Types of Number (Cambridge O Level Maths)

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
• You may see the symbol ℝ used to denote 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
• You may see the symbol ℤ used to denote 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
• You may see the symbol ℕ used to denote natural numbers

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 are factors, multiples and prime numbers?

• 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
• Every number has at least two factors, itself and 1
• 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
• A prime number is a number which has exactly two factors, itself and 1
• 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
• 2, 3, 5, 7, 11, 13, 17, 19, 23, 29

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 fifteen square numbers
• 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225
• 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
• You should remember that 10³ = 1000
• 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 one
• The reciprocal of  is 
• The reciprocal of   is 
• The reciprocal of a number, n, may also be written as n^-1_.

What is an irrational number?

• An irrational number is a number that cannot be written in the form , where and  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
  
  • most calculators will often leave irrational numbers as a surd

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 negative numbers?

• Negative numbers are any number less than zero
• They appear in lots of places from numerical calculations to algebra 
• You might come across them in real-life problems such as temperature or debt

What are the rules for working with negative numbers?

• When multiplying and dividing with negative numbers
  • Two numbers with the same sign makes a positive
    • e.g and 
  • Two numbers with different signs makes a negative
    • e.g.  and
  • For multiplication and division, it's often easier to calculate ignoring any signs, then making a decision about whether the answer should be positive or negative
• When adding and subtracting with negative numbers
  • Subtracting a negative is the same as adding the positive number
    • e.g.
  • Adding a negative is the same as subtracting the positive number
    • e.g.