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Concept
Positive real numbers
Summary
In mathematics, the set of positive real numbers, is the subset of those real numbers that are greater than zero. The non-negative real numbers, also include zero. Although the symbols and are ambiguously used for either of these, the notation or for and or for has also been widely employed, is aligned with the practice in algebra of denoting the exclusion of the zero element with a star, and should be understandable to most practicing mathematicians.
In a complex plane, is identified with the positive real axis, and is usually drawn as a horizontal ray. This ray is used as reference in the polar form of a complex number. The real positive axis corresponds to complex numbers with argument
The set is closed under addition, multiplication, and division. It inherits a topology from the real line and, thus, has the structure of a multiplicative topological group or of an additive topological semigroup.
For a given positive real number the sequence of its integral powers has three different fates: When the limit is zero; when the sequence is constant; and when the sequence is unbounded.
and the multiplicative inverse function exchanges the intervals. The functions floor, and excess, have been used to describe an element as a continued fraction which is a sequence of integers obtained from the floor function after the excess has been reciprocated. For rational the sequence terminates with an exact fractional expression of and for quadratic irrational the sequence becomes a periodic continued fraction.
The ordered set forms a total order but is a well-ordered set. The doubly infinite geometric progression where is an integer, lies entirely in and serves to section it for access. forms a ratio scale, the highest level of measurement. Elements may be written in scientific notation as where and is the integer in the doubly infinite progression, and is called the decade. In the study of physical magnitudes, the order of decades provides positive and negative ordinals referring to an ordinal scale implicit in the ratio scale.
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Ontological neighbourhood
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In mathematics, the classical groups are defined as the special linear groups over the reals R, the complex numbers C and the quaternions H together with special automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or skew-Hermitian sesquilinear forms defined on real, complex and quaternionic finite-dimensional vector spaces. Of these, the complex classical Lie groups are four infinite families of Lie groups that together with the exceptional groups exhaust the classification of simple Lie groups.
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In mathematics, specifically in group theory, the direct product is an operation that takes two groups G and H and constructs a new group, usually denoted G × H. This operation is the group-theoretic analogue of the Cartesian product of sets and is one of several important notions of direct product in mathematics. In the context of abelian groups, the direct product is sometimes referred to as the direct sum, and is denoted .
Log semiring
In mathematics, in the field of tropical analysis, the log semiring is the semiring structure on the logarithmic scale, obtained by considering the extended real numbers as logarithms. That is, the operations of addition and multiplication are defined by conjugation: exponentiate the real numbers, obtaining a positive (or zero) number, add or multiply these numbers with the ordinary algebraic operations on real numbers, and then take the logarithm to reverse the initial exponentiation. Such operations are also known as, e.