Types of Number (Cambridge International IGCSE Mathematics)

Types of number

You will come across vocabulary such as

• Integers and natural numbers
• Rational and irrational numbers
• Multiples
• Factors
• Prime numbers
• Squares, cubes and roots
• Reciprocals

Knowing what each of these terms mean is essential.

What are integers and natural numbers?

• Integers are whole numbers;

  • They can be positive, negative and zero
  • For example, -3, -2, -1, 0, 1, 2, 3 are all integers

• Natural numbers are the positive integers

  • They can be thought of as counting numbers

  • 1, 2, 3, 4, … are the natural numbers

    • Notice that 0 is not included

What are rational and irrational numbers?

• A rational number is a number that can be written as a fraction in its simplest form

  • A rational number can be written in the form $\frac{a}{b}$

    • where $a$ and $b$* *are both integers, and$b$ is not zero

  • All terminating decimals are rational numbers

    • e.g. $0.32=\frac{32}{100}$

  • All recurring decimals are rational numbers

    • e.g. $0.4444...=\frac{4}{9}$
    • Recurring digits are indicated by dots or bars e.g. $0.4 \dot{5} =0.4 \bar{5} =0.455 555 555 ...$ e.g. $0. \dot{5} 82 \dot{4} =0.582 \bar{4} =0.5824 5824 5824 ...$

• An irrational number cannot be written as a fraction

  • All non-terminating, non-recurring decimals are irrational numbers

  • $π$ (pi) is an irrational number

    • It has an endless amount of decimal places and there is no pattern or recurring digits
    • Any (simplified) fraction involving $π$, such as$\frac{π}{2}$, will also be irrational (as $a$ in $\frac{a}{b}$ is not an integer)

Exam Hint

• In the calculator paper, use your calculator to your advantage!

  • To determine whether a number is rational or irrational, type it into your calculator and see if it can be displayed as a fraction (with integers)

Worked Example Explain why $0. \dot{3} 49 \dot{2}$ is a rational number.

Answer:

The dots indicate the first and last of the recurring digits

$0. \dot{3} 49 \dot{2} =0.3492 3492 3492 ...$

Final answer: $0 \dot{3} 49 \dot{2}$ is a rational number as it is a recurring decimal