1945 lines
50 KiB
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1945 lines
50 KiB
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<title>Logic and Advanced Math · Learn Lisp The Hard Way</title>
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<li class="header">LEARN LISP THE HARD WAY</li>
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<li class="chapter " data-level="1.1" data-path="./">
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Second Draft (in-progress)
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CHANGELOG
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TODO
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<li class="header">PREFACE</li>
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TANSTAAFL
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<li class="chapter " data-level="2.2" data-path="preface-part-two.html">
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The Hard Way is Easier
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</a>
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Who Is This Book For?
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</a>
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<li class="chapter " data-level="2.4" data-path="introduction.html">
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<a href="introduction.html">
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Lisp: A Future History
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</a>
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</li>
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<li class="chapter " data-level="2.5" data-path="acknowledgements.html">
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<a href="acknowledgements.html">
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Acknowledgements
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</a>
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</li>
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<li class="header">PART ONE</li>
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<li class="chapter " data-level="3.1" data-path="1-0-0-overview.html">
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<a href="1-0-0-overview.html">
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Grokking Lisp
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</a>
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<ul class="articles">
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<li class="chapter " data-level="3.1.1" data-path="1-01-00-lisp-bootcamp.html">
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<a href="1-01-00-lisp-bootcamp.html">
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Common Lisp Bootcamp
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</a>
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<ul class="articles">
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<li class="chapter " data-level="3.1.1.1" data-path="1-01-01-syntax-overview.html">
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<a href="1-01-01-syntax-overview.html">
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Syntax Overview in 5 Minutes
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</a>
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</li>
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<li class="chapter " data-level="3.1.1.2" data-path="1-01-02-repl.html">
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<a href="1-01-02-repl.html">
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The REPL
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</a>
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<li class="chapter " data-level="3.1.1.3" data-path="1-01-03-expressions.html">
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<a href="1-01-03-expressions.html">
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Expressions, Parentheses, and Return Values
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</a>
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</li>
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<li class="chapter " data-level="3.1.1.4" data-path="1-01-04-lists-cons-cells.html">
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<a href="1-01-04-lists-cons-cells.html">
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Lists, Cons-Cells, and Memory
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</a>
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</li>
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<li class="chapter " data-level="3.1.1.5" data-path="1-01-05-symbols.html">
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<a href="1-01-05-symbols.html">
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Symbols and Namespaces
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</a>
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</li>
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<li class="chapter " data-level="3.1.1.6" data-path="1-01-06-prefix-notation.html">
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<a href="1-01-06-prefix-notation.html">
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Prefix Notation
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</a>
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</li>
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<li class="chapter " data-level="3.1.1.7" data-path="1-01-07-style-guide.html">
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<a href="1-01-07-style-guide.html">
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Common Lisp Style Guide
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</a>
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</li>
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<li class="chapter " data-level="3.1.1.8" data-path="1-01-08-configuration.html">
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<a href="1-01-08-configuration.html">
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Configuring Your Development Environment
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</a>
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</li>
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</ul>
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</li>
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<li class="chapter " data-level="3.1.2" data-path="1-02-00-input-output.html">
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<a href="1-02-00-input-output.html">
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Printing, Streams, and Strings
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</a>
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<ul class="articles">
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<li class="chapter " data-level="3.1.2.1" data-path="1-02-01-strings.html">
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<a href="1-02-01-strings.html">
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Strings
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</a>
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</li>
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<li class="chapter " data-level="3.1.2.2" data-path="1-02-02-more-strings.html">
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<a href="1-02-02-more-strings.html">
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More Strings
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</a>
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</li>
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<li class="chapter " data-level="3.1.2.3" data-path="1-02-03-unicode.html">
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<a href="1-02-03-unicode.html">
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Unicode and Strings
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</a>
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</li>
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<li class="chapter " data-level="3.1.2.4" data-path="1-02-04-chars.html">
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<a href="1-02-04-chars.html">
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Characters
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</a>
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</li>
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<li class="chapter " data-level="3.1.2.5" data-path="1-02-05-more-chars.html">
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<a href="1-02-05-more-chars.html">
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More Characters
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</a>
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</li>
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<li class="chapter " data-level="3.1.2.6" data-path="1-02-06-char-codes.html">
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<a href="1-02-06-char-codes.html">
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Character Codes
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</a>
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</li>
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<li class="chapter " data-level="3.1.2.7" data-path="1-02-07-strings-from-chars.html">
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<a href="1-02-07-strings-from-chars.html">
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Strings from Chars
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</a>
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</li>
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<li class="chapter " data-level="3.1.2.8" data-path="1-02-08-printing.html">
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<a href="1-02-08-printing.html">
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Printing
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</a>
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</li>
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<li class="chapter " data-level="3.1.2.9" data-path="1-02-09-more-printing.html">
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<a href="1-02-09-more-printing.html">
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More Printing
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</a>
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</li>
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<li class="chapter " data-level="3.1.2.10" data-path="1-02-10-prin1.html">
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<a href="1-02-10-prin1.html">
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Printing With prin1
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</a>
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</li>
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<li class="chapter " data-level="3.1.2.11" data-path="1-02-11-princ.html">
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<a href="1-02-11-princ.html">
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Printing With princ
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</a>
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</li>
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<li class="chapter " data-level="3.1.2.12" data-path="1-02-12-format.html">
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<a href="1-02-12-format.html">
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A Brief Introduction to Format
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</a>
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</li>
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<li class="chapter " data-level="3.1.2.13" data-path="1-02-13-more-format.html">
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<a href="1-02-13-more-format.html">
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A Little Bit More on Format
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</a>
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</li>
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<li class="chapter " data-level="3.1.2.14" data-path="1-02-14-pathnames.html">
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<a href="1-02-14-pathnames.html">
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Pathnames
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</a>
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</li>
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<li class="chapter " data-level="3.1.2.15" data-path="1-02-15-streams.html">
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<a href="1-02-15-streams.html">
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Streams
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</a>
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</li>
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<li class="chapter " data-level="3.1.2.16" data-path="1-02-16-file-streams.html">
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<a href="1-02-16-file-streams.html">
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File Streams
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</a>
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</li>
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<li class="chapter " data-level="3.1.2.17" data-path="1-02-17-binary-streams.html">
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<a href="1-02-17-binary-streams.html">
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Binary Streams
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</a>
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</li>
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<li class="chapter " data-level="3.1.2.18" data-path="1-02-18-prompting-users.html">
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<a href="1-02-18-prompting-users.html">
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Prompting Users
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</a>
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</li>
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<li class="chapter " data-level="3.1.2.19" data-path="1-02-19-pretty-printing.html">
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<a href="1-02-19-pretty-printing.html">
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Pretty-Printing
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</a>
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</li>
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</ul>
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</li>
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<li class="chapter " data-level="3.1.3" data-path="1-03-0-getting-input-from-users.html">
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<a href="1-03-0-getting-input-from-users.html">
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Extra Credit: Getting Input from Users
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</a>
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</li>
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<li class="chapter " data-level="3.1.4" data-path="1-04-0-lists.html">
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<a href="1-04-0-lists.html">
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Lists and List-Operations
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</a>
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</li>
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<li class="chapter " data-level="3.1.5" data-path="1-05-0-lookups-trees.html">
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<a href="1-05-0-lookups-trees.html">
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Extra Credit: Look-up Lists and Trees
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</a>
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</li>
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<li class="chapter " data-level="3.1.6" data-path="1-06-0-math.html">
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<a href="1-06-0-math.html">
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Numbers and Math
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</a>
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<ul class="articles">
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<li class="chapter " data-level="3.1.6.1" data-path="1-06-01-integers.html">
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<a href="1-06-01-integers.html">
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Integers
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</a>
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</li>
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<li class="chapter " data-level="3.1.6.2" data-path="1-06-02-more-integers.html">
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<a href="1-06-02-more-integers.html">
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More Integers
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</a>
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</li>
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<li class="chapter " data-level="3.1.6.3" data-path="1-06-03-hexadecimal-notation.html">
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<a href="1-06-03-hexadecimal-notation.html">
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Hexadecimal Integer Notation
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</a>
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</li>
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<li class="chapter " data-level="3.1.6.4" data-path="1-06-04-octal-notation.html">
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<a href="1-06-04-octal-notation.html">
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Octal Integer Notation
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</a>
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</li>
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<li class="chapter " data-level="3.1.6.5" data-path="1-06-05-binary-notation.html">
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<a href="1-06-05-binary-notation.html">
|
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|
|
Binary Integer Notation
|
|
|
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</a>
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</li>
|
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|
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<li class="chapter " data-level="3.1.6.6" data-path="1-06-06-ratios.html">
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|
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|
<a href="1-06-06-ratios.html">
|
|
|
|
|
|
Ratios and Rational Numbers
|
|
|
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</a>
|
|
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|
|
|
|
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</li>
|
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|
<li class="chapter " data-level="3.1.6.7" data-path="1-06-07-floating-point.html">
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|
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<a href="1-06-07-floating-point.html">
|
|
|
|
|
|
Floating-point Numbers
|
|
|
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</a>
|
|
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|
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|
|
|
</li>
|
|
|
|
<li class="chapter " data-level="3.1.6.8" data-path="1-06-08-constants.html">
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<a href="1-06-08-constants.html">
|
|
|
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|
|
Numeric Constants
|
|
|
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</a>
|
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|
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</li>
|
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|
|
<li class="chapter " data-level="3.1.6.9" data-path="1-06-09-complex-numbers.html">
|
|
|
|
<a href="1-06-09-complex-numbers.html">
|
|
|
|
|
|
Complex Numbers
|
|
|
|
</a>
|
|
|
|
|
|
|
|
</li>
|
|
|
|
<li class="chapter " data-level="3.1.6.10" data-path="1-06-10-arithmetic.html">
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|
|
<a href="1-06-10-arithmetic.html">
|
|
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|
|
Arithmetic
|
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|
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</a>
|
|
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</li>
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|
|
<li class="chapter " data-level="3.1.6.11" data-path="1-06-11-more-arithmetic.html">
|
|
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|
<a href="1-06-11-more-arithmetic.html">
|
|
|
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|
|
More Arithmetic
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|
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</a>
|
|
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|
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</li>
|
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|
|
<li class="chapter " data-level="3.1.6.12" data-path="1-06-12-even-more-arithmetic.html">
|
|
|
|
<a href="1-06-12-even-more-arithmetic.html">
|
|
|
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|
|
Even More Arithmetic
|
|
|
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</a>
|
|
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|
|
|
|
|
</li>
|
|
|
|
<li class="chapter " data-level="3.1.6.13" data-path="1-06-13-exponents.html">
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|
<a href="1-06-13-exponents.html">
|
|
|
|
|
|
Exponents
|
|
|
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</a>
|
|
|
|
|
|
|
|
</li>
|
|
|
|
<li class="chapter " data-level="3.1.6.14" data-path="1-06-14-logarithms.html">
|
|
|
|
<a href="1-06-14-logarithms.html">
|
|
|
|
|
|
Logarithms
|
|
|
|
</a>
|
|
|
|
|
|
|
|
</li>
|
|
|
|
<li class="chapter " data-level="3.1.6.15" data-path="1-06-15-trigonometry.html">
|
|
|
|
<a href="1-06-15-trigonometry.html">
|
|
|
|
|
|
Trigonometry
|
|
|
|
</a>
|
|
|
|
|
|
|
|
</li>
|
|
|
|
<li class="chapter " data-level="3.1.6.16" data-path="1-06-16-psuedorandom-numbers.html">
|
|
|
|
<a href="1-06-16-psuedorandom-numbers.html">
|
|
|
|
|
|
Pseudo-Random Numbers
|
|
|
|
</a>
|
|
|
|
|
|
|
|
</li>
|
|
|
|
|
|
</ul>
|
|
|
|
</li>
|
|
|
|
<li class="chapter " data-level="3.1.7" data-path="1-07-0-arrays.html">
|
|
|
|
<a href="1-07-0-arrays.html">
|
|
|
|
|
|
Extra Credit: Arrays and Vectors
|
|
|
|
</a>
|
|
|
|
|
|
|
|
</li>
|
|
|
|
<li class="chapter " data-level="3.1.8" data-path="1-08-0-variables.html">
|
|
|
|
<a href="1-08-0-variables.html">
|
|
|
|
|
|
Variables, Parameters, and Constants
|
|
|
|
</a>
|
|
|
|
|
|
|
|
</li>
|
|
|
|
<li class="chapter " data-level="3.1.9" data-path="1-09-0-closures.html">
|
|
|
|
<a href="1-09-0-closures.html">
|
|
|
|
|
|
Extra Credit: Closures
|
|
|
|
</a>
|
|
|
|
|
|
|
|
</li>
|
|
|
|
<li class="chapter " data-level="3.1.10" data-path="1-10-0-functions.html">
|
|
|
|
<a href="1-10-0-functions.html">
|
|
|
|
|
|
Functions and Macros
|
|
|
|
</a>
|
|
|
|
|
|
|
|
</li>
|
|
|
|
<li class="chapter " data-level="3.1.11" data-path="1-11-0-text-adventure.html">
|
|
|
|
<a href="1-11-0-text-adventure.html">
|
|
|
|
|
|
Extra Credit: A Simple Text Adventure
|
|
|
|
</a>
|
|
|
|
|
|
|
|
</li>
|
|
|
|
<li class="chapter " data-level="3.1.12" data-path="1-12-0-namespaces.html">
|
|
|
|
<a href="1-12-0-namespaces.html">
|
|
|
|
|
|
Namespaces, Symbols, Packages, and Systems
|
|
|
|
</a>
|
|
|
|
|
|
|
|
</li>
|
|
|
|
<li class="chapter " data-level="3.1.13" data-path="1-13-0-simple-web-app.html">
|
|
|
|
<a href="1-13-0-simple-web-app.html">
|
|
|
|
|
|
Extra Credit: A Simple Web Application
|
|
|
|
</a>
|
|
|
|
|
|
|
|
</li>
|
|
|
|
<li class="chapter " data-level="3.1.14" data-path="1-14-0-conditionals.html">
|
|
|
|
<a href="1-14-0-conditionals.html">
|
|
|
|
|
|
Conditionals
|
|
|
|
</a>
|
|
|
|
|
|
|
|
</li>
|
|
|
|
<li class="chapter " data-level="3.1.15" data-path="1-15-0-command-line-utility.html">
|
|
|
|
<a href="1-15-0-command-line-utility.html">
|
|
|
|
|
|
Extra Credit: Command-Line Utilities
|
|
|
|
</a>
|
|
|
|
|
|
|
|
</li>
|
|
|
|
<li class="chapter " data-level="3.1.16" data-path="1-16-0-map-loop.html">
|
|
|
|
<a href="1-16-0-map-loop.html">
|
|
|
|
|
|
Mapping and Looping
|
|
|
|
</a>
|
|
|
|
|
|
|
|
</li>
|
|
|
|
<li class="chapter " data-level="3.1.17" data-path="1-17-0-iterate.html">
|
|
|
|
<a href="1-17-0-iterate.html">
|
|
|
|
|
|
Extra Credit: Revisiting Loops with Iterate
|
|
|
|
</a>
|
|
|
|
|
|
|
|
</li>
|
|
|
|
<li class="chapter " data-level="3.1.18" data-path="1-18-0-format.html">
|
|
|
|
<a href="1-18-0-format.html">
|
|
|
|
|
|
Format Strings
|
|
|
|
</a>
|
|
|
|
|
|
|
|
</li>
|
|
|
|
<li class="chapter " data-level="3.1.19" data-path="1-19-0-dsl.html">
|
|
|
|
<a href="1-19-0-dsl.html">
|
|
|
|
|
|
Extra Credit: Domain Specific Languages
|
|
|
|
</a>
|
|
|
|
|
|
|
|
</li>
|
|
|
|
<li class="chapter " data-level="3.1.20" data-path="1-20-0-review.html">
|
|
|
|
<a href="1-20-0-review.html">
|
|
|
|
|
|
Part One in Review
|
|
|
|
</a>
|
|
|
|
|
|
|
|
</li>
|
|
|
|
|
|
</ul>
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
<li class="header">PART TWO</li>
|
|
|
|
|
|
|
|
<li class="chapter " data-level="4.1" data-path="2-0-0-overview.html">
|
|
|
|
<a href="2-0-0-overview.html">
|
|
|
|
|
|
The Suffusion of Blue
|
|
|
|
</a>
|
|
|
|
|
|
|
|
<ul class="articles">
|
|
|
|
|
|
<li class="chapter " data-level="4.1.1" data-path="2-01-0-programming-paradigms.html">
|
|
|
|
<a href="2-01-0-programming-paradigms.html">
|
|
|
|
|
|
Programming Paradigms
|
|
|
|
</a>
|
|
|
|
|
|
|
|
</li>
|
|
|
|
<li class="chapter " data-level="4.1.2" data-path="2-02-0-regex.html">
|
|
|
|
<a href="2-02-0-regex.html">
|
|
|
|
|
|
Extra Credit: Regular Expressions
|
|
|
|
</a>
|
|
|
|
|
|
|
|
</li>
|
|
|
|
<li class="chapter " data-level="4.1.3" data-path="2-03-0-objects-control.html">
|
|
|
|
<a href="2-03-0-objects-control.html">
|
|
|
|
|
|
Objects and Control Structures
|
|
|
|
</a>
|
|
|
|
|
|
|
|
</li>
|
|
|
|
<li class="chapter " data-level="4.1.4" data-path="2-04-0-data-persistence.html">
|
|
|
|
<a href="2-04-0-data-persistence.html">
|
|
|
|
|
|
Extra Credit: Persistence and Databases
|
|
|
|
</a>
|
|
|
|
|
|
|
|
</li>
|
|
|
|
<li class="chapter " data-level="4.1.5" data-path="2-05-0-extended-types.html">
|
|
|
|
<a href="2-05-0-extended-types.html">
|
|
|
|
|
|
Extended Types
|
|
|
|
</a>
|
|
|
|
|
|
|
|
</li>
|
|
|
|
<li class="chapter " data-level="4.1.6" data-path="2-06-0-threads-memos-parallel.html">
|
|
|
|
<a href="2-06-0-threads-memos-parallel.html">
|
|
|
|
|
|
Extra Credit: Concurrency and Memoization
|
|
|
|
</a>
|
|
|
|
|
|
|
|
</li>
|
|
|
|
<li class="chapter active" data-level="4.1.7" data-path="2-07-0-logic-and-more-math.html">
|
|
|
|
<a href="2-07-0-logic-and-more-math.html">
|
|
|
|
|
|
Logic and Advanced Math
|
|
|
|
</a>
|
|
|
|
|
|
|
|
</li>
|
|
|
|
<li class="chapter " data-level="4.1.8" data-path="2-08-0-number-theory.html">
|
|
|
|
<a href="2-08-0-number-theory.html">
|
|
|
|
|
|
Extra Credit: Number Theory
|
|
|
|
</a>
|
|
|
|
|
|
|
|
</li>
|
|
|
|
<li class="chapter " data-level="4.1.9" data-path="2-09-0-binary-octets-bits.html">
|
|
|
|
<a href="2-09-0-binary-octets-bits.html">
|
|
|
|
|
|
Binary Streams, Octet-Vectors, and Bit-Vectors
|
|
|
|
</a>
|
|
|
|
|
|
|
|
</li>
|
|
|
|
<li class="chapter " data-level="4.1.10" data-path="2-10-0-improved-text-adventure-engine.html">
|
|
|
|
<a href="2-10-0-improved-text-adventure-engine.html">
|
|
|
|
|
|
Extra Credit: An Improved Text Adventure Engine
|
|
|
|
</a>
|
|
|
|
|
|
|
|
</li>
|
|
|
|
<li class="chapter " data-level="4.1.11" data-path="2-11-0-conditions.html">
|
|
|
|
<a href="2-11-0-conditions.html">
|
|
|
|
|
|
Conditions and Error Handling
|
|
|
|
</a>
|
|
|
|
|
|
|
|
</li>
|
|
|
|
<li class="chapter " data-level="4.1.12" data-path="2-12-0-2d-game.html">
|
|
|
|
<a href="2-12-0-2d-game.html">
|
|
|
|
|
|
Extra Credit: Write a 2D Game
|
|
|
|
</a>
|
|
|
|
|
|
|
|
</li>
|
|
|
|
<li class="chapter " data-level="4.1.13" data-path="2-13-0-compiler.html">
|
|
|
|
<a href="2-13-0-compiler.html">
|
|
|
|
|
|
The Compiler
|
|
|
|
</a>
|
|
|
|
|
|
|
|
</li>
|
|
|
|
<li class="chapter " data-level="4.1.14" data-path="2-14-0-tree-shaker.html">
|
|
|
|
<a href="2-14-0-tree-shaker.html">
|
|
|
|
|
|
Extra Credit: Write a Tree-Shaker
|
|
|
|
</a>
|
|
|
|
|
|
|
|
</li>
|
|
|
|
<li class="chapter " data-level="4.1.15" data-path="2-15-0-docs-and-inspection.html">
|
|
|
|
<a href="2-15-0-docs-and-inspection.html">
|
|
|
|
|
|
Documentation and Inspection
|
|
|
|
</a>
|
|
|
|
|
|
|
|
</li>
|
|
|
|
<li class="chapter " data-level="4.1.16" data-path="2-16-0-foreign-libs.html">
|
|
|
|
<a href="2-16-0-foreign-libs.html">
|
|
|
|
|
|
Extra Credit: Foreign Libraries in Lisp
|
|
|
|
</a>
|
|
|
|
|
|
|
|
</li>
|
|
|
|
<li class="chapter " data-level="4.1.17" data-path="2-17-0-debugging-testing.html">
|
|
|
|
<a href="2-17-0-debugging-testing.html">
|
|
|
|
|
|
Debugging and Unit Testing
|
|
|
|
</a>
|
|
|
|
|
|
|
|
</li>
|
|
|
|
<li class="chapter " data-level="4.1.18" data-path="2-18-0-ffi.html">
|
|
|
|
<a href="2-18-0-ffi.html">
|
|
|
|
|
|
Extra Credit: Write a Foreign Function Interface
|
|
|
|
</a>
|
|
|
|
|
|
|
|
</li>
|
|
|
|
<li class="chapter " data-level="4.1.19" data-path="2-19-0-essential-libs.html">
|
|
|
|
<a href="2-19-0-essential-libs.html">
|
|
|
|
|
|
Essential Lisp Libraries
|
|
|
|
</a>
|
|
|
|
|
|
|
|
</li>
|
|
|
|
<li class="chapter " data-level="4.1.20" data-path="2-20-0-packaging-libs.html">
|
|
|
|
<a href="2-20-0-packaging-libs.html">
|
|
|
|
|
|
Extra Credit: Packaging Lisp Libraries
|
|
|
|
</a>
|
|
|
|
|
|
|
|
</li>
|
|
|
|
<li class="chapter " data-level="4.1.21" data-path="2-21-0-review.html">
|
|
|
|
<a href="2-21-0-review.html">
|
|
|
|
|
|
Detailed Syntax Review
|
|
|
|
</a>
|
|
|
|
|
|
|
|
</li>
|
|
|
|
|
|
</ul>
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
<li class="header">PART THREE</li>
|
|
|
|
|
|
|
|
<li class="chapter " data-level="5.1" data-path="3-00-00-overview.html">
|
|
|
|
<a href="3-00-00-overview.html">
|
|
|
|
|
|
Lisp So(u)rcery
|
|
|
|
</a>
|
|
|
|
|
|
|
|
<ul class="articles">
|
|
|
|
|
|
<li class="chapter " data-level="5.1.1" data-path="3-01-00-web-apps.html">
|
|
|
|
<a href="3-01-00-web-apps.html">
|
|
|
|
|
|
Real-world Web Apps
|
|
|
|
</a>
|
|
|
|
|
|
|
|
</li>
|
|
|
|
<li class="chapter " data-level="5.1.2" data-path="3-02-00-typesetting.html">
|
|
|
|
<a href="3-02-00-typesetting.html">
|
|
|
|
|
|
Typesetting
|
|
|
|
</a>
|
|
|
|
|
|
|
|
</li>
|
|
|
|
<li class="chapter " data-level="5.1.3" data-path="3-03-00-mobile.html">
|
|
|
|
<a href="3-03-00-mobile.html">
|
|
|
|
|
|
Native Mobile Applications
|
|
|
|
</a>
|
|
|
|
|
|
|
|
</li>
|
|
|
|
<li class="chapter " data-level="5.1.4" data-path="3-04-00-gui.html">
|
|
|
|
<a href="3-04-00-gui.html">
|
|
|
|
|
|
Cross-platform Desktop Applications
|
|
|
|
</a>
|
|
|
|
|
|
|
|
</li>
|
|
|
|
<li class="chapter " data-level="5.1.5" data-path="3-05-00-system-utils.html">
|
|
|
|
<a href="3-05-00-system-utils.html">
|
|
|
|
|
|
Drivers, Daemons, and System-Utilities
|
|
|
|
</a>
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A Simple LispOS
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Build Your Own Lisp Machine
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<!-- Title -->
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<h1>
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<a href="index.html" >Logic and Advanced Math</a>
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<h1 id="chapter-27">Chapter 2.7</h1>
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<h2 id="logic-and-advanced-math">Logic and Advanced Math</h2>
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<blockquote>
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<p>"...Mathematics need not have use; it was a game, like chess but more fun."</p>
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<footer>Robert A. Heinlein, <em>Citizen of the Galaxy</em></footer>
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</blockquote>
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<p><strong>Revision Note:</strong> <em>Intro text needs complete rewrite</em>.</p>
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<p>We've dabbled in basic math already, and chances are high that in every-day programming, that might be all you ever need. However, there are many problem domains that require a sophisticated understanding of advanced logic, Lisp's built-in tools for modular arithmetic and trigonometry, and the implementation of functions to support number theory, linear algebra, and calculus; for example, game development, artificial intelligence, computational physics, cryptography, FINTECH, and everything in the Crypto-Currency space. Even as "just a web developer", on-line security is becoming ever more important as developers are pushed to get the web experience ever closer to the desktop experience---and cryptography plays a central role in that. As the saying goes, "<em>the more you know...</em>"</p>
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<p>Of course, by this point you should no longer be afraid of math, since every line of code you write in Lisp, no matter the paradigm, is formally expressed as an S-Expression in the Lambda Calculus. While what you're doing may not conform to the conception of <em>pure functional programming</em>, every expression you write that is evaluated as a proper form returns some result. At the end of the day, that's all math really is---a set of formally expressed rules that operate on quantitative data. And <em>everything</em> in a computer is quantitative, whether you would normally think of it that way or not. That Lisp adapts a purely mathematical model underneath in its implementation, and allows qualitative logic to be built on top of a quantifiable system is a feature---and an exciting one. Lisp demonstrates how all qualitative systems can be reduced to quantitative. In the age-old debate of philisophy vs. science, the importance of this cannot be understated. And this should also serve as an explanation as to why Lisp Hackers always say that <em>programming is math</em>, while developers from other languages simply fail to see this basic tenet of computer science.</p>
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<p>But Math is more important than even that, to all programmers. Every program that can be written can be abstracted and generalized to a formal mathematical definition, and in so doing, can be simplified and made more elegant. Consider the problem of 3D Graphics, whether for gaming, simulation, virtual reality, augmented reality, or immersive interfaces. You can use 3D Graphics applications such as CAD or Blender, or the myriad of robust proprietary alternatives, to visually create your 3D artwork and assets---there's nothing wrong with this, as for artists it is quite intuitive. However, a mathematician can achieve the same end with pure code. These 3D assets are all collections of vertices in cartesian space, and they can be generated procedurally just as easily---and sometimes more easily---than they can be created with a 3D graphics application. It just depends on how well you understand the math behind 3D graphics.</p>
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<p>Lisp is particularly good for generating 3D graphics without having to use assets created in a third-party application. In fact, that was one of the greatest strengths of the fabled Lisp Machines. In Part Three we will be tackling this problem head-on, to create a complete 3D game without any third-party assets. Just code. <em>This</em> chapter will provide you with all the math you'll need to achieve that feat---any many, many more.</p>
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<h3 id="exercise-271">Exercise 2.7.1</h3>
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<p><strong>Formal Logic</strong></p>
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<pre><code class="lang-lisp">
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<h3 id="exercise-272">Exercise 2.7.2</h3>
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<p><strong>Boolean Logic on Integers</strong></p>
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<pre><code class="lang-lisp">
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<h3 id="exercise-273">Exercise 2.7.3</h3>
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<p><strong>Boolean Logic on Bit-Vectors</strong></p>
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<h3 id="exercise-274">Exercise 2.7.4</h3>
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<p><strong>Prime Numbers</strong></p>
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<h3 id="exercise-275">Exercise 2.7.5</h3>
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<p><strong>Prime Factorization</strong></p>
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<h3 id="exercise-276">Exercise 2.7.6</h3>
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<p><strong>Modular Arithmetic</strong></p>
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<pre><code class="lang-lisp">
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<h3 id="exercise-277">Exercise 2.7.7</h3>
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<p><strong>Imaginary Numbers</strong></p>
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<pre><code class="lang-lisp">
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<h3 id="exercise-278">Exercise 2.7.8</h3>
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<p><strong>Complex Numbers</strong></p>
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<pre><code class="lang-lisp">
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</code></pre>
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<h3 id="exercise-279">Exercise 2.7.9</h3>
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<p><strong>Trigonometry</strong></p>
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<pre><code class="lang-lisp">
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<h3 id="exercise-2710">Exercise 2.7.10</h3>
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<p><strong>More Trigonometry</strong></p>
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<pre><code class="lang-lisp">
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<h3 id="exercise-2711">Exercise 2.7.11</h3>
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<p><strong>Geometry</strong></p>
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<pre><code class="lang-lisp">
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<h3 id="exercise-2712">Exercise 2.7.12</h3>
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<p><strong>More Geometry</strong></p>
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<pre><code class="lang-lisp">
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<h3 id="exercise-2713">Exercise 2.7.13</h3>
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<p><strong>Even More Geometry</strong></p>
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<pre><code class="lang-lisp">
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<h3 id="exercise-2714">Exercise 2.7.14</h3>
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<p><strong>The GSLL Library</strong></p>
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<pre><code class="lang-lisp">
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<h3 id="exercise-2715">Exercise 2.7.15</h3>
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<p><strong>Linear Algebra with GSLL</strong></p>
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<pre><code class="lang-lisp">
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</code></pre>
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<h3 id="exercise-2716">Exercise 2.7.16</h3>
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<p><strong>More Linear Algebra</strong></p>
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<pre><code class="lang-lisp">
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</code></pre>
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<h3 id="exercise-2717">Exercise 2.7.17</h3>
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<p><strong>Even More Linear Algebra</strong></p>
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<pre><code class="lang-lisp">
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</code></pre>
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<h3 id="exercise-2718">Exercise 2.7.18</h3>
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<p><strong>Calculus with GSLL</strong></p>
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<pre><code class="lang-lisp">
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</code></pre>
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<h3 id="exercise-2719">Exercise 2.7.19</h3>
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<p><strong>More Calculus</strong></p>
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<pre><code class="lang-lisp">
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</code></pre>
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<h3 id="exercise-2720">Exercise 2.7.20</h3>
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<p><strong>Even More Calculus</strong></p>
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<pre><code class="lang-lisp">
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</code></pre>
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<h3 id="exercise-2721">Exercise 2.7.21</h3>
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<p><strong>Further Explorations with GSLL</strong></p>
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<pre><code class="lang-lisp">
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</code></pre>
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