Infinite Decimals (Part 1)

A. Gardiner
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A. Gardiner: University of Birmingham, Department of Mathematics

Chapter Chapter II.6 in Infinite Processes, 1982, pp 70-80 from Springer

Abstract: Abstract Those who introduce us to counting numbers usually go to great lengths both to invest these mathematical entities with ordinary meaning and to clarify the mathematical idea behind the usual base 10 representation of numbers (that is, the idea of place value corresponding to increasing powers of 10). Similar efforts are made when the time comes to introduce negative whole numbers, fractions, and the decimal representation of decimal fractions. But though the long division process, which we use to transform an ordinary fraction into a decimal fraction, frequently gives rise to infinite decimals, little if any time or effort is devoted either to investing these curious entities with ordinary meaning, or to clarifying the mathematical idea which justifies their representation as never ending decimals. Experience suggests that many undergraduates complete their studies of sequences, series, and limits in the calculus without ever realising the light they shed on infinite decimals. But since the very essence of the calculus lies in the careful use it makes of infinite processes to supplement the familiar processes of ordinary arithmetic, and since infinite decimals constitute the most familiar example of such infinite processes, it seems rather obvious that these should be the very first candidates for analysis: this is precisely the aim of the next few chapters.

Date: 1982
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DOI: 10.1007/978-1-4612-5654-0_8

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