Introduction to Mathematical Reasoning (9781139633581) by Eccles Peter J

Author:Eccles, Peter J.
Language: eng
Format: epub
Publisher: Cambridge Univ Pr
Published: 2013-01-10T00:00:00+00:00

Subtracting we obtain

999x = 12793.17 - 12.79 = 12780.38 = 1278038/100

and so x = 1278038/99900, a rational number as required.

We will see later on that the converse of this theorem is also true (see Problems IV, Question 5). This means that the rational numbers correspond precisely to the recurring decimals (a finite decimal can be thought of as a recurring infinite decimal:

Exercises

13.1 Prove that there does not exist a rational number whose square is 3.

[You may assume that a2 is divisible by 3 if and only if a is divisible by 3 (proved later as Proposition 15.2.1).]

13.2 Find the first two decimal places of an infinite decimal representing

13.3 Try to mimic the proof of Theorem 13.2.1 to show that there does not exist a rational number whose square is 4. Where does this go wrong?

13.4 Prove from Definition 13.3.2 that

13.5 Find the rational number equal to the recurring infinite decimal

† To be precise, the Greeks did not regard 1 as a number. Euclid (Book Seven of the Elements) refers to 1 as a unit and then defines a number to be a multitude composed of units.

‡ See, for example, C.B. Boyer and U.C. Merzbach, A history of mathematics, Wiley, Second edition 1989.

† See for example R. Haggerty, Fundamentals of mathematical analysis, Addison-Wesley, Second edition 1993.

† Alternatively, we place a dot over each digit which recurs, or a line over the repeating block,

† An excellent discussion of the arithmetic of infinite decimals, a topic bypassed in most treatments of infinite decimals, may be found in A. Gardiner, Infinite processes, background to analysis, Springer-Verlag, 1982.

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.