the groups of real numbers

Proposition 1.

The additive groupMathworldPlanetmath of real number R,+ is isomorphicPlanetmathPlanetmathPlanetmath to the multiplicative group of positive real numbers R+,×.


Let f(x)=ex. This maps the group ,+ to the group +,×. As f has an inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath f-1(x)=lnx we observe f is invertible. Furthermore, f(x+y)=ex+y=exey=f(x)f(y) so f is a homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath. Thus f is an isomorphism. ∎

Corollary 2.

The multiplicative group of non-zero real number R× is isomorphic to Z2R,+.


Use the map f:2,+× defined by f(s,r)=(-1)ser.11We write (-1)s to mean (-1)s for any integer s representative of the equivalence classMathworldPlanetmathPlanetmath of s in 2. Then


so that f is a homomorphism. Furthermore, f-1(r)=(signr,ln|r|) is the inverse of f so that f is bijectiveMathworldPlanetmath and thus an isomorphism of groups. ∎

Title the groups of real numbers
Canonical name TheGroupsOfRealNumbers
Date of creation 2013-03-22 16:08:50
Last modified on 2013-03-22 16:08:50
Owner Algeboy (12884)
Last modified by Algeboy (12884)
Numerical id 5
Author Algeboy (12884)
Entry type Result
Classification msc 54C30
Classification msc 26-00
Classification msc 12D99
Related topic ExponentialFunction
Related topic GroupHomomorphism
Related topic GroupsInField