A number is a mathematical object used in one of three ways:
- Cardinal - A cardinal number refers to a quantity or the size of a group.
- Ordinal - An ordinal number refers to a position in a series or a ranking.
- Nominal - A nominal number is used for identification only, such as an identifying label, and the numerical value is irrelevant.
Common sets of numbers (or number systems):
- N - The Natural or counting numbers (1, 2, 3 ...) are familiar to everyone. The result of addition or multiplication of elements of this set produces a value within the set; the set is closed to those operations. It is not closed to subtraction or division.
- Z - Integers include zero and negative numbers (Z from the German word Zahl, meaning "number"). This set is closed to addition, multiplication and subtraction but not division.
- Q - Rational numbers, from the root word ratio, include all numbers which can be expressed as a quotient or fraction. This set is closed to the four elementary arithmetic operations, making this set a field.
- R - Real numbers include irrational numbers, those that cannot be written as a ratio of two integers such as
√ 2
or π.
- C - Complex Numbers are composed of a real and imaginary part of the form a + bi, where a and b are real numbers and i is the imaginary unit, i.e., i =
√ -1
or i2 = -1.
Each set above is a proper subset of the next set: N⊂Z⊂Q⊂R⊂C. The natural numbers, integers and rational numbers are countable sets (or denumerable), whereas the sets containing the real numbers are uncountable (or nondenumerable).
Other number sets are:
- A - Algebraic numbers are any number that is a solution to a polynomial equation with rational coefficients. Algebraic numbers include all rational numbers and some irrational numbers. Numbers that are not algebraic are called transcendental numbers.
- I - Imaginary numbers are numbers that when squared give a negative result; they are a subset of the complex numbers: I⊂C.
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Last updated Tuesday September 22nd 2015