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Structure

  • Geoff C. Smith
  • Olga M. Tabachnikova
Part of the Springer Undergraduate Mathematics Series book series (SUMS)

Abstract

Consider the famous map from the real numbers to the positive real numbers exp: ℝ + → ℝ defined by rer where e has suddenly (but temporarily) become the base of natural logarithms. The reals comprise a group under addition and the positive reals ℝ+ form a group under multiplication. Our map, is bijective, and its inverse is in: ℝ+→ℝ .These maps respect the group structures in the following sense:
$$\exp \left( {x + y} \right) = \exp \left( x \right)\exp \left( y \right)\forall x,y \in \mathbb{R}$$
and
$$\ln \left( {ab} \right) = \ln \left( a \right) + \ln \left( b \right)\forall a,b \in {{\mathbb{R}}^{ + }}$$

Keywords

Abelian Group Normal Subgroup Conjugacy Class Cyclic Group Semidirect Product 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag London 2000

Authors and Affiliations

  • Geoff C. Smith
    • 1
  • Olga M. Tabachnikova
    • 1
  1. 1.School of Mathematical SciencesUniversity of BathBathUK

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