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Analysis for Computer Scientists pp 1–11Cite as

Numbers

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Part of the book series: Undergraduate Topics in Computer Science ((UTICS))

Abstract

The commonly known rational numbers (fractions) are not sufficient for a rigorous foundation of mathematical analysis. The historical development shows that for issues concerning analysis, the rational numbers have to be extended to the real numbers. For clarity we introduce the real numbers as decimal numbers with an infinite number of decimal places. We illustrate exemplarily how the rules of calculation and the order relation extend from the rational to the real numbers in a natural way.

A further section is dedicated to floating point numbers, which are implemented in most programming languages as approximations to the real numbers. In particular, we will discuss optimal rounding and in connection with this the relative machine accuracy.

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Notes

  1. 1.

    We will rarely use the term mapping in such generality. The special case of real-valued functions, which is important for us, will be discussed thoroughly in Chap. 2.

  2. 2.

    G. Cantor, 1845–1918.

References

Textbooks

  1. E. Hairer, G. Wanner: Analysis by Its History. Springer, New York 1996.

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Further Reading

  1. M.L. Overton: Numerical Computing with IEEE Floating Point Arithmetic. SIAM, Philadelphia 2001.

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Author information

Authors and Affiliations

  1. Institute of Basic Sciences in Civil Eng, University of Innsbruck, Technikerstrasse 13, Innsbruck, 6020, Austria

    Michael Oberguggenberger

  2. Department of Mathematics, University of Innsbruck, Technikerstrasse 13/7, Innsbruck, 6020, Austria

    Alexander Ostermann

Authors
  1. Michael Oberguggenberger
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  2. Alexander Ostermann
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Corresponding author

Correspondence to Michael Oberguggenberger .

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© 2011 Springer-Verlag London Limited

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Oberguggenberger, M., Ostermann, A. (2011). Numbers. In: Analysis for Computer Scientists. Undergraduate Topics in Computer Science. Springer, London. https://doi.org/10.1007/978-0-85729-446-3_1

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  • DOI: https://doi.org/10.1007/978-0-85729-446-3_1

  • Publisher Name: Springer, London

  • Print ISBN: 978-0-85729-445-6

  • Online ISBN: 978-0-85729-446-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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