Positive real numbers

In mathematics, the set of positive real numbers, R > 0 = { x ∈ R ∣ x > 0 } {displaystyle mathbb {R} _{>0}=left{xin mathbb {R} mid x>0 ight}} , is the subset of those real numbers that are greater than zero. The non-negative real numbers, R ≥ 0 = { x ∈ R ∣ x ≥ 0 } {displaystyle mathbb {R} _{geq 0}=left{xin mathbb {R} mid xgeq 0 ight}} , also include zero. The symbols R + {displaystyle mathbb {R} _{+}} and R + {displaystyle mathbb {R} ^{+}} are ambiguously used for either of these, so it safer to always specify which. In mathematics, the set of positive real numbers, R > 0 = { x ∈ R ∣ x > 0 } {displaystyle mathbb {R} _{>0}=left{xin mathbb {R} mid x>0 ight}} , is the subset of those real numbers that are greater than zero. The non-negative real numbers, R ≥ 0 = { x ∈ R ∣ x ≥ 0 } {displaystyle mathbb {R} _{geq 0}=left{xin mathbb {R} mid xgeq 0 ight}} , also include zero. The symbols R + {displaystyle mathbb {R} _{+}} and R + {displaystyle mathbb {R} ^{+}} are ambiguously used for either of these, so it safer to always specify which. In a complex plane, R > 0 {displaystyle mathbb {R} _{>0}} 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 z = | z | e i φ {displaystyle z=|z|mathrm {e} ^{mathrm {i} varphi }} with argument φ = 0 {displaystyle varphi =0} . The set R > 0 {displaystyle mathbb {R} _{>0}} 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 x, the sequence {xn} of its integral powers has three different fates: When x ∈ (0, 1) the limit is zero and when x ∈ (1, ∞) the limit is infinity, while the sequence is constant for x = 1. The case x > 1 thus leads to an unbounded sequence. R > 0 = ( 0 , 1 ) ∪ { 1 } ∪ ( 1 , ∞ ) {displaystyle mathbb {R} _{>0}=(0,1)cup {1}cup (1,infty )} and the multiplicative inverse function exchanges the intervals. The functions floor, floor : [ 1 , ∞ ) → N , x ↦ ⌊ x ⌋ {displaystyle operatorname {floor} :[1,infty ) o mathbb {N} ,,xmapsto lfloor x floor } , and excess, excess : [ 1 , ∞ ) → ( 0 , 1 ) , x ↦ x − ⌊ x ⌋ {displaystyle operatorname {excess} :[1,infty ) o (0,1),,xmapsto x-lfloor x floor } , have been used to describe an element x ∈ R > 0 {displaystyle xin mathbb {R} _{>0}} as a continued fraction [ n 0 ; n 1 , n 2 , … ] {displaystyle } which is a sequence of integers obtained from the floor function after the excess has been reciprocated. For x ∈ Q {displaystyle xin mathbb {Q} } the sequence terminates with an exact fractional expression of x {displaystyle x} , and for quadratic irrational x {displaystyle x} the sequence becomes a periodic continued fraction. In the study of classical groups, for every n ∈ N {displaystyle nin mathbb {N} } , the determinant gives a map from n × n {displaystyle n imes n} matrices over the reals to the real numbers M ( n , R ) → R . {displaystyle mathrm {M} (n,mathbf {R} ) o mathbf {R} .} Restricting to invertible matrices gives a map from the general linear group to non-zero real numbers: G L ( n , R ) → R × {displaystyle mathrm {GL} (n,mathbf {R} ) o mathbf {R} ^{ imes }} . Restricting to matrices with a positive determinant gives the map G L + ( n , R ) → R > 0 {displaystyle mathrm {GL} ^{+}(n,mathbf {R} ) o mathbf {R} _{>0}} ; interpreting the image as a quotient group by the normal subgroup relation SL(n,ℝ) ◁ GL+(n,ℝ) expresses the positive reals as a Lie group. If [ a , b ] ⊆ R > 0 {displaystyle subseteq mathbb {R} _{>0}} is an interval, then μ ( [ a , b ] ) = log ⁡ ( b / a ) = log ⁡ b − log ⁡ a {displaystyle mu ()=log(b/a)=log b-log a} determines a measure on certain subsets of R > 0 {displaystyle mathbb {R} _{>0}} , 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 [ a , b ] → [ a z , b z ] {displaystyle o } by a z ∈ R > 0 {displaystyle zin mathbb {R} _{>0}} , 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. The non-negative reals serve as the range for metrics, norms, and measures in mathematics.