Subject: Mathematics | Level: GCSE | Exam Board: WJEC
The Number topic is the foundational bedrock of GCSE Mathematics, covering everything from basic arithmetic and prime factorisation to standard form and surds. Mastering this topic is essential because it not only secures significant marks in its own right, but also underpins every other area of the specification, from algebra to statistics.
Revision Notes & Key Concepts
Revision Podcast Transcript
Welcome to your GCSE Mathematics revision podcast. I'm your tutor, and today we're diving into one of the most fundamental topics in the entire specification: Number. Whether you're sitting Foundation or Higher tier, this topic is absolutely everywhere in your exam — and getting it right is the difference between dropping easy marks and picking them up confidently. So let's get started. In this episode, we'll cover the core concepts you need to know, walk through some exam-style examples, talk about the mistakes that cost students marks every single year, and finish with a quick-fire quiz to test your recall. Grab a pen — you might want to jot a few things down. Let's begin with the big picture. The Number topic covers everything from basic arithmetic all the way through to surds and standard form. Think of it as the language of mathematics — if you don't speak Number fluently, every other topic becomes harder. Algebra, ratio, statistics — they all rely on your number skills. So this is time very well spent. Right. Core concepts. Let's go through them one by one. First up: Types of Numbers. You need to know the difference between natural numbers, integers, rational numbers, and irrational numbers. Natural numbers are your counting numbers: 1, 2, 3, 4 — positive whole numbers. Integers include all whole numbers, both positive and negative, and zero. So minus 5, minus 1, 0, 7, 100 — all integers. Rational numbers can be written as a fraction — a ratio of two integers. So 0.5 is rational because it equals one half. Even recurring decimals like 0.333 repeating are rational, because they equal one third. Irrational numbers cannot be written as exact fractions. Pi is irrational. The square root of 2 is irrational. Surds — which we'll come back to — are irrational. On the Higher tier, you need to be able to prove a number is irrational, so make sure you understand this distinction deeply. Next: Order of Operations — BIDMAS. This is Brackets, Indices, Division, Multiplication, Addition, Subtraction. Always work in this order. The most common mistake? Candidates add before they multiply. For example: 3 plus 4 times 5. The answer is NOT 35. You must do the multiplication first: 4 times 5 is 20, then add 3, giving 23. Examiners see this error constantly. Remember: Division and Multiplication have equal priority, so work left to right. Same for Addition and Subtraction — left to right when they appear together. Now let's talk about Prime Numbers and Prime Factorisation. A prime number has exactly two factors: 1 and itself. The first few primes are 2, 3, 5, 7, 11, 13, 17, 19, 23. Note: 1 is NOT a prime number — this is a very common error. Every integer greater than 1 can be written as a product of prime factors — this is called prime factorisation. The method you should use is a factor tree or repeated division. For example, 60 equals 2 times 2 times 3 times 5, which we write as 2 squared times 3 times 5. Prime factorisation is the foundation for finding the Highest Common Factor and the Lowest Common Multiple. The HCF — Highest Common Factor — is the largest number that divides exactly into two or more numbers. To find it, write both numbers as products of prime factors, then multiply together the primes they share. For example, HCF of 24 and 36: 24 is 2 cubed times 3, and 36 is 2 squared times 3 squared. The shared primes are 2 squared and 3 to the power 1, so HCF equals 4 times 3, which is 12. The LCM — Lowest Common Multiple — is the smallest number that is a multiple of two or more numbers. To find it, write both numbers as products of prime factors, then multiply together each prime factor at its highest power. Using the same example: LCM of 24 and 36 is 2 cubed times 3 squared, which is 8 times 9, equals 72. Moving on to Rounding and Significant Figures. You need to be able to round to a given number of decimal places or significant figures. Decimal places are easy — count from the decimal point. Significant figures start from the first non-zero digit. So 0.00456 to 2 significant figures is 0.0046 — the significant figures are the 4 and the 5, and we round up because the next digit is 6. A key exam skill is estimation: round all numbers to 1 significant figure first, then calculate. This gives you a quick check on whether your answer is sensible. Now let's tackle Standard Form. This is used to write very large or very small numbers concisely. A number in standard form is written as A times 10 to the power n, where A is between 1 and 10 — that means 1 is less than or equal to A, which is less than 10 — and n is an integer. For example, 3,400,000 in standard form is 3.4 times 10 to the power 6. And 0.000052 is 5.2 times 10 to the power negative 4. To multiply numbers in standard form, multiply the A values and add the powers. To divide, divide the A values and subtract the powers. Always check your final answer is still in correct standard form — if A is 10 or greater, you need to adjust. Now for Surds — this is Higher tier content. A surd is an irrational root that cannot be simplified to a whole number or fraction. The square root of 2, the square root of 3, the square root of 5 — these are all surds. You need to be able to simplify surds, add and subtract like surds, and multiply and divide them. To simplify: look for square factors. The square root of 48 equals the square root of 16 times 3, which equals 4 times the square root of 3. To multiply surds: the square root of a times the square root of b equals the square root of a times b. To rationalise the denominator — this means removing the surd from the bottom of a fraction — multiply top and bottom by the surd. So 1 over the square root of 3 becomes the square root of 3 over 3. Finally, Limits of Accuracy. When a measurement is rounded, the true value lies within a range. If a length is given as 7.4 centimetres to 1 decimal place, the lower bound is 7.35 and the upper bound is 7.45. The key rule: lower bound equals the rounded value minus half the unit of rounding. Upper bound equals the rounded value plus half the unit of rounding. For calculations involving bounds: to find the maximum of a sum, use upper plus upper. For a maximum of a product, use upper times upper. For a maximum of a quotient, use upper divided by lower. And for a minimum, reverse these. This is a classic Higher tier question worth 3 to 4 marks. Right, let's move on to exam tips and common mistakes — this is where marks are won and lost. Tip one: always show your working. In mathematics, method marks are available even when the final answer is wrong. If you write down the correct method but make an arithmetic slip, you can still earn 2 out of 3 marks. Never just write the answer — always show the steps. Tip two: read the question carefully for the level of accuracy required. Does it say give your answer to 3 significant figures? Or give an exact answer? If it says exact, do not round — leave it as a surd or a fraction. Candidates lose marks every year by rounding when the question said exact. Tip three: on non-calculator papers, practice your written methods. Long multiplication, long division, adding and subtracting fractions — these need to be automatic. Use formal column methods and show every step. Tip four: with negative numbers, be extra careful. Minus 3 times minus 4 is positive 12 — two negatives multiplied give a positive. But minus 3 plus minus 4 is minus 7 — you're adding a negative, which means going further left on the number line. Draw a number line if you need to. Tip five: for standard form on a calculator paper, use the EXP or times 10 to the power button on your calculator. Make sure you know how your specific calculator handles standard form — this is worth practising before the exam. Common mistake number one: treating 1 as a prime number. It is not. It has only one factor — itself — so it doesn't meet the definition. Common mistake number two: confusing HCF and LCM. Remember: HCF is smaller, LCM is larger. HCF for sharing into equal groups; LCM for finding when events coincide again. Common mistake number three: with bounds, using the wrong combination. For a maximum value of a fraction, you want the upper bound of the numerator divided by the lower bound of the denominator — not upper divided by upper. Common mistake number four: when simplifying surds, not spotting the largest square factor. The square root of 72 — some candidates write it as 2 times the square root of 18, which is partially simplified. The fully simplified form is 6 times the square root of 2, because 72 equals 36 times 2. Now for our quick-fire recall quiz. I'll ask a question, pause, then give the answer. See how many you can get. Question one: What is the HCF of 18 and 24? ... The answer is 6. Question two: Write 0.000034 in standard form. ... The answer is 3.4 times 10 to the power negative 5. Question three: Simplify the square root of 50. ... The answer is 5 times the square root of 2. Question four: A value is rounded to 2 decimal places and gives 6.73. What is the lower bound? ... The answer is 6.725. Question five: Calculate 5 plus 3 squared times 2 minus 1. ... Using BIDMAS: indices first, 3 squared is 9. Then multiplication: 9 times 2 is 18. Then left to right: 5 plus 18 is 23, minus 1 is 22. The answer is 22. How did you do? If you got all five, brilliant — you're in great shape. If you missed any, go back and review that concept before your exam. Let's wrap up with a quick summary of the key things to remember. One: BIDMAS — Brackets, Indices, Division, Multiplication, Addition, Subtraction. Always follow this order. Two: Prime factorisation is your tool for HCF and LCM. Write both numbers as products of primes, then use the Venn diagram or comparison method. Three: Standard form requires A to be between 1 and 10. Positive powers for large numbers, negative powers for small numbers. Four: Surds are irrational. Simplify by finding square factors. Rationalise by multiplying by the surd. Five: Bounds — lower bound is half the rounding unit below, upper bound is half above. Choose the right combination for max or min calculations. Six: Always show working. Always check the required accuracy. Never round too early in multi-step calculations. This topic underpins everything in GCSE Mathematics. Master it, and you'll find every other topic easier. Good luck with your revision — you've got this. See you in the next episode.
Key Terms & Definitions
- Integer
- A whole number that can be positive, negative, or zero.
- Prime Number
- A number with exactly two distinct factors: 1 and itself.
- Rational Number
- A number that can be expressed as a fraction a/b, where a and b are integers and b is not zero.
- Irrational Number
- A number that cannot be written as a simple fraction; its decimal goes on forever without repeating.
- Significant Figure
- The digits in a number that carry meaning contributing to its precision, starting from the first non-zero digit.
- Standard Form
- A way of writing numbers as A × 10^n, where 1 ≤ A < 10 and n is an integer.
Worked Examples
Worked Example
Question: Calculate $\frac{4.5 \times 10^7}{5 \times 10^{-3}}$. Give your answer in standard form. (3 marks)
Solution: Step 1: Divide the numbers: $4.5 \div 5 = 0.9$ Step 2: Divide the powers of 10 by subtracting indices: $10^7 \div 10^{-3} = 10^{7 - (-3)} = 10^{10}$ Step 3: Combine: $0.9 \times 10^{10}$ Step 4: Adjust to standard form ($1 \le A < 10$): $0.9 = 9 \times 10^{-1}$ Step 5: Final answer: $9 \times 10^{-1} \times 10^{10} = 9 \times 10^9$
Worked Example
Question: Rationalise the denominator of $\frac{12}{\sqrt{3}}$ and simplify fully. (2 marks)
Solution: Step 1: Multiply numerator and denominator by $\sqrt{3}$. $\frac{12 \times \sqrt{3}}{\sqrt{3} \times \sqrt{3}}$ Step 2: Simplify the denominator. $\frac{12\sqrt{3}}{3}$ Step 3: Cancel the whole numbers ($12 \div 3 = 4$). Final answer: $4\sqrt{3}$
Worked Example
Question: A rectangle has a length of 8.4 cm and a width of 5.3 cm, both measured to 1 decimal place. Calculate the upper bound for the area of the rectangle. (3 marks)
Solution: Step 1: Find the upper bound of the length. $8.4 + 0.05 = 8.45$ cm. Step 2: Find the upper bound of the width. $5.3 + 0.05 = 5.35$ cm. Step 3: To find the maximum area, multiply the two upper bounds. Area = $8.45 \times 5.35$ Final answer: $45.2075$ cm²
Practice Questions
Question: Write 84 as a product of its prime factors. (2 marks)
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Question: Estimate the value of $\frac{41.3 \times 29.6}{0.198}$. (3 marks)
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Question: Expand and simplify $(2\sqrt{3} - 1)(\sqrt{3} + 4)$. (3 marks)
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Question: A runner completes a 100m race in 12.4 seconds, measured to the nearest 0.1 seconds. The track is 100m long, measured to the nearest metre. Calculate the upper bound for the runner's average speed. (4 marks)
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Question: Prove algebraically that the recurring decimal $0.\dot{2}1\dot{5}$ can be written as the fraction $\frac{215}{999}$. (3 marks)
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