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

Dition 

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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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