Positive real numbers

In mathematics, the set of positive real numbers, formula_1, is the subset of those real numbers that are greater than zero. The non-negative real numbers, formula_2, also include zero. Although the symbols formula_3 and formula_4 are ambiguously used for either of these, the notation formula_3 or formula_4 for formula_7 and formula_8 or formula_9 for formula_10 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, formula_11 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 formula_12, with argument formula_13.
Properties.
The set formula_11 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 formula_15, the sequence formula_16 of its integral powers has three different fates: When formula_17, the limit is zero; when formula_18, the sequence is constant; and when formula_19, the sequence is unbounded.
formula_20 and the multiplicative inverse function exchanges the intervals. The functions floor, formula_21, and excess, formula_22, have been used to describe an element formula_23 as a continued fraction formula_24, which is a sequence of integers obtained from the floor function after the excess has been reciprocated. For rational formula_15, the sequence terminates with an exact fractional expression of formula_15, and for quadratic irrational formula_15, the sequence becomes a periodic continued fraction.
The ordered set (formula_11, >) forms a total order but is "not" a well-ordered set. The doubly infinite geometric progression 10n, where "n" is an integer, lies entirely in (formula_11, >) and serves to section it for access. formula_11 forms a ratio scale, the highest level of measurement. Elements may be written in scientific notation as "a" × 10n, where 1 ≤ "a" < 10 and "b" 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 formula_31, the determinant gives a map from matrices over the reals to the real numbers: formula_32 Restricting to invertible matrices gives a map from the general linear group to non-zero real numbers: formula_33. Restricting to matrices with a positive determinant gives the map formula_34; interpreting the image as a quotient group by the normal subgroup, relation expresses the positive reals as a Lie group.
Logarithmic measure.
If formula_35 is an interval, then formula_36 determines a measure on certain subsets of formula_11, 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 formula_38 by a formula_39, 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.
The non-negative reals serve as the range for metrics, norms, and measures in mathematics.
Including 0, the set formula_40 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.
Let formula_41 the first quadrant of the Cartesian plane. The quadrant itself is divided into four parts by the line formula_42 and the standard hyperbola formula_43
The "L" ∪ "H" forms a trident while "L" ∩ "H" = (1,1) is the central point. It is the identity element of two one-parameter groups that intersect there:
Since formula_46 is a group, "Q" is a direct product of groups. The one-parameter subgroups "L" and "H" in "Q" profile the activity in the product, and "L" × "H" 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.