Pythagoras' Revenge: A Mathematical Mystery

By Arturo Sangalli | Go to book overview

Appendix 3
Random Sequences

In 1919, the Austrian-born mathematician Richard von Mises proposed the following definition: an infinite sequence s, say, of os and 1s is random if

(a) s satisfies the law of large numbers, that is, ‘there are as many os as there are 1s,’ or, more precisely, the limiting value of x/n, where x is the number of os among the first n terms of the sequence, is 0.5, and

(b) Every subsequence that can be extracted from s by reasonable means also satisfies the law of large numbers.

Applying von Mises definition to the sequence 0 1 0 1 0 1 0 1… of alternating 0s and 1s would confirm that it is not random, for the subsequence of even bits (the 2nd, 4th, etc.), that is, 1 1 1 1 1 1…, clearly does not satisfy condition (a). Likewise, many other sequences which appear intuitively to be nonrandom fail to satisfy von Mises’ conditions, and hence they are not random also in the technical sense.

Unfortunately, the definition proposed by the Austrian mathematician suffered from a fundamental defect: it did not specify which means for extracting a subsequence are “reasonable” means. To remedy this situation Alonzo Church, an American mathematician, suggested in 1940 that condition (b) of von Mises’ definition should only apply to computable subsequences, that is, to those subsequences whose terms could be defined by a computer program. Although Church’s idea had the merit of making the definition precise, examples were subsequently found of sequences that are intuitively

-175-

Notes for this page

Add a new note
If you are trying to select text to create highlights or citations, remember that you must now click or tap on the first word, and then click or tap on the last word.
One moment ...
Default project is now your active project.
Project items

Items saved from this book

This book has been saved
Highlights (0)
Some of your highlights are legacy items.

Highlights saved before July 30, 2012 will not be displayed on their respective source pages.

You can easily re-create the highlights by opening the book page or article, selecting the text, and clicking “Highlight.”

Citations (0)
Some of your citations are legacy items.

Any citation created before July 30, 2012 will labeled as a “Cited page.” New citations will be saved as cited passages, pages or articles.

We also added the ability to view new citations from your projects or the book or article where you created them.

Notes (0)
Bookmarks (0)

You have no saved items from this book

Project items include:
  • Saved book/article
  • Highlights
  • Quotes/citations
  • Notes
  • Bookmarks
Notes
Cite this page

Cited page

Style
Citations are available only to our active members.
Sign up now to cite pages or passages in MLA, APA and Chicago citation styles.

(Einhorn, 1992, p. 25)

(Einhorn 25)

1

1. Lois J. Einhorn, Abraham Lincoln, the Orator: Penetrating the Lincoln Legend (Westport, CT: Greenwood Press, 1992), 25, http://www.questia.com/read/27419298.

Cited page

Bookmark this page
Pythagoras' Revenge: A Mathematical Mystery
Table of contents

Table of contents

  • Title Page iii
  • Contents v
  • Preface ix
  • List of Main Characters (Chapter in Which They Are Introduced) xi
  • Prologue xiii
  • Part I- A Time Capsule? 1
  • Chapter 1- The Fifteen Puzzle 3
  • Chapter 2- The Impossible Manuscript 10
  • Chapter 3- Game over 19
  • Chapter 4- A Trip to London 25
  • Chapter 5- A Letter from the Past 32
  • Chapter 6- Found and Lost 38
  • Chapter 7- A Death in the Family 46
  • Part II- An Extraordinarily Gifted Man 51
  • Chapter 8- The Mission 53
  • Chapter 9- Norton Thorp 63
  • Chapter 10- Random Numbers 69
  • Chapter 11- Randomness Everywhere 76
  • Chapter 12- Vanished 82
  • Part III- A Sect of Neo­ Pythagoreans 83
  • Chapter 13- The Mandate 85
  • Chapter 14- The Beacon 87
  • Chapter 15- The Team 98
  • Chapter 16- The Hunt 106
  • Chapter 17- The Symbol of the Serpent 115
  • Chapter 18- A Professional Job 122
  • Chapter 19- with a Little Help from Your Sister 126
  • Part IV- Pythagoras' Mission 137
  • Chapter 20- All Roads Lead to Rome 139
  • Chapter 21- Kidnapped 152
  • Chapter 22- The Last Piece of the Puzzle 158
  • Epilogue 169
  • Appendix 1- Jule's Solution 171
  • Appendix 2- Infinitely Many Primes 173
  • Appendix 3- Random Sequences 175
  • Appendix 4- A Simple Visual Proof of the Pythagorean Theorem 177
  • Appendix 5- Perfect and Figured Numbers 178
  • Notes, Credits, and Bibliographical Sources 181
Settings

Settings

Typeface
Text size Smaller Larger Reset View mode
Search within

Search within this book

Look up

Look up a word

  • Dictionary
  • Thesaurus
Please submit a word or phrase above.
Print this page

Print this page

Why can't I print more than one page at a time?

Full screen
/ 183

matching results for page

Cited passage

Style
Citations are available only to our active members.
Sign up now to cite pages or passages in MLA, APA and Chicago citation styles.

"Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences." (Einhorn, 1992, p. 25).

"Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences." (Einhorn 25)

"Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences."1

1. Lois J. Einhorn, Abraham Lincoln, the Orator: Penetrating the Lincoln Legend (Westport, CT: Greenwood Press, 1992), 25, http://www.questia.com/read/27419298.

Cited passage

Welcome to the new Questia Reader

The Questia Reader has been updated to provide you with an even better online reading experience.  It is now 100% Responsive, which means you can read our books and articles on any sized device you wish.  All of your favorite tools like notes, highlights, and citations are still here, but the way you select text has been updated to be easier to use, especially on touchscreen devices.  Here's how:

1. Click or tap the first word you want to select.
2. Click or tap the last word you want to select.

OK, got it!

Thanks for trying Questia!

Please continue trying out our research tools, but please note, full functionality is available only to our active members.

Your work will be lost once you leave this Web page.

For full access in an ad-free environment, sign up now for a FREE, 1-day trial.

Already a member? Log in now.