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Positive numbers: Real numbers that are greater than zero. Negative numbers: Real numbers that are less than zero. Because zero itself has no sign, neither the positive numbers nor the negative numbers include zero. When zero is a possibility, the following terms are often used: Non-negative numbers: Real numbers that are greater than or equal ...
In mathematics real is used as an adjective, meaning that the underlying field is the field of the real numbers (or the real field). For example, real matrix, real polynomial and real Lie algebra. The word is also used as a noun, meaning a real number (as in "the set of all reals").
In mathematics, a negative number is the opposite of a positive real number. [1] Equivalently, a negative number is a real number that is less than zero. Negative numbers are often used to represent the magnitude of a loss or deficiency. A debt that is owed may be thought of as a negative asset.
These are provable properties of rational numbers and positional number systems and are not used as definitions in mathematics. Irrational numbers can also be expressed as non-terminating continued fractions (which in some cases are periodic), and in many other ways. As a consequence of Cantor's proof that the real numbers are uncountable and ...
An imaginary number is the product of a real number and the imaginary unit i, [note 1] which is defined by its property i 2 = −1. [1] [2] The square of an imaginary number bi is −b 2. For example, 5i is an imaginary number, and its square is −25. The number zero is considered to be both real and imaginary. [3]
In computing, NaN (/ n æ n /), standing for Not a Number, is a particular value of a numeric data type (often a floating-point number) which is undefined as a number, such as the result of 0/0. Systematic use of NaNs was introduced by the IEEE 754 floating-point standard in 1985, along with the representation of other non-finite quantities ...
This allows using them in any area of mathematics, without having to recall their definition. For example, if one encounters in combinatorics, one should immediately know that this denotes the real numbers, although combinatorics does not study the real numbers (but it uses them for many proofs).
An axiomatic definition of the real numbers consists of defining them as the elements of a complete ordered field. [2] [3] [4] This means the following: The real numbers form a set, commonly denoted , containing two distinguished elements denoted 0 and 1, and on which are defined two binary operations and one binary relation; the operations are called addition and multiplication of real ...