Edexcel IGCSE Mathematics: Numbers and the Number System — Revision Notes

Condensed for the final weeks. For the full explanation, use the Numbers and the Number System study guide.

Types of number and factorisation

Primes have exactly two factors, so 1 is not prime and 2 is the only even prime.

Prime factorisation gives HCF and LCM:

• HCF — product of the lowest power of each common prime.
• LCM — product of the highest power of every prime present.

Rational numbers can be written as a fraction; irrational numbers (√2, π) cannot. A recurring decimal is rational, and converting one to a fraction is a standard question: let x equal the decimal, multiply by a power of 10 to align the recurring part, subtract, and solve.

Fractions

Multiply across; divide by multiplying by the reciprocal; add and subtract with a common denominator. Convert mixed numbers to improper fractions before multiplying or dividing.

Ratio

Divide by the total number of parts, then multiply.

Share 240 in the ratio 3 : 5     ->  8 parts, 240/8 = 30  ->  90 : 150

Read carefully whether the question gives the total, one share, or the difference between shares. Which is given determines how you find the value of one part, and misreading it is the most common error.

Percentages

increase by 25%:   x 1.25        decrease by 25%:   x 0.75
percentage change = (change / ORIGINAL) x 100
reverse percentage: DIVIDE by the multiplier
compound growth:   P x (multiplier)^n

Two errors dominate this topic:

1. Dividing by the new value instead of the original in percentage change.
2. Subtracting instead of dividing in reverse percentages. If a price is $90 after a 20% increase, the original is 90 ÷ 1.2 = $75, not 90 × 0.8 = $72.

Note also that a 20% rise followed by a 20% fall does not return to the start: 1.2 × 0.8 = 0.96, a 4% net loss.

Simple versus compound interest: simple is calculated on the original amount each year; compound is calculated on the running total, so the gap widens with time.

Worked example. £2000 is invested at 3% compound interest for 4 years. Find the total interest earned.

2000 x 1.03^4 = 2251.02 (2 d.p.)
interest = 2251.02 - 2000 = 251.02

Carry full accuracy through the calculation and round only the final answer — rounding the multiplier early is a common way to lose the last mark.

Indices and standard form

a^m x a^n = a^(m+n)     a^m / a^n = a^(m-n)     (a^m)^n = a^(mn)
a^0 = 1                 a^-n = 1/a^n            a^(m/n) = (n-th root of a)^m

A negative index means reciprocal, not a negative answer: 2⁻³ = 1/8. For a negative fractional index, flip first: (4/9)^(−1/2) = (9/4)^(1/2) = 3/2.

standard form: A x 10^n     with 1 <= A < 10

After multiplying or dividing, re-check A is between 1 and 10 and adjust the power.

Surds

sqrt(a) x sqrt(b) = sqrt(ab)        sqrt(50) = 5 sqrt(2)

Rationalise a single term by multiplying top and bottom by the surd; for a + √b, multiply by the conjugate a − √b, since (a+√b)(a−√b) = a² − b.

√a + √b ≠ √(a+b) — check with √9 + √16 = 7, not √25 = 5.

Bounds

For a value rounded to the nearest unit u, bounds are ± u/2.

mass 3.7 kg to 1 d.p.:   3.65 <= m < 3.75

Want	Add	Subtract	Multiply	Divide
Maximum	UB + UB	UB − LB	UB × UB	UB ÷ LB
Minimum	LB + LB	LB − UB	LB × LB	LB ÷ UB

Subtraction and division cross over. That single fact is most of the topic.

Set notation

The universal set 𝓔 contains every element under consideration. A′ denotes the complement of set A (every element not in A). A ∩ B denotes intersection (elements in both A and B). A ∪ B denotes union (elements in A, B, or both).

Venn diagrams translate these symbols visually. Exam questions typically ask you to shade a described region, or to read off the number of elements satisfying a given combination of conditions directly from a labelled diagram.

Exam traps

• Treating 1 as prime.
• Dividing by the new value in percentage change.
• Subtracting instead of dividing in reverse percentages.
• Assuming a rise then an equal fall cancels out.
• Reading a negative index as a negative answer.
• Using UB ÷ UB for a maximum quotient.
• Rounding partway through — round only at the end.
• Confusing intersection (∩, “and”) with union (∪, “or”) when shading a Venn diagram.

Self-test

1. Find the HCF and LCM of 36 and 60 by prime factorisation.
2. A price is $90 after a 20% increase. What was it before?
3. Evaluate (4/9)^(−1/2).
4. Rationalise 2/(5 − √3).
5. How do you find the maximum value of a ÷ b from bounds?
6. What does A ∩ B mean, and how does it differ from A ∪ B?

Answers: 1. 36 = 2²×3², 60 = 2²×3×5; HCF = 2²×3 = 12, LCM = 2²×3²×5 = 180. 2. 90 ÷ 1.2 = $75. 3. Flip to (9/4)^(1/2) = 3/2. 4. Multiply top and bottom by (5 + √3) to get 2(5 + √3)/22 = (5 + √3)/11. 5. Upper bound of a divided by the lower bound of b. 6. A ∩ B is the intersection — elements in both A and B; A ∪ B is the union — elements in A, B, or both.