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<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
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<title>Overview</title>
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<link rel="stylesheet" href="../boostbook.css" type="text/css">
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<meta name="generator" content="DocBook XSL Stylesheets V1.78.1">
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<link rel="home" href="../index.html" title="Math Toolkit">
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<link rel="up" href="../octonions.html" title="Chapter 9. Octonions">
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<link rel="prev" href="../octonions.html" title="Chapter 9. Octonions">
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<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
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<a accesskey="p" href="../octonions.html"><img src="../images/prev.png" alt="Prev"></a><a accesskey="u" href="../octonions.html"><img src="../images/up.png" alt="Up"></a><a accesskey="h" href="../index.html"><img src="../images/home.png" alt="Home"></a><a accesskey="n" href="header0.html"><img src="../images/next.png" alt="Next"></a>
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<div class="section math_toolkit_overview0">
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<div class="titlepage"><div><div><h2 class="title" style="clear: both">
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<a name="math_toolkit.overview0"></a><a class="link" href="overview0.html" title="Overview">Overview</a>
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</h2></div></div></div>
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<p>
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Octonions, like <a href="../../../quaternion/html/index.html" target="_top">quaternions</a>,
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are a relative of complex numbers.
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</p>
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<p>
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Octonions see some use in theoretical physics.
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</p>
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<p>
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In practical terms, an octonion is simply an octuple of real numbers (α,β,γ,δ,ε,ζ,η,θ), which
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we can write in the form <span class="emphasis"><em><code class="literal">o = α + βi + γj + δk + εe' + ζi' + ηj' + θk'</code></em></span>, where
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<span class="emphasis"><em><code class="literal">i</code></em></span>, <span class="emphasis"><em><code class="literal">j</code></em></span>
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and <span class="emphasis"><em><code class="literal">k</code></em></span> are the same objects as for quaternions,
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and <span class="emphasis"><em><code class="literal">e'</code></em></span>, <span class="emphasis"><em><code class="literal">i'</code></em></span>,
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<span class="emphasis"><em><code class="literal">j'</code></em></span> and <span class="emphasis"><em><code class="literal">k'</code></em></span>
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are distinct objects which play essentially the same kind of role as <span class="emphasis"><em><code class="literal">i</code></em></span>
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(or <span class="emphasis"><em><code class="literal">j</code></em></span> or <span class="emphasis"><em><code class="literal">k</code></em></span>).
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</p>
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<p>
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Addition and a multiplication is defined on the set of octonions, which generalize
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their quaternionic counterparts. The main novelty this time is that <span class="bold"><strong>the multiplication is not only not commutative, is now not even
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associative</strong></span> (i.e. there are octonions <span class="emphasis"><em><code class="literal">x</code></em></span>,
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<span class="emphasis"><em><code class="literal">y</code></em></span> and <span class="emphasis"><em><code class="literal">z</code></em></span>
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such that <span class="emphasis"><em><code class="literal">x(yz) ≠ (xy)z</code></em></span>). A way of remembering
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things is by using the following multiplication table:
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../../octonion/graphics/octonion_blurb17.jpeg"></span>
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</p>
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<p>
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Octonions (and their kin) are described in far more details in this other
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<a href="../../../quaternion/TQE.pdf" target="_top">document</a> (with <a href="../../../quaternion/TQE_EA.pdf" target="_top">errata
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and addenda</a>).
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</p>
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<p>
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Some traditional constructs, such as the exponential, carry over without too
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much change into the realms of octonions, but other, such as taking a square
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root, do not (the fact that the exponential has a closed form is a result of
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the author, but the fact that the exponential exists at all for octonions is
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known since quite a long time ago).
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</p>
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</div>
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<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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<td align="left"></td>
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<td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012, 2013 Paul A. Bristow, Christopher Kormanyos,
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Hubert Holin, Bruno Lalande, John Maddock, Johan Råde, Gautam Sewani, Benjamin
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Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
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Distributed under the Boost Software License, Version 1.0. (See accompanying
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file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
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</p>
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</div></td>
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</tr></table>
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<hr>
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