Types of Number (CIE IGCSE Maths: Core)

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

What are multiples?

• A multiple is a number that is in the times table of another number
  • Every non-zero number has an infinite number of multiples, they go on forever
  • For example, the multiples of 3 are 3, 6, 9, 12, 15, 18 and so on

What are factors?

• A factor is a number that divides into another number exactly
  • For example, the factors of 18 are 1, 2, 3, 6, 9, and 18
    • We could say that 18 is divisible by 1, 2, 3, 6, 9, and 18
  • Every number has at least two factors; itself and 1
• Factors will always come in pairs, so when listing factors for a number, do so in pairs
  • This will help you not to miss any
  • e.g. the factors of 24 are
    • 1, 24
    • 2, 12
    • 3, 8
    • 4, 6

How can I test if a number is a factor?

• If you are unsure if a number is a factor, you can use your calculator to divide it and check if the result is an integer
  • e.g. If you want to test if 5 is a factor of 24, find 24÷5
  • 24÷5 = 4.8, which is not an integer
  • Therefore 5 is not a factor of 24
• Other than using your calculator, there are some common "divisibility tests" you can use
• A number is divisible by 2 if the last digit is even
  • 126 Yes
  • 135 No
• A number is divisible by 3 if the sum of the digits is divisible by 3
  • 123
    • 1 + 2 + 3 = 6, which is a multiple of 3, so 123 is a multiple of 3
  • 134
    • 1+3+4=8, which is not a multiple of 3, so 134 is not a multiple of 3
• A number is divisible by 4 if the last 2 digits are divisible by 4
  • 128 Yes (28÷4=7)
  • 134 No (34÷4=8.5)
  • Another way is to halve the number, twice, and see if the answer is an integer
    • 128÷2=64   64÷2=32 Yes
    • 134÷2=67   67÷2=33.5 No
• A number is divisible by 5 if the last digit is a 5 or 0
  • 165 Yes
  • 160 Yes
  • 162 No
• A number is divisible by 6 if it is divisible by both 2 and 3 (using the rules above)
• A number is divisible by 7 if you get a multiple of 7 when you double the last digit, and subtract it from the remaining part of the number
  • 245
    • Double 5 is 10
    • 24 - 10 = 14 which is a multiple of 7, so 245 is a multiple of 7
  • 469
    • Double 9 is 18
    • 46-18=28 which is a multiple of 7, so 469 is a multiple of 7
  • 906
    • Double 6 is 12
    • 90-12=78 which is not a multiple of 7, so 906 is not a multiple of 7
• A number is divisible by 8 if the last 3 digits are divisible by 8, or if it can be halved 3 times and the result is an integer (similar to the rule for 4)
• A number is divisible by 9 if the sum of the digits is divisible by 9 (similar to the rule for 3)
• A number is divisible by 10 if the last digit is a zero
• A number is divisible by 11 if you get an answer of 0 or a multiple of 11 when you alternately add and subtract the digits
  • 319
    • +3-1+9=11 Yes
  • 1364
    • +1-3+6-4=0 Yes
  • 428
    • +4-2+8=10 No
• A number is divisible by 12 if it is divisible by both 4 and 3 using the rules above

What are prime numbers?

• A prime number is a number which has exactly two factors; itself and 1
  • The first 10 prime numbers are
    • 2, 3, 5, 7, 11, 13, 17, 19, 23, 29
  • 1 is not a prime number, as it only has one factor
  • 2 is the only even prime number
  • You should remember at least the first ten prime numbers
• If a number has any factors other than itself and 1, it is not a prime number
  • 27 is often mistaken for a prime number, but it is divisible by 3 and 9 so it is not a prime number

What are squares, cubes and indices?

• A square number is the number derived from multiplying a number by itself
• For example, 3 × 3 = 9, so 9 is a square number
• a × a can be denoted a²
• You should remember at least the first twelve square numbers
• 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144
• A cube number is the number derived from multiplying a number by itself twice
• For example, 3 × 3 × 3 = 27, so 27 is a cube number
• a × a × a can be denoted a³
• You should remember at least the first five cube numbers
• 1, 8, 27, 64, 125
• An index (indices plural) is a way of writing a string of multiplications of the same number neatly
• They are often called powers, and sometimes exponents
• For example, 3 × 3 × 3 × 3 is the number 3 multiplied by itself 4 times and can be written 3⁴
• a × a × a × a × b × b × b × b × b can be written in index form as a⁴ × b⁵

What is a reciprocal?

• The reciprocal of a number is the result of dividing 1 by that number
  • Any number multiplied by its reciprocal will be equal to 1
• The reciprocal of 3 is 
  • The reciprocal of  is 3
• The reciprocal of  is 
  • The reciprocal of  is 
• The reciprocal of a is 
  • The reciprocal of   is a
• The reciprocal of a number, n, may also be written as n^-1