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Published in:
2018 | OriginalPaper | Chapter
1. Numbers
Authors : Michael Oberguggenberger, Alexander Ostermann
Published in: Analysis for Computer Scientists
Publisher: Springer International Publishing
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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.