Dr. Michael Angel Carter, GED: New Applications
https://www.maplesoft.com/applications/author.aspx?mid=8554
en-us2021 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemTue, 19 Oct 2021 15:27:24 GMTTue, 19 Oct 2021 15:27:24 GMTNew applications published by Dr. Michael Angel Carter, GEDhttps://www.maplesoft.com/images/Application_center_hp.jpgDr. Michael Angel Carter, GED: New Applications
https://www.maplesoft.com/applications/author.aspx?mid=8554
The CayleyDickson Algebra from 4D to 256D
https://www.maplesoft.com/applications/view.aspx?SID=35420&ref=Feed
<p>There are higher dimensional numbers besides complex numbers. There are also hypercomplex numbers, such as, quaternions (4 D), octonions (8 D), sedenions (16 D), pathions (32 D), chingons (64 D), routons (128 D), voudons (256 D), and so on, without end. These names were coined by Robert P.C. de Marrais and Tony Smith. It is an alternate naming system providing relief from the difficult Latin names, such as:<br /> trigintaduonions (32 D), sexagintaquatronions (64 D), centumduodetrigintanions (128 D), and ducentiquinquagintasexions (256 D).</p><img src="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="The CayleyDickson Algebra from 4D to 256D" style="max-width: 25%;" align="left"/><p>There are higher dimensional numbers besides complex numbers. There are also hypercomplex numbers, such as, quaternions (4 D), octonions (8 D), sedenions (16 D), pathions (32 D), chingons (64 D), routons (128 D), voudons (256 D), and so on, without end. These names were coined by Robert P.C. de Marrais and Tony Smith. It is an alternate naming system providing relief from the difficult Latin names, such as:<br /> trigintaduonions (32 D), sexagintaquatronions (64 D), centumduodetrigintanions (128 D), and ducentiquinquagintasexions (256 D).</p>https://www.maplesoft.com/applications/view.aspx?SID=35420&ref=FeedFri, 23 Apr 2010 04:00:00 ZMichael CarterMichael CarterQuaternions, Octonions and Sedenions
https://www.maplesoft.com/applications/view.aspx?SID=35196&ref=Feed
<p>This Hypercomplex package provides the algebra of the quaternion, octonion and sedenion hypercomplex numbers.</p><img src="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="Quaternions, Octonions and Sedenions" style="max-width: 25%;" align="left"/><p>This Hypercomplex package provides the algebra of the quaternion, octonion and sedenion hypercomplex numbers.</p>https://www.maplesoft.com/applications/view.aspx?SID=35196&ref=FeedFri, 16 Apr 2010 04:00:00 ZDr. Michael Angel Carter
Dr. Michael Angel Carter
Quaternions
https://www.maplesoft.com/applications/view.aspx?SID=4886&ref=Feed
Overview on Hamilton Quaternions
A Hamilton Quaternion is a hypercomplex number with one real part (the scalar) and three imaginary parts (the vector).
This is an extension of the concept of numbers. We have found that a real number is a one-part number that can be represented on a number line and a complex number is a two-part number that can be represented on a plane. Extending that logic, we have also found that we can produce more numbers by adding more parts.
Quaternion --> a + b*i + c*j + d*k, where the coefficients a, b, c, d are elements of the reals
The hypercomplex number-quaternion-is a non-commutative division ring. This is what we call a four-dimensional number. Here is an example of a quaternion: 5 + 2i + 3j + 4k. The first term is called the scalar term; it is simply a real number. The other terms consisting of, i, j, k. are called the imaginary terms. As a group, they are called the vector of the quaternion. Vector algebra uses the same name vector as the quaternion number, but with a different meaning. Although vector algebra is an offspring of quaternions, vectors are not numbers. Starting with complex numbers, we lost the permanence of trichotomy. The permanencies we lose with quaternion numbers are the trichotomy property and the commutative property under multiplication. In addition, the imaginary units are anti-commutative under multiplication. Anti-commutative means the sign of the imaginary unit changes when we transpose the two operands under multiplication, e.g., i*j = -(j*i). The imaginary elements, i, j, and k, give cyclic permutations with each other (see Behaviors of Quaternions);.
If we set the coefficients of the imaginary elements j and k to zero, the quaternion number becomes an ordinary complex number. We can deduce all of the other algebraic numbers and/or the transcendental numbers from the quaternions simply by setting all the coefficients of the imaginary elements to zero.
This is an update from an earlier version that was updated in March 2005.<img src="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="Quaternions" style="max-width: 25%;" align="left"/>Overview on Hamilton Quaternions
A Hamilton Quaternion is a hypercomplex number with one real part (the scalar) and three imaginary parts (the vector).
This is an extension of the concept of numbers. We have found that a real number is a one-part number that can be represented on a number line and a complex number is a two-part number that can be represented on a plane. Extending that logic, we have also found that we can produce more numbers by adding more parts.
Quaternion --> a + b*i + c*j + d*k, where the coefficients a, b, c, d are elements of the reals
The hypercomplex number-quaternion-is a non-commutative division ring. This is what we call a four-dimensional number. Here is an example of a quaternion: 5 + 2i + 3j + 4k. The first term is called the scalar term; it is simply a real number. The other terms consisting of, i, j, k. are called the imaginary terms. As a group, they are called the vector of the quaternion. Vector algebra uses the same name vector as the quaternion number, but with a different meaning. Although vector algebra is an offspring of quaternions, vectors are not numbers. Starting with complex numbers, we lost the permanence of trichotomy. The permanencies we lose with quaternion numbers are the trichotomy property and the commutative property under multiplication. In addition, the imaginary units are anti-commutative under multiplication. Anti-commutative means the sign of the imaginary unit changes when we transpose the two operands under multiplication, e.g., i*j = -(j*i). The imaginary elements, i, j, and k, give cyclic permutations with each other (see Behaviors of Quaternions);.
If we set the coefficients of the imaginary elements j and k to zero, the quaternion number becomes an ordinary complex number. We can deduce all of the other algebraic numbers and/or the transcendental numbers from the quaternions simply by setting all the coefficients of the imaginary elements to zero.
This is an update from an earlier version that was updated in March 2005.https://www.maplesoft.com/applications/view.aspx?SID=4886&ref=FeedMon, 26 Mar 2007 04:00:00 ZMichael CarterMichael Carter