Movatterモバイル変換

[0]ホーム
URL:

Jump to content
WikipediaThe Free Encyclopedia
Search

Positive real numbers

From Wikipedia, the free encyclopedia
Subset of real numbers that are greater than zero

Inmathematics, theset ofpositive real numbers,R>0={x∈R∣x>0},{\displaystyle \mathbb {R} _{>0}=\left\{x\in \mathbb {R} \mid x>0\right\},} is the subset of thosereal numbers that are greater than zero. Thenon-negative real numbers,R≥0={x∈R∣x≥0},{\displaystyle \mathbb {R} _{\geq 0}=\left\{x\in \mathbb {R} \mid x\geq 0\right\},} also include zero. Although the symbolsR+{\displaystyle \mathbb {R} _{+}} andR+{\displaystyle \mathbb {R} ^{+}} are ambiguously used for either of these, the notationR+{\displaystyle \mathbb {R} _{+}} orR+{\displaystyle \mathbb {R} ^{+}} for{x∈R∣x≥0}{\displaystyle \left\{x\in \mathbb {R} \mid x\geq 0\right\}} andR+∗{\displaystyle \mathbb {R} _{+}^{*}} orR∗+{\displaystyle \mathbb {R} _{*}^{+}} for{x∈R∣x>0}{\displaystyle \left\{x\in \mathbb {R} \mid x>0\right\}} 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 acomplex plane,R>0{\displaystyle \mathbb {R} _{>0}} is identified with thepositive real axis, and is usually drawn as a horizontalray. This ray is used as reference in thepolar form of a complex number. The real positive axis corresponds tocomplex numbersz=|z|eiφ,{\displaystyle z=|z|\mathrm {e} ^{\mathrm {i} \varphi },} withargumentφ=0.{\displaystyle \varphi =0.}

Properties

[edit]

The setR>0{\displaystyle \mathbb {R} _{>0}} isclosed under addition, multiplication, and division. It inherits atopology from thereal line and, thus, has the structure of a multiplicativetopological group or of an additivetopological semigroup.

For a given positive real numberx,{\displaystyle x,} thesequence{xn}{\displaystyle \left\{x^{n}\right\}} of its integral powers has three different fates: Whenx∈(0,1),{\displaystyle x\in (0,1),} thelimit is zero; whenx=1,{\displaystyle x=1,} the sequence is constant; and whenx>1,{\displaystyle x>1,} the sequence isunbounded.

R>0=(0,1)∪{1}∪(1,∞){\displaystyle \mathbb {R} _{>0}=(0,1)\cup \{1\}\cup (1,\infty )} and themultiplicative inverse function exchanges the intervals. The functionsfloor,floor:[1,∞)→N,x↦⌊x⌋,{\displaystyle \operatorname {floor} :[1,\infty )\to \mathbb {N} ,\,x\mapsto \lfloor x\rfloor ,} andexcess,excess:[1,∞)→(0,1),x↦x−⌊x⌋,{\displaystyle \operatorname {excess} :[1,\infty )\to (0,1),\,x\mapsto x-\lfloor x\rfloor ,} have been used to describe an elementx∈R>0{\displaystyle x\in \mathbb {R} _{>0}} as acontinued fraction[n0;n1,n2,…],{\displaystyle \left[n_{0};n_{1},n_{2},\ldots \right],} which is a sequence of integers obtained from the floor function after the excess has been reciprocated. For rationalx,{\displaystyle x,} the sequence terminates with an exact fractional expression ofx,{\displaystyle x,} and forquadratic irrationalx,{\displaystyle x,} the sequence becomes aperiodic continued fraction.

The ordered set(R>0,>){\displaystyle \left(\mathbb {R} _{>0},>\right)} forms atotal order but isnot awell-ordered set. Thedoubly infinitegeometric progression10n,{\displaystyle 10^{n},} wheren{\displaystyle n} is aninteger, lies entirely in(R>0,>){\displaystyle \left(\mathbb {R} _{>0},>\right)} and serves to section it for access.R>0{\displaystyle \mathbb {R} _{>0}} forms aratio scale, the highestlevel of measurement. Elements may be written inscientific notation asa×10b,{\displaystyle a\times 10^{b},} where1≤a<10{\displaystyle 1\leq a<10} andb{\displaystyle b} is the integer in the doubly infinite progression, and is called thedecade. 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 ofclassical groups, for everyn∈N,{\displaystyle n\in \mathbb {N} ,} thedeterminant gives a map fromn×n{\displaystyle n\times n} matrices over the reals to the real numbers:M(n,R)→R.{\displaystyle \mathrm {M} (n,\mathbb {R} )\to \mathbb {R} .} Restricting to invertible matrices gives a map from thegeneral linear group to non-zero real numbers:GL(n,R)→R×.{\displaystyle \mathrm {GL} (n,\mathbb {R} )\to \mathbb {R} ^{\times }.} Restricting to matrices with a positive determinant gives the mapGL+⁡(n,R)→R>0{\displaystyle \operatorname {GL} ^{+}(n,\mathbb {R} )\to \mathbb {R} _{>0}}; interpreting the image as aquotient group by thenormal subgroupSL⁡(n,R)◃GL+⁡(n,R),{\displaystyle \operatorname {SL} (n,\mathbb {R} )\triangleleft \operatorname {GL} ^{+}(n,\mathbb {R} ),} called thespecial linear group, expresses the positive reals as aLie group.

Ratio scale

[edit]

Among thelevels of measurement, theratio scale provides the finest detail. Thedivision function takes a value of one whennumerator anddenominator are equal. Other ratios are compared to one by logarithms, oftencommon logarithm using base 10. The ratio scale then segments byorders of magnitude used in science and technology, expressed in variousunits of measurement.

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

Logarithmic measure

[edit]

If[a,b]⊆R>0{\displaystyle [a,b]\subseteq \mathbb {R} _{>0}} is aninterval, thenμ([a,b])=log⁡(b/a)=log⁡b−log⁡a{\displaystyle \mu ([a,b])=\log(b/a)=\log b-\log a} determines ameasure on certain subsets ofR>0,{\displaystyle \mathbb {R} _{>0},} corresponding to thepullback of the usualLebesgue measure on the real numbers under the logarithm: it is the length on thelogarithmic scale. In fact, it is aninvariant measure with respect to multiplication[a,b]→[az,bz]{\displaystyle [a,b]\to [az,bz]} by az∈R>0,{\displaystyle z\in \mathbb {R} _{>0},} just as the Lebesgue measure is invariant under addition. In the context of topological groups, this measure is an example of aHaar measure.

The utility of this measure is shown in its use for describingstellar magnitudes and noise levels indecibels, among other applications of thelogarithmic scale. For purposes of international standardsISO 80000-3, the dimensionless quantities are referred to aslevels.

Applications

[edit]

The non-negative reals serve as theimage formetrics,norms, andmeasures in mathematics.

Including 0, the setR≥0{\displaystyle \mathbb {R} _{\geq 0}} has asemiring structure (0 being theadditive identity), known as theprobability semiring; taking logarithms (with a choice of base giving alogarithmic unit) gives anisomorphism with thelog semiring (with 0 corresponding to−∞{\displaystyle -\infty }), and its units (the finite numbers, excluding−∞{\displaystyle -\infty }) correspond to the positive real numbers.

Square

[edit]

LetQ=R>0×R>0,{\displaystyle Q=\mathbb {R} _{>0}\times \mathbb {R} _{>0},} the first quadrant of the Cartesian plane. The quadrant itself is divided into four parts by the lineL={(x,y):x=y}{\displaystyle L=\{(x,y):x=y\}} and the standard hyperbolaH={(x,y):xy=1}.{\displaystyle H=\{(x,y):xy=1\}.}

TheL∪H{\displaystyle L\cup H} forms a trident whileL∩H=(1,1){\displaystyle L\cap H=(1,1)} is the central point. It is the identity element of twoone-parameter groups that intersect there:{{(ea, ea):a∈R},×} on L and {{(ea, e−a):a∈R},×} on H.{\displaystyle \{\left\{\left(e^{a},\ e^{a}\right):a\in R\right\},\times \}{\text{ on }}L\quad {\text{ and }}\quad \{\left\{\left(e^{a},\ e^{-a}\right):a\in R\right\},\times \}{\text{ on }}H.}

SinceR>0{\displaystyle \mathbb {R} _{>0}} is agroup,Q{\displaystyle Q} is adirect product of groups. The one-parameter subgroupsL andH inQ profile the activity in the product, andL×H{\displaystyle L\times 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 tohyperbolic coordinates inQ. Motion against theL axis indicates a change in thegeometric meanxy,{\displaystyle {\sqrt {xy}},} while a change alongH indicates a newhyperbolic angle.

See also

[edit]

References

[edit]
  1. ^"positive number in nLab".ncatlab.org. Retrieved2020-08-11.
  2. ^E. J. Dijksterhuis (1961)Mechanization of the World-Picture, page 51, viaInternet Archive

Bibliography

[edit]
Basic concepts
Sets
Types ofmeasures
Particular measures
Maps
Main results
Other results
ForLebesgue measure
Applications & related
Authority control databases: NationalEdit this at Wikidata
Retrieved from "https://en.wikipedia.org/w/index.php?title=Positive_real_numbers&oldid=1283043424"
Categories:
Hidden categories:
[8]ページ先頭
©2009-2025 Movatter.jp