Types of Number

At GCSE level you will come across vocabulary such as real numbers, integers, natural numbers, indices, factors, multiples, prime, square and cube numbers, reciprocals, rational and irrational numbers. Knowing what all of this means is essential.

What are real numbers, integers and natural numbers?

• Real numbers are the set of all numbers, including integers, fractions, rational and irrational numbers
• All numbers dealt with at GCSE level are considered real numbers
• Integers are all whole numbers, they can be positive, negative or zero
• For example, …, -3, -2, -1, 0, 1, 2, 3, … are all integers
• Natural numbers are the set of all positive integers
• They are sometimes thought of as the counting numbers
• For example, 1, 2, 3, … are the natural numbers

What are even and odd numbers?

• If a number can be divided by 2 and the answer is an integer (a whole number), then it is said to be even
  • 2, 4, 6, 8, 10, 12, 14, ... are all even numbers
  • -2, -4, -6, -8, -10, -12, -14, ... are also all even numbers
  • Even numbers are multiples of 2
• If this is not true, then the number is odd
  • 1, 3, 5, 7, 9, 11, 13, .... are all odd numbers
  • -1, -3, -5, -7, -9, -11, -13, ... are also all odd numbers
  • Odd numbers are not multiples of 2

What is a rational number?

• A rational number is a number that can be written as a fraction in its simplest form
• It must be possible to write in the form , where a and b are both whole numbers
• This includes all terminating and recurring decimals

What is an irrational number?

• An irrational number is a number that cannot be written in the form , where a and b are whole numbers (or integers)
• Any non-terminating and non-recurring decimal is an irrational number
• The number √n, where n is not a square number, is an irrational number
• The square root of a non-square integer is also called a surd
• you may come across this term
• most calculators will often leave irrational numbers as a surd
•  is rational, as it is equal to 8
•  however is irrational

What irrational numbers should I know?

• You may be asked to identify an irrational number from a list
• Irrational numbers that you should know are π, ,  
• Any multiple of these is also irrational
• For example  are irrational
• Most modern calculators will show irrational numbers in their exact form rather than as a decimal where possible
• These means as either a multiple of π or √n, where n is not a square number
• If the calculator cannot show the exact form, it will show the number rounded to 9 or 10 decimal places