Number Toolkit Notes

Cambridge IGCSE Maths: Revision notes

Key facts

  • Know the number types: natural, integer, prime, square, cube, rational, irrational, reciprocal.
  • Use BIDMAS for every multi-step calculation.
  • Two negatives multiplied or divided give a positive; one of each gives a negative.
  • Recall squares 1² to 15² and cubes of 1, 2, 3, 4, 5 and 10 without a calculator.
  • Convert between numbers in words and figures by tracking place value.

Types of number

Each type of number has a precise definition that you must be able to recognise.

Natural numbers are the counting numbers 1, 2, 3, ... (Cambridge's set N\mathbb{N} also includes 0). Integers include zero and negatives. A prime has exactly two factors, 1 and itself. Rational numbers can be written as ab\dfrac{a}{b}; irrational numbers such as π\pi and 2\sqrt{2} cannot. The reciprocal of 5 is 15\dfrac{1}{5}.

−3−2.5−2−1.5−1−0.500.511.522.533.54−2 (integer)1/2 (rational)√2 (irrational)π (irrational)

Examples

Square numbers:
1, 4, 9, 16 (square)
Cube numbers:
1, 8, 27, 64 (cube)
Prime numbers:
2, 3, 5, 7, 11 (prime)

Definition

Square numbers:
An integer multiplied by itself
Cube numbers:
An integer multiplied by itself three times
Prime numbers:
Greater than 1 with exactly two factors

Which number is irrational?

Words and figures

Work in groups of three digits, using zeros as placeholders for any group with no value.

Each group of three digits from the right is thousand, million, billion in turn. When converting words to figures, write out each group carefully and use zeros as placeholders.

  1. 1

    Split into groups

    3 million | 40 thousand | 7

  2. 2

    Write each group in three digits

    3 | 040 | 007

  3. 3

    Use zeros as placeholders

    every group after the first has three digits

  4. 4

    Join them

    3 040 007

Converting words to figures for three million, forty thousand and seven.

In words

  • Six billion
  • Four hundred and five thousand
  • Three million, forty thousand and seven

In figures

  • 6 000 000 000
  • 405 000
  • 3 040 007

Write "two million and five" in figures.

The four operations

Follow BIDMAS and the sign rules, and apply them to fractions, decimals and real contexts.

Adding a negative is the same as subtracting; subtracting a negative is the same as adding. For example, 5−(−3)=5+3=85 - (-3) = 5 + 3 = 8.

  1. 1

    Brackets

    work out brackets first

  2. 2

    Indices

    powers and roots

  3. 3

    Division and multiplication

    left to right

  4. 4

    Addition and subtraction

    left to right

Order of operations: BIDMAS

Same signs

  • Multiply or divide to get a positive

Different signs

  • Multiply or divide to get a negative

Worked example

Work out 3+4×23 + 4 \times 2.

Work out −3×−4-3 \times -4.

Squares, cubes and roots

Recall squares up to 15² and cubes of 1, 2, 3, 4, 5 and 10, with their roots, without a calculator.

Recall square numbers from 121^2 to 15215^2 (1 to 225) and their square roots, and the cube numbers 1, 8, 27, 64, 125 and 1000 and their cube roots. Beyond these, use a calculator where permitted. Memorising them saves time on the non-calculator paper.

050100150200123456789101112131415nn squared
The square numbers from 1² to 15²

What is 1253\sqrt[3]{125}?

Numbers in context

A rise means adding and a fall means subtracting; check that your answer makes sense.

Real-life problems often disguise integer and fraction operations. For temperatures treat a rise as adding and a fall as subtracting. Convert mixed numbers and improper fractions to one form before calculating. Re-read the context at the end to check the sign and size.

−8−7−6−5−4−3−2−10123456−6°C4°C

Worked example

The temperature was −6°C and rises by 10°C. Find the new temperature.

The temperature is 3°C and falls by 8°C. What is it now?

Try an exam question

(a) Write down the value of 169\sqrt{169}. (b) Work out 5−3×(−4)5 - 3 \times (-4).

[3 marks]

That's the notes covered.

Carry on to the next subtopic.