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.[1]

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

Properties

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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 not 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.

In the study of classical groups, for every   the determinant gives a map from   matrices over the reals to the real numbers:   Restricting to invertible matrices gives a map from the general linear group to non-zero real numbers:   Restricting to matrices with a positive determinant gives the map  ; interpreting the image as a quotient group by the normal subgroup   called the special linear group, expresses the positive reals as a Lie group.

Ratio scale

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Among the levels of measurement the ratio scale provides the finest detail. The division function takes a value of one when numerator and denominator are equal. Other ratios are compared to one by logarithms, often common logarithm using base 10. The ratio scale then segments by orders of magnitude used in science and technology, expressed in various units of measurement.

An early expression of ratio scale was articulated geometrically by Eudoxus: "it was ... in geometrical language that the general theory of proportion of Eudoxus was developed, which is equivalent to a theory of positive real numbers."[2]

Logarithmic measure

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If   is an interval, then   determines a measure on certain subsets of   corresponding to the pullback of the usual Lebesgue measure on the real numbers under the logarithm: it is the length on the logarithmic scale. In fact, it is an invariant measure with respect to multiplication   by a   just as the Lebesgue measure is invariant under addition. In the context of topological groups, this measure is an example of a Haar measure.

The utility of this measure is shown in its use for describing stellar magnitudes and noise levels in decibels, among other applications of the logarithmic scale. For purposes of international standards ISO 80000-3, the dimensionless quantities are referred to as levels.

Applications

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The non-negative reals serve as the image for metrics, norms, and measures in mathematics.

Including 0, the set   has a semiring structure (0 being the additive identity), known as the probability semiring; taking logarithms (with a choice of base giving a logarithmic unit) gives an isomorphism with the log semiring (with 0 corresponding to  ), and its units (the finite numbers, excluding  ) correspond to the positive real numbers.

Square

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Let   the first quadrant of the Cartesian plane. The quadrant itself is divided into four parts by the line   and the standard hyperbola  

The   forms a trident while   is the central point. It is the identity element of two one-parameter groups that intersect there:

 

Since   is a group,   is a direct product of groups. The one-parameter subgroups L and H in Q profile the activity in the product, and   is a resolution of the types of group action.

The realms of business and science abound in ratios, and any change in ratios draws attention. The study refers to hyperbolic coordinates in Q. Motion against the L axis indicates a change in the geometric mean   while a change along H indicates a new hyperbolic angle.

See also

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References

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  1. ^ "positive number in nLab". ncatlab.org. Retrieved 2020-08-11.
  2. ^ E. J. Dijksterhuis (1961) Mechanization of the World-Picture, page 51, via Internet Archive

Bibliography

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