Number ToolkitCambridge IGCSE Maths: Revision notes

Section 1

What types of number do I need to recognise?

Natural numbers: positive whole numbers used for counting (1, 2, 3, ...)
Integers: positive, zero and negative whole numbers (..., -2, -1, 0, 1, 2, ...)
Prime numbers: numbers greater than 1 with exactly two factors, 1 and itself, e.g. 2, 3, 5, 7, 11
Square numbers: the result of multiplying an integer by itself, e.g. 1, 4, 9, 16
Cube numbers: the result of multiplying an integer by itself three times, e.g. 1, 8, 27, 64
Rational numbers: numbers that can be written as a fraction $\frac{a}{b}$ where $a$ and $b$ are integers
Irrational numbers: numbers that cannot be written as an exact fraction, e.g. $\pi$ or $\sqrt{2}$
Reciprocal: 1 divided by a number, or the number 'flipped' if it's a fraction, e.g. the reciprocal of 5 is $\frac{1}{5}$

Key termsnatural numberprime numberrational numberirrational numberreciprocal

Section 2

Large numbers must be converted accurately between figures and words, keeping track of place value.

Example: six billion is written 6 000 000 000. Each 'group of three' from the right represents thousand, million, billion in turn.

When converting words to figures, write out each place value group carefully and use zeros as placeholders for any group with no stated value.

Key termsplace value

Section 3

Follow the correct order of operations: Brackets, Indices, Division/Multiplication, Addition/Subtraction (BIDMAS)
With negative numbers: adding a negative is the same as subtracting; subtracting a negative is the same as adding, e.g. $5 - (-3) = 5 + 3 = 8$
Multiplying/dividing two negatives gives a positive; multiplying/dividing one negative and one positive gives a negative
Apply these same rules consistently to fractions and decimals, and to practical contexts like temperature changes (e.g. a rise of 8°C from -3°C gives -3 + 8 = 5°C)

Key termsorder of operationsBIDMAS

Common mistake

Forgetting BIDMAS order — e.g. calculating 3 + 4 × 2 as 14 instead of the correct 11 — is a very common error.

Section 4

You are expected to recall without a calculator:

Square numbers from $1^2$ to $15^2$ (1 to 225) and their corresponding square roots
Cube numbers of 1, 2, 3, 4, 5 and 10 (1, 8, 27, 64, 125, 1000) and their corresponding cube roots

Beyond these recalled values, you can calculate squares, square roots, cubes, cube roots and other powers/roots using a calculator where permitted.

Key termssquare rootcube root

Exam tip

Memorising squares 1–15 and cubes of 1,2,3,4,5,10 saves valuable time on the non-calculator paper.

Section 5

Real-life problems often disguise integer and fraction operations:

Temperature changes: treat a rise as addition and a fall as subtraction of a positive or negative value
Improper fractions and mixed numbers: convert to a consistent form before adding, subtracting, multiplying or dividing
Always re-read the context at the end to check your answer makes sense (e.g. a temperature answer should have a sensible sign and size)

Example

The temperature was -6°C and rises by 10°C. New temperature = -6 + 10 = 4°C.

Know the number types: natural, integer, prime, square, cube, rational, irrational, reciprocal
Convert confidently between numbers and words, tracking place value carefully
Apply BIDMAS to every multi-step calculation
Two negatives multiplied/divided give a positive; one negative and one positive give a negative
Recall squares 1–15 and their roots, and cubes of 1, 2, 3, 4, 5, 10 and their roots, without a calculator
Apply integer and fraction rules correctly to real-life contexts like temperature change

That's the notes covered.

Carry on to the next subtopic.