At GCSE level you will come across vocabulary such as real numbers, integers, natural numbers, indices, factors, multiples, prime, square and cube 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 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 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