Number Systems

• Each number system can be defined as a set. A set is an unordered collection of distinct elements (each element occurs only once).
• The set of all natural numbers is represented by the mathematical symbol ℕ.

  ℕ is an infinite set:

  ℕ ={0,1,2,3,...}

  Natural numbers can be defined as counting numbers.
• The set of all integers is represented by the mathematical symbol ℤ.

  ℤ is an infinite set of whole numbers:

  ℤ = {...,-2,-1,0,1,2,...}

  It includes negative numbers, positive numbers, and 0.
• The set of all real numbers is represented by the mathematical symbol ℝ.

  A real number is any positive or negative number, or 0. The set includes numbers with a fractional part (rational numbers) and numbers defined by infinite decimal expansions (irrational numbers).

  The set of real numbers consists of all points on a number line.
• The set of all rational numbers is represented by the mathematical symbol ℚ.

  A rational number can be expressed as the ratio between two integers. This ratio can be represented as a fraction, e.g. 1/2, with a numerator at the top and a denominator at the bottom, or as a decimal number, e.g. 0.5.

  The numerator and denominator of a rational number can be any positive or negative integer.
• Irrational numbers don't have a special symbol. They can be defined as ℝ - ℚ, which is the set of all real numbers minus the set of all rational numbers. An irrational number can be calculated to an infinite number of decimal places, without ever slipping into a repeating pattern, so cannot be accurately represented as a fraction. 
• An ordinal number is a natural number that describes the numerical position of a value, e.g. first, second. Ordinal numbers are used for ordering.