12.5 cm. The actual length, x, can be written as an error interval: 11.5 ≤ x < 12.5. Note the use of '<' for the upper bound, as the value can be 11.5 but must be less than 12.5.

Lower Bound ≤ Actual Value < Upper Bound

Key term

Error Interval: The range of values a number could have taken before being rounded, expressed as an inequality.

Examiner insight

For calculations involving bounds, marks are awarded for correctly identifying the upper and lower bounds of each individual measurement before performing the calculation. Do not round your final answer unless instructed to.

Common pitfall

Using the wrong inequality sign for the upper bound. The actual value must be strictly less than the upper bound, so use '<' and not '≤'.

Worked example 12 marks

A bag of flour weighs 1.5 kg, correct to 1 decimal place. Write down the error interval for the weight of the bag.

  1. 1

    Step 1: Identify the measurement and the degree of accuracy. The measurement is 1.5 kg, and the accuracy is to 1 decimal place (0.1 kg).

  2. 2

    Step 2: Find the unit of accuracy and halve it. The unit is 0.1 kg. Half of this is 0.1 / 2 = 0.05 kg.

  3. 3

    Step 3: Calculate the lower bound by subtracting this value from the measurement. Lower Bound = 1.5 - 0.05 = 1.45 kg.

  4. 4

    Step 4: Calculate the upper bound by adding this value to the measurement. Upper Bound = 1.5 + 0.05 = 1.55 kg.

  5. 5

    Step 5: Write the error interval for the weight (w). 1.45 ≤ w < 1.55 kg.

Worked example 23 marks

A rectangular field measures 20m by 30m, both to the nearest metre. Calculate the upper bound for the area of the field.

  1. 1

    Step 1: Find the bounds for each measurement. Both are to the nearest metre, so the unit is 1m. Half of this is 0.5m.

  2. 2

    Step 2: For the length of 30m: Lower Bound = 29.5m, Upper Bound = 30.5m.

  3. 3

    Step 3: For the width of 20m: Lower Bound = 19.5m, Upper Bound = 20.5m.

  4. 4

    Step 4: To find the upper bound for the area (Area = Length × Width), we need to multiply the upper bound of the length by the upper bound of the width.

  5. 5

    Step 5: Upper Bound of Area = 30.5m × 20.5m = 625.25 m².

Recap

  • All measurements have a degree of error.
  • To find the bounds, halve the unit of accuracy and add/subtract from the measurement.
  • The error interval is written as Lower Bound ≤ x < Upper Bound.
  • For calculations, use the correct combination of bounds to find the maximum or minimum possible value.
  • To maximise a product or quotient, use the largest possible numerator and smallest possible denominator, and vice versa.

Quick check

  1. A crowd of 25,000 is reported, correct to the nearest thousand. What is the lower bound for the crowd size?1 mark
  2. A number, y, is 8.4 when rounded to one decimal place. State the error interval for y.2 marks

End-of-chapter exercise

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

  1. From the list of numbers: √100, -3.5, √8, 17/3, π+2, 0.2̇7̇, identify all the irrational numbers.2 marks
  2. Calculate: (-4) × (5 - 9) + (-20) ÷ 43 marks
  3. Simplify √125 + √20. Give your answer in the form a√5.3 marks
  4. Arrange the following values in ascending order: 3/8, 0.3, 35%, 1/3.3 marks
  5. A car travels a distance of 150 km (to the nearest 10 km) in a time of 2.5 hours (to the nearest 0.1 hours). Calculate the lower bound for the average speed of the car. Speed = Distance / Time.4 marks
  6. A number, x, is 5.3 when rounded to one decimal place. A number, y, is 4 when rounded to the nearest integer. Find the upper bound of x - y.3 marks
  7. In a school, 2/5 of students are in the lower school. 1/4 of the remaining students are in the sixth form. What fraction of the students are in the middle school?3 marks
  8. a) Estimate the value of √90. b) Write down an irrational number that lies between 9 and 10.3 marks
  9. Show that the recurring decimal 0.454545... can be written as the fraction 5/11.2 marks
  10. A plank of wood is 3.5m long, correct to one decimal place. A carpenter cuts off a piece that is 150cm long, correct to the nearest cm. What is the maximum possible length of the remaining piece of wood, in cm?4 marks

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