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'''Carol Yager''' ([[1960]]-[[1994]]) holds the distinction of having been the [[obesity|most obese]] person ever to live. When she died in 1994 at the age of 34, she weighed about 1200 [[pound (mass)|pounds]]. Some estimates place her weight at as much as 1600 pounds at her peak, but these are unverified. At death, she was 5'7" tall, and able to fit through her custom-built 48" wide front door, although some sources claim she was more than 5 feet wide.
[[Category:Group theory]]
A '''quaternion''' is a mathematical concept discovered by [[William Rowan Hamilton]] of [[Ireland]] in [[1843]]. The idea captured the popular imagination for a time because it involves relatively simple calculations that abandon the [[commutative law]], one of the basic rules of arithmetic. As such, it seemed to undermine one of the tenets of scientific knowledge.
Like others in the 900+ pound weight class, Yager was not able to stand or walk, as her [[muscle]]s were not strong enough to lift her due to [[atrophy]].
Specifically, a quaternion is a [[commutative|non-commutative]] extension of the [[complex number]]s. As a [[vector space]] over the [[real number]]s, the quaternions have [[Hamel dimension|dimension]] 4, whereas the complex numbers have dimension 2.
She lived in Mt Morris Township, near [[Flint, Michigan]], and was cared for by health care professionals, friends, her daughter Heather, and other family members, many of whom visited daily.
== Definition ==
Yager claimed to have started her massive weight gain deliberately as a child to discourage the sexual attacks of a "close family member," although in later interviews, she indicated that there were other contributing factors, or "skeletons in my closet", and "monsters" as she was quoted.
While the complex numbers are obtained by adding the element ''i'' to the real numbers which satisfies ''i''<sup>2</sup> = −1, the quaternions are obtained by adding the elements ''i'', ''j'' and ''k'' to the real numbers which satisfy the following relations.
In January, 1993, she was admitted to Hurley Medical Center, weighing-in at 1189 lbs. She suffered from [[cellulitis]] (her skin was breaking down due to the stress of holding in her mass). She stayed in the hospital for three months, where she was restricted to a 1200 [[calorie]] diet, and while there, lost 519 pounds, though most of this was fluid. (Massively obese people often suffer from [[edema]], and their weight can fluctuate with astonishing speed as fluid is taken up or released.) Yager sufferred from many other obesity-related health problems as well, including breathing difficulty, a dangerously high sugar level, and stress on her heart and other organs. Yager's death certificate lists kidney failure as the cause of death, with obesity and multiple organ failure as contributing causes.
:''i''<sup>2</sup> = ''j''<sup>2</sup> = ''k''<sup>2</sup> = ''ijk'' = −1
It took a lot of teamwork among as many as 15 - 20 fire fighters and ambulance workers to convey Yager to the [[ambulance]], in relay fashion. One team inside the house would pass her through the doorway to another team on the outside, who would in turn pass her off to another team inside the ambulance, where she would ride on the floor, for her many trips to the hospital (13 times in two years). Eventually, she was moved into the [[nursing home]] where she lived after leaving the hospital. She appeared on the [[Jerry Springer Show]], and was the subject of attention from several [[dieting]] gurus.
Every quaternion is a real [[linear combination]] of the '''unit quaternions''' 1, ''i'', ''j'', and ''k'', i.e. every quaternion is uniquely expressible in the form ''a'' + ''bi'' + ''cj'' + ''dk''.
A short time before her death, Yager's latest boyfriend, Larry Maxwell, who was characterized by her family as being 'an opportunist who courted media attention for money-making possibilities', married her friend, Felicia White. Maxwell had claimed that the only donation in Yager's name he ever received was for $20.00, although numerous talk shows, newspapers, radio stations, and other national and international media are reported to have offered her cash and other gifts in exchange for interviews, pictures, etc. Diet maven [[Richard Simmons]] is said to have been 'angry that Yager's story was actively peddled to tabloid and television media by Maxwell and others'.
Addition of quaternions is accomplished by adding corresponding coefficients, as with the complex numbers. By linearity, multiplication of quaternions is completely determined by the [[multiplication table]] for the unit quaternions; this table is given below.
Yager was buried privately, with about 90 friends and family members attending memorial services.
<table border cellspacing="0" cellpadding="5" bgcolor="#DDEEFF" >
<tr>
<td align="center" bgcolor="#FFFFFF">·
<td align="center" bgcolor="#EEEEEE">1
<td align="center" bgcolor="#EEEEEE">i
<td align="center" bgcolor="#EEEEEE">j
<td align="center" bgcolor="#EEEEEE">k
<tr>
<td align="center" bgcolor="#EEEEEE">1
<td align="center">1
<td align="center">i
<td align="center">j
<td align="center">k
<tr>
<td align="center" bgcolor="#EEEEEE">i
<td align="center">i
<td align="center">−1
<td align="center">k
<td align="center">−j
<tr>
<td align="center" bgcolor="#EEEEEE">j
<td align="center">j
<td align="center">−k
<td align="center">−1
<td align="center">i
<tr>
<td align="center" bgcolor="#EEEEEE">k
<td align="center">k
<td align="center">j
<td align="center">−i
<td align="center">−1
</table>
Under this multiplication, the unit quaternions form the [[quaternion group]] of order 8, ''Q''<sub>8</sub>.
== Example ==
Let
:''x'' = 3 + ''i''
:''y'' = 5''i'' + ''j'' − 2''k''
Then
:''x'' + ''y'' = 3 + 6''i'' + ''j'' − 2''k''
:''xy'' = (3 + ''i'')(5''i'' + ''j'' − 2''k'')
::= 15''i'' + 3''j'' − 6''k'' + 5''i''<sup>2</sup> + ''ij'' − 2''ik''
::= 15''i'' + 3''j'' − 6''k'' − 5 + ''k'' + 2''j''
::= − 5 + 15''i'' + 5''j'' − 5''k''
== Properties ==
Unlike real or complex numbers, multiplication of quaternions is not [[commutative]]: e.g. ''ij'' = ''k'', ''ji'' = −''k'', ''jk'' = ''i'', ''kj'' = −''i'', ''ki'' = ''j'', ''ik'' = −''j''. The quaternions are an example of a [[division ring]], an algebraic structure similar to a [[field (mathematics)|field]] except for commutativity of multiplication. In particular, multiplication is still [[associative]] and every non-zero element has a unique inverse.
Quaternions form a 4-dimensional [[associative algebra]] over the reals (in fact a [[division algebra]]) and contain the complex numbers, but they do not form an associative algebra over the complex numbers. The quaternions, along with the complex and real numbers, are the only finite-dimensional associative division algebras over the field of real numbers. The non-commutativity of multiplication has some unexpected consequences, among them that [[polynomial]] equations over the quaternions can have more distinct solutions than the degree of the polynomial.
The equation ''z''<sup>2</sup> + 1 = 0, for instance, has the infinitely many quaternion solutions ''z'' = ''bi'' + ''cj'' + ''dk'' with ''b''<sup>2</sup> + ''c''<sup>2</sup> + ''d''<sup>2</sup> = 1. The ''conjugate'' of the quaternion ''z'' = ''a'' + ''bi'' + ''cj'' + ''dk'' is defined as
<center>
''z''<sup>*</sup> = ''a'' − ''bi'' − ''cj'' − ''dk''
</center>
and the ''absolute value'' of ''z'' is the non-negative real number defined by
<center>
<math>|z| = \sqrt{z\times{}z^*} = \sqrt{a^2+b^2+c^2+d^2}</math>
</center>
Note that (''wz'')<sup>*</sup>= ''z''<sup>*</sup>''w''<sup>*</sup>, which is not in general equal to ''w''<sup>*</sup>''z''<sup>*</sup>. The multiplicative inverse of the non-zero quaternion ''z'' can be conveniently computed as ''z''<sup>−1</sup> = ''z''<sup>*</sup> / |''z''|<sup>2</sup>.
By using the [[distance function]] ''d''(''z'', ''w'') = |''z'' − ''w''|, the quaternions form a [[metric space]] (isometric to the usual Euclidean metric on '''R'''<sup>4</sup>) and the arithmetic operations are continuous. We also have |''zw''| = |''z''| |''w''| for all quaternions ''z'' and ''w''. Using the absolute value as norm, the quaternions form a real [[Banach algebra]].
==Group rotation==
As is explained in more detail in [[quaternions and spatial rotation]], the multiplicative group of non-zero quaternions acts by conjugation on the copy of '''R'''<sup>3</sup> consisting of quaternions with real part equal to zero. The conjugation by a unit quaternion (a quaternion of absolute value 1) with real part cos(''t'') is a rotation by an angle 2''t'', the axis of the rotation being the direction of the imaginary part. The advantages of Quaternions are:
# Non singular representation (compared with [[Euler angles]] for example)
# More compact (and faster) than [[matrix (mathematics)|matrices]]
# Pairs of unit quaternions can represent a rotation in [[4d]] space.
The set of all unit quaternions forms a [[3-sphere|3-dimensional sphere]] ''S''<sup>3</sup> and a [[group (mathematics)|group]] (a [[Lie group]]) under multiplication. ''S''<sup>3</sup> is the double cover of the group ''SO''(3,'''R''') of real orthogonal 3×3 [[matrix|matrices]] of [[determinant]] 1 since ''two'' unit quaternions correspond to every rotation under the above correspondence. The group ''S''<sup>3</sup> is isomorphic to ''SU''(2), the group of complex [[unitary matrix|unitary]] 2×2 matrices of [[determinant]] 1. Let ''A'' be the set of quaternions of the form ''a'' + ''bi'' + ''cj'' + ''dk'' where ''a'', ''b'', ''c'' and ''d'' are either all [[integer]]s or all [[rational number]]s with odd numerator and denominator 2. The set ''A'' is a [[ring (mathematics)|ring]] and a [[lattice]]. There are 24 unit quaternions in this ring, and they are the vertices of a [[24-cell|24-cell regular polytope]] with [[Schläfli symbol]] {3,4,3}.
== Representing quaternions by matrices ==
There are at least two ways of representing quaternions as [[matrix (mathematics)|matrices]], in such a way that quaternion addition and multiplication correspond to matrix addition and [[matrix multiplication]] (i.e., quaternion-matrix [[homomorphism]]s).
One is to use 2×2 [[complex number|complex]] matrices, and the other is to use 4×4 [[real number|real]] matrices.
In the first way, the quaternion ''a'' + ''bi'' + ''cj'' + ''dk'' is represented as
: <math>\begin{pmatrix} a-di & -b+ci \\ b+ci & \;\; a+di \end{pmatrix}</math>
This representation has several nice properties.
* All [[complex number]]s (''c'' = ''d'' = 0) correspond to matrices with only real entries.
* The square of the absolute value of a quaternion is the same as the [[determinant]] of the corresponding matrix.
* The conjugate of a quaternion corresponds to the [[conjugate transpose]] of the matrix.
* Restricted to unit quaternions, this representation provides the [[group isomorphism|isomorphism]] between [[3-sphere|''S''<sup>3</sup>]] and SU(2). The latter group is important in [[quantum mechanics]] when dealing with [[spin (physics)|spin]]; see also [[Pauli matrices]].
In the second way, the quaternion ''a'' + ''bi'' + ''cj'' + ''dk'' is represented as
: <math>\begin{pmatrix}
\;\; a & -b & \;\; d & -c \\
\;\; b & \;\; a & -c & -d \\
-d & \;\; c & \;\; a & -b \\
\;\; c & \;\; d & \;\; b & \;\; a
\end{pmatrix}</math>
In this representation, the conjugate of a quaternion corresponds to the [[transpose]] of the matrix.
== History ==
Quaternions were discovered by [[William Rowan Hamilton]] of [[Ireland]] in [[1843]]. Hamilton was looking for ways of extending [[complex number]]s (which can be viewed as [[point]]s on a [[plane]]) to higher spatial dimensions. He could not do so for 3-dimensions, but 4-dimensions produce quaternions. According to the story Hamilton told, on [[October 16]] Hamilton was out walking along the Royal Canal in [[Dublin]] with his wife when the solution in the form of the equation
<center>
'''<i>i</i><sup>2</sup> = <i>j</i><sup>2</sup> = <i>k</i><sup>2</sup> = <i>ijk</i> = -1 '''
</center>
suddenly occurred to him; Hamilton then promptly carved this equation into the side of the nearby Brougham Bridge (now called Broom Bridge). This involved abandoning the commutative law, a radical step for the time. Vector algebra and matrices were still in the future.
Not only this, but Hamilton had in a sense invented the cross and dot products of vector algebra. Hamilton also described a quaternion as an ordered quadruple (4-tuple) of real numbers, and described the first coordinate as the 'scalar' part, and the remaining three as the 'vector' part. If two quaternions with zero scalar parts are multiplied, the scalar part of the product is the negative of the dot product of the vector parts, while the vector part of the product is the cross product. But the significance of these was still to be discovered. Hamilton proceeded to popularize quaternions with several books, the last of which, ''Elements of Quaternions'', had 800 pages and was published shortly after his death.
===Use controversy ===
Even by this time there was controversy about the use of quaternions. Some of Hamilton's supporters vociferously opposed the growing fields of [[vector algebra]] and [[vector calculus]] (developed by [[Oliver Heaviside]] and [[Willard Gibbs]] among others), maintaining that quaternions provided a superior notation. While this is debatable in three dimensions, quaternions cannot be used in other dimensions (though extensions like [[octonion]]s and [[Clifford algebra]]s may be more applicable).
Vector notation has nearly universally replaced quaternions in [[science]] and [[engineering]] by the mid-20th century.
=== Recent years ===
Quaternions see use in [[computer graphics]], [[control theory]], [[signal processing]], [[attitude control]], [[physics]], and [[orbital mechanics]], mainly for representing rotations/orientations in three dimensions. For example, it is common for spacecraft attitude-control systems to be commanded in terms of quaternions, which are also used to telemeter their current attitude. The rationale is that combining many quaternion transformations is more numerically stable than combining many matrix transformations.
Since 1989, the [[National University of Ireland, Maynooth]] has organized a pilgrimage, where mathematicians (including [[Murray Gell-Mann]] in 2002 and [[Andrew Wiles]] in 2003) take a walk from [[Dunsink observatory]] to the Royal Canal bridge where, unfortunately no trace of the Hamilton's carving remains.
== Generalizations ==
If ''F'' is any [[field (mathematics)|field]] and ''a'' and ''b'' are elements of ''F'', one may define a four-dimensional unitary [[associative algebra]] over ''F'' by using two generators ''i'' and ''j'' and the relations ''i''<sup>2</sup> = ''a'', ''j''<sup>2</sup> = ''b'' and ''ij'' = −''ji''. These algebras are either isomorphic to the algebra of 2×2 [[matrix (mathematics)|matrices]] over ''F'', or they are [[division algebra]]s over ''F''. They are called [[quaternion algebra]]s.
== See also ==
* [[List of the most obese humans]]
== Sources ==
* [[Complex number]]
* [http://www.dimensionsmagazine.com/dimtext/kjn/people/heaviest.htm Dimensions Magazine, people known to have weighed more than 900 pounds]
* [[Octonion]]
* ''Bizarre'' magazine 64, p. 81
* [[Hypercomplex number]]
* [http://www.mlive.com/fljournal/ The Flint Journal]
* [[Division algebra]]
* ''The Flint [Michigan] Journal'', Wednesday, August 18, 1993, page A1, "Weight loss brings star status" by Mike Stobbe (Journal health writer)
* [[Associative algebra]]
* ''The Flint Journal'', Tuesday, May 24, 1994, page C1, "Obese woman's losing bid to lose hits TV show"
* [[Quarternion (Gubaidulina)|Quarternion]] by [[Sofia Gubaidulina]]
* ''The Flint Journal'', Friday, June 17, 1994, page A1, "What next for 1,200-pound woman?" by Marcia Mattson (Journal staff writer)
* ''The Flint Journal'', Tuesday, July 19, 1994, page A1, "1,200-lb Woman dies" by Marcia Mattson
== Related resources ==
* ''The Flint Journal'', Wednesday, July 20, 1994, page B1, "Richard Simmons mourns Yager" by Marcia Mattson
* ''The Flint Journal'', Sunday, July 24, 1994, page B1, "1,200-lb. woman more than curiosity" by Ken Palmer (Journal staff writer)
* [http://world.std.com/~sweetser/quaternions/qindex/qindex.html Doing Physics with Quaternions]
* ''The Flint Journal'', Monday, July 25, 1994, page A6, "Americans must work harder to overcome weight problems"
* [http://theworld.com/~sweetser/java/qcalc/qcalc.html Quaternion Calculator] [Java]
* [http://arxiv.org/pdf/math-ph/0201058 The Physical Heritage of Sir W. R. Hamilton] (PDF)
* Kuipers, Jack (2002). ''Quaternions and Rotation Sequences: A Primer With Applications to Orbits, Aerospace, and Virtual Reality'' (Reprint edition). Princeton University Press. ISBN 0691102988
[[Category:Number]]
[[Category:Vector spaces]]
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[[zh-cn:四元數]]
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[[Category:World record holders|Yager, Carol]]
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[[Category:Obesity|Yager, Carol]]
[[Category:1960 births|Yager, Carol]]
[[Category:1994 deaths|Yager, Carol]]
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