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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").
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 ...
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 ...
For example, the imaginary number is undefined within the set of real numbers. So it is meaningless to reason about the value, solely within the discourse of real numbers. However, defining the imaginary number i {\displaystyle i} to be equal to − 1 {\displaystyle {\sqrt {-1}}} , allows there to be a consistent set of mathematics referred to ...
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).
The real numbers have various lattice-theoretic properties that are absent in the complex numbers. Also, the real numbers form an ordered field, in which sums and products of positive numbers are also positive. Moreover, the ordering of the real numbers is total, and the real numbers have the least upper bound property:
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 ...
Example: Let a and b be nonzero real numbers. Then the subgroup of the real numbers R generated by a is commensurable with the subgroup generated by b if and only if the real numbers a and b are commensurable, in the sense that a/b is rational. Thus the group-theoretic notion of commensurability generalizes the concept for real numbers.