Number Manipulation Explained: Edexcel IGCSE Mathematics Specification B Complete Breakdown

Number manipulation is the first section of the Edexcel IGCSE (4MB1) Mathematics Specification B specification, and it is tested with remarkable consistency. The skills here are foundational: every other topic in the course draws on your ability to handle numbers confidently and precisely.

Key facts

• Number manipulation is among the most commonly examined areas on both papers.
• It covers the four operations with brackets, prime numbers, factors and multiples (HCF and LCM), index notation and laws, surds, rationalising the denominator, and the classification of number types.
• Questions frequently combine several of these skills in a single problem.
• The edexcel igcse number manipulation content requires both conceptual understanding and procedural fluency.

Prime numbers, factors and multiples

A prime number has exactly two factors: 1 and itself. The first ten primes are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29. Note that 1 is not a prime and 2 is the only even prime.

The Highest Common Factor (HCF) of two numbers is the largest number that divides both exactly. The Lowest Common Multiple (LCM) is the smallest number that both divide into exactly.

Worked example: Find the HCF and LCM of 36 and 48.

Prime factorise each number:

36 = 2² x 3²

48 = 2⁴ x 3

HCF: take the lowest power of each common prime factor: 2² x 3 = 12

LCM: take the highest power of each prime factor that appears: 2⁴ x 3² = 144

Index notation and index laws

Index laws are central to number manipulation explained at this level. The rules apply to all bases, whether numerical or algebraic, and you need them for integer, fractional and negative powers.

Worked example: Evaluate 32^-2/5

Step 1: Root first. 32^1/5 = 2 (the fifth root of 32 is 2)

Step 2: Power. 2² = 4, so 32^2/5 = 4

Step 3: Negative index. 32^-2/5 = 1/4

Surds

A surd is an irrational root left in exact form: sqrt(2), sqrt(5), 3sqrt(7) and so on. The exam expects you to simplify surds, multiply and divide them, and use them for exact answers where a decimal would lose precision.

Simplifying surds

Find the largest perfect square factor and extract its root.

Worked example: Simplify sqrt(72)

72 = 36 x 2

sqrt(72) = sqrt(36) x sqrt(2) = 6sqrt(2)

Multiplying surds

sqrt(a) x sqrt(b) = sqrt(ab)

Worked example: Simplify sqrt(3) x sqrt(12)

sqrt(3) x sqrt(12) = sqrt(36) = 6

Adding and subtracting surds

You can only combine surds with the same radicand (the number under the root).

3sqrt(5) + 2sqrt(5) = 5sqrt(5)

3sqrt(5) + 2sqrt(3) cannot be simplified further.

Worked example: Simplify sqrt(50) + sqrt(18)

sqrt(50) = 5sqrt(2), sqrt(18) = 3sqrt(2)

5sqrt(2) + 3sqrt(2) = 8sqrt(2)

Rationalising the denominator

The edexcel igcse mathematics specification b explained approach to rationalisation is to eliminate the surd from the denominator by multiplying top and bottom by a suitable expression.

Simple case: Rationalise 5/sqrt(3)

Multiply by sqrt(3)/sqrt(3): 5sqrt(3)/3

Harder case (conjugate): Rationalise 4/(3 + sqrt(2))

Multiply by (3 - sqrt(2))/(3 - sqrt(2)):

Numerator: 4(3 - sqrt(2)) = 12 - 4sqrt(2)

Denominator: (3 + sqrt(2))(3 - sqrt(2)) = 9 - 2 = 7

Result: (12 - 4sqrt(2))/7

Natural numbers, integers, rational and irrational numbers

The what is number manipulation igcse specification also requires you to classify numbers correctly.

Key distinction: sqrt(9) = 3 is rational (it simplifies to an integer). sqrt(10) is irrational (it does not simplify to a fraction). The question of whether a square root is rational depends entirely on whether the number under the root is a perfect square.

Exam question patterns

The edexcel igcse mathematics specification b definition of number manipulation spans a wide range of question styles.

• "Express in the form a sqrt(b)": Simplify a surd. The answer must have the smallest possible integer under the root sign.
• "Rationalise the denominator": Remove the surd from the denominator. Show the multiplication step clearly.
• "Evaluate": Calculate the numerical value. For fractional and negative indices, show each stage.
• "Write as a product of prime factors": Use a factor tree or repeated division. Express the answer using index notation.
• "Find the HCF and LCM": Start from the prime factorisation of each number. The method should be visible.

Common mistakes

• Confusing negative index with a negative number. 2^-3 = 1/8, not -8 and not -6.
• Adding surds with different radicands. sqrt(2) + sqrt(3) is not sqrt(5). You cannot add surds unless the number under the root is the same.
• Forgetting to simplify surds before adding. sqrt(8) + sqrt(2) = 2sqrt(2) + sqrt(2) = 3sqrt(2). If you leave sqrt(8) as is, you cannot see that it combines with sqrt(2).
• Using the wrong operation for HCF and LCM. HCF uses the lowest powers, LCM uses the highest. Mixing them up gives the wrong answer and loses both marks.

Self-check questions

1. Express 180 as a product of its prime factors in index form.
   Answer: 180 = 2² x 3² x 5
2. Find the HCF and LCM of 60 and 84.
   Answer: 60 = 2² x 3 x 5, 84 = 2² x 3 x 7. HCF = 2² x 3 = 12. LCM = 2² x 3 x 5 x 7 = 420.
3. Simplify sqrt(200)
   Answer: sqrt(200) = sqrt(100 x 2) = 10sqrt(2)
4. Rationalise the denominator: 6/(sqrt(5) - 1)
   Answer: Multiply by (sqrt(5) + 1)/(sqrt(5) + 1): 6(sqrt(5) + 1)/(5 - 1) = 6(sqrt(5) + 1)/4 = 3(sqrt(5) + 1)/2
5. Evaluate 81^-3/4
   Answer: 81^1/4 = 3, 3³ = 27, so 81^3/4 = 27, and 81^-3/4 = 1/27.

Standard form

Standard form is closely linked to number manipulation, and the exam tests it regularly. A number in standard form is written as A x 10ⁿ, where 1 <= A < 10 and n is an integer.

Worked example: Write 0.00045 in standard form.

Move the decimal point four places to the right: 4.5

Since we moved right, the power is negative: 4.5 x 10^-4

Worked example: Write 3.2 x 10⁵ as an ordinary number.

Move the decimal point five places to the right: 320000

For calculations in standard form, use the index laws. When multiplying, multiply the A values and add the powers. When dividing, divide the A values and subtract the powers. If the resulting A value is not between 1 and 10, adjust it.

Worked example: Calculate (4 x 10³) x (3 x 10⁵)

4 x 3 = 12, and 10³ x 10⁵ = 10⁸

12 x 10⁸ = 1.2 x 10⁹ (adjusting 12 to 1.2 and adding 1 to the power)

Fractions, decimals, ratios and percentages

The specification expects you to move fluently between fractions, decimals, ratios, proportions and percentages, selecting the most appropriate form for a given context. This is a skill that appears not only in number questions but embedded throughout other topics.

Worked example: Express 3/8 as a decimal and a percentage.

3 / 8 = 0.375

0.375 x 100 = 37.5%

Worked example: Increase 240 by 15%.

Method 1: 240 x 0.15 = 36, so 240 + 36 = 276

Method 2 (multiplier): 240 x 1.15 = 276

The multiplier method is faster and less error-prone, especially for repeated percentage changes. A 15% increase uses 1.15; a 15% decrease uses 0.85. For compound percentage changes, raise the multiplier to the appropriate power.

Self-check questions

1. Explain the difference between theoretical probability and experimental probability, and give an example where they might differ significantly.
2. A bag contains 4 red balls, 3 blue balls and 2 green balls. Two balls are drawn without replacement. Draw a tree diagram and find the probability that both balls are the same colour.
3. The mean of five numbers is 12. Four of the numbers are 8, 10, 14 and 16. Find the fifth number.

The edexcel igcse mathematics specification b notes on number manipulation cover a section that appears in almost every exam session. Fluency with these techniques directly supports your performance across the rest of the specification, from algebra through to calculus. Practise on the Green Bridge CBT platform with targeted exam questions, and review the mark scheme after each attempt to understand exactly how examiners award credit.