Skip to main content Accessibility help
×
Hostname: page-component-7c8c6479df-p566r Total loading time: 0 Render date: 2024-03-28T14:12:51.222Z Has data issue: false hasContentIssue false

3 - Numeration Systems

Published online by Cambridge University Press:  13 October 2009

Jean-Paul Allouche
Affiliation:
Université de Paris XI
Jeffrey Shallit
Affiliation:
University of Waterloo, Ontario
Get access

Summary

In this chapter, we discuss how numbers can be represented by strings over a finite alphabet. Our emphasis is on the representation of integers, although we briefly discuss representations for real numbers in Section 3.4.

We start with the classical base-k representation, and then discuss less familiar representations such as representation in base —k, Fibonacci representation, and representation in complex bases.

Numeration Systems

A numeration system is a way of expressing an integer n (or, more generally, an element of a given semiring S) as a finite linear combination of base elements ui. The ai are called the generalized digits, or just digits. The finite string of digits arar-1a1a0 is then said to be a representation of the number n. Note that our convention is to write representations starting with the most significant digit, although admittedly this choice is somewhat arbitrary.

For example, in ordinary decimal notation the base elements are the powers of 10. As is certainly familiar to most readers, every non-negative integer can be expressed as a non-negative integer linear combination with 0 ≤ ai < 10.

The leading-zeros problem is a minor annoyance we must deal with. For example, each of the strings 101, 0101, 00101, … is a different way to express the number 5 in base 2.

Type
Chapter
Information
Automatic Sequences
Theory, Applications, Generalizations
, pp. 70 - 127
Publisher: Cambridge University Press
Print publication year: 2003

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×