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Dead Reckoning: Calculating Without Instruments

Appendix: Finding Rational Approximations to Precomputed Constants

Background and Topics

Often we encounter multi-digit numbers, such as scaling factors, that are difficult to use directly as a divisor or multiplier because of the number of digits involved. For example, converting from radians to degrees is done by multiplying by 180/pi = 57.29578..., so in this book I approximate this multiplication by 401/7 = 57.2857..., which is accurate to almost 4 digits and is a more convenient multiplier for mental work. (Here the perfectionist tendencies in me whisper to just add 1/100 of the radian value...) This appendix describes the general procedure (using continued fractions) for generating whole-number fractions that approximate multi-digit numbers to a required accuracy and in practice involve multiplication and division by reasonably small whole numbers.

Notes and Errata (If you have any more to contribute, please email me! This is a compendium of feedback)  A printer-friendly summary table for all chapters is found here.

Color Code
Type
Meaning
Note
Note
A clarification or elaboration of the text.
Typo
Typo
A simple mistake that does not affect the method presented.
Error
Error
An error that may affect a method or the reader interpretation of it.
 
Page
Code
Explanation
All
Note A graphical software program for generating rational approximations to a decimal number entered by the user can be found on the webpage of Online Material.
172
Typo

In the top paragraph, 1/2n in 10^(1/2n) should read 1/2^n.

173
Typo
The phrase “a0 greater than 1” should read “an greater than 1”.
176
Note
To be consistent with my general philosophy of rounding the last digit even when an ellipsis (…) exists at the end, the value 1.1555555… should read 1.1555556…

 

Additional Materials Related to Topics in This Chapter

I wrote a useful, interactive software program to generate rational approximations to entered decimal numbers based on the method in this Appendix.  Please see the Online Materials webpage to obtain this free program.

Cited Reference Materials

In the near future, this section will contain relevant sections of some of the references cited at the end of this chapter.

Acknowledgements of Contributors to this Page

John McIntosh provided the errata and notes you see on this page.

 
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