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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 272 27 "Dr. Sarah'
s Quaternion Demo" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "Quaternions \+
are an extension of complex numbers. " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 220 "Instead of just i, we have three dif
ferent numbers that are all square roots of -1 labelled i, j, and k. T
he operations that define multiplication between them are i^2=j^2=k^2=
-1, ij=k, ji=-k ,jk=i, kj=-i, ki=j, ik=-j.\n " }}}{SECT 0 {PARA 3 "" 
0 "" {TEXT -1 27 "Applications of Quaternions" }}{SECT 0 {PARA 4 "" 0 
"" {TEXT -1 38 "Beyond complex Numbers By Girish Joshi" }}{PARA 4 "" 
0 "" {TEXT -1 1 "I" }{TEXT 256 238 "N NATURE, there is a deep connecti
on between exceptional mathematical structures and the laws of micro- \+
and macro-physics --- Quaternions\nand Octonions have played an import
ant role in the recent development of pure and applied physics.\n\n" }
{TEXT 267 30 "Discovered by Hamilton in 1843" }{TEXT 268 156 ", Quater
nions' main use in the 19th century\nconsisted in expressing physical \+
theories in \"Quaternionic notation\". An important work\nwhere this w
as done was " }{TEXT 265 48 "Maxwell's treatise on electricity and mag
netism." }{TEXT 266 226 " Towards the end\nof the century, the value o
f their use in electromagnetic theories led to a heated debate\ndubbed
 \"The Great Quaternionic War\".\n\nIn a 1936 paper, Birkhoff and von \+
Neumann presented a propositional calculus for\n" }{TEXT 270 18 "Quant
um Mechanics " }{TEXT 271 757 "and showed that a concrete realisation \+
leads to the general result that a\nQuantum Mechanical system may be r
epresented as a vector space over the Real, Complex,\nand Quaternionic
 fields. Since then this area has remained active, aiming to extend\nC
omplex Quantum Mechanics (CQM) by generalising the complex unit in CQM
 to\nQuaternions and to find observable effects of QQM. Jordan Algebra
s were proposed by\nJordan, Neumann and Wigner in formulating non-asso
ciative Quantum Mechanics, where\nquantisation is achieved through ass
ociators rather than commutators. This formulation\nallows mixing of s
pace-time and internal symmetries. Another attractive feature of Jorda
n\nAlgebra is that critical dimensions of 10 and 26 arise naturally, s
uggesting a connection to\n" }{TEXT 269 13 "string theory" }{TEXT 264 
62 ".\n\nAway from physics, Quaternions have recently been used for " 
}{TEXT 260 15 "robotic control" }{TEXT 261 2 ", " }{TEXT 262 71 "compu
ter\ngraphics, vision theory, spacecraft orientation and geophysics" }
{TEXT 263 6 ". The " }{TEXT 257 22 "space shuttle's flight" }{TEXT 
258 34 "\nsoftware uses Quaternions in its " }{TEXT 259 52 "guidance n
avigation and flight control computations." }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 47 "Quaternions are an Abe
lian Group Under Addition" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 273 41 "To \+
show that the quaternions H are closed" }{TEXT -1 1 " " }{TEXT 275 15 
"under addition," }{TEXT -1 26 " let u,v be quaternions.  " }}{PARA 0 
"" 0 "" {TEXT -1 38 "We will show that u+v is a quaternion." }}{PARA 
0 "" 0 "" {TEXT -1 61 "By definition of H, we know there exist real a,
b,c,d, e,f,g,h" }}{PARA 0 "" 0 "" {TEXT -1 43 "so that u=a+bi+cj+dk, v
=e+fi+gj+hk.  Then, " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "Then, u+v
=  (I'm defining a procedure to add two quaternions)" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 27 "plus:=proc(a,b,c,d,e,f,g,h)" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 21 "      a+e,b+f,c+g,d+h" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 4 "end;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 169 "Now plus
 is a procedure that inputs the coefficients of two quaternions  and o
utputs the new coefficients of the sum of the two quaternions, with co
mmas separating them:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "pl
us(a,b,c,d,e,f,g,h);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 191 "So u+v=(
a+e) + (b+f)i + (c+g)j + (d+h)k.  These coefficients are real since th
e sum of real numbers is still real.  Thus u+v is a quaternion, and so
 the quaternions are closed under addition." }}}{EXCHG {PARA 256 "" 0 
"" {TEXT -1 64 "To show that the quaternions (H) are associative under
 addition," }}{PARA 0 "" 0 "" {TEXT -1 56 " let u,v, and w be in H be \+
arbitrary.  We must show that" }}{PARA 0 "" 0 "" {TEXT -1 90 "(u+v)+w \+
= u+(v+w).  By definition of H, we know there exist real a,b,c,d, e,f,
g,h, q,r,s,t" }}{PARA 0 "" 0 "" {TEXT -1 61 "so that u=a+bi+cj+dk, v=e
+fi+gj+hk, and w=q+ri+sj+tk.  Then, " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 14 "Then, (u+v)+w=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "pl
us(plus(a,b,c,d,e,f,g,h),q,r,s,t);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 90 "\n     Hence (u+v)+w=         (a + e + q)   + ( b + f + r)i + (
 c + g + s)j + (d + h + t)k\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "
Also, u+(v+w)=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "plus(a,b,
c,d,plus(e,f,g,h,q,r,s,t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "      u+(v+w)=      (a + e
 + q)   + ( b + f + r)i + ( c + g + s)j + (d + h + t)k." }}{PARA 0 "" 
0 "" {TEXT -1 120 "Since these are the same, we see that (u+v)+w=u+(v+
w), as desired.  Hence H satisfies the additive associative property.
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 274 38 "To show the additive identit
y property" }{TEXT -1 55 ", we must produce 0 in H so that for all u i
n H, 0+u=u." }}{PARA 0 "" 0 "" {TEXT -1 86 "So, take 0=0+0i+0j+0k, whi
ch is an element of H since 0 is real.  Then, let u be in H." }}{PARA 
0 "" 0 "" {TEXT -1 74 "By definition of H, we know there exist real a,
b,c,d so that u=a+bi+cj+dk." }}{PARA 0 "" 0 "" {TEXT -1 10 "Then, 0+u=
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plus(0,0,0,0,a,b,c,b);
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "=a+bi+cj+dk, as desired.  Hen
ce H satisfies the additive identity property." }}}{EXCHG {PARA 0 "" 
0 "" {TEXT 286 50 "To show that the quaternions have additive inverse
" }{TEXT -1 49 "s, let u be in H.  We must produce z in H so that" }}
{PARA 0 "" 0 "" {TEXT -1 8 "u+z=0.  " }}{PARA 0 "" 0 "" {TEXT -1 98 "B
y definition of H, we know there exist real a,b,c,d so that u=a+bi+cj+
dk.  Take z=-a -bi -cj -dk." }}{PARA 0 "" 0 "" {TEXT -1 72 "We see tha
t z is an element of H since -a,-b,-c,-d are also real.  Now, " }}
{PARA 0 "" 0 "" {TEXT -1 4 "u+z=" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "plus(a,b,c,d,-a,-b,-c,-d);" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 74 "=0+0i+0j+0k, as desired.  Hence H satisfies the additiv
e inverse property." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 276 40 "To show t
hat H is abelian under addition" }{TEXT -1 48 ", let u,v be in H.  We \+
must show that u+v=v+w.  " }}{PARA 0 "" 0 "" {TEXT -1 63 "By definitio
n of H, we know there exist real a,b,c,d, e,f,g,h, " }}{PARA 0 "" 0 "
" {TEXT -1 47 "so that u=a+bi+cj+dk, and v=e+fi+gj+hk.  Then, " }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Then, u+v = " }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 22 "plus(a,b,c,d,e,f,g,h);" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 8 "And v+u=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
22 "plus(e,f,g,h,a,b,c,d);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "Sin
ce these are equal, then we know that the quaternions are abelian unde
r addition." }}}{EXCHG {PARA 261 "" 0 "" {TEXT -1 0 "" }{TEXT 277 57 "
Hence the quaternions are an abelian group under addition" }}}{SECT 0 
{PARA 3 "" 0 "" {TEXT -1 29 "Multiplication of Quaternions" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 168 "Mult is a procedure that inputs the coef
ficients of two quaternions and outputs the new coefficients of the pr
oduct of the two quaternions, with commas separating them:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "mult:=proc(a,b,c,d,e,f,g,h)" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "      a*e-b*f-c*g-d*h,a*f+e*b+c*h-d
*g,a*g-b*h+c*e+d*f,a*h+b*g-c*f+d*e" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
4 "end;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "To show that ij=k, we
 will look at i=(0,1,0,0) and j=(0,0,1,0)  We must show that the produ
ct is k=(0,0,0,1)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "mult(0
,1,0,0,0,0,1,0);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "To show that \+
i^2=-1, we'll show that i*i=-1=(-1,0,0,0)" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 22 "mult(0,1,0,0,0,1,0,0);" }}}{EXCHG {PARA 258 "" 0 "
" {TEXT -1 73 "Write down similar Maple commands to show that the othe
r 7 commands hold:" }}{PARA 259 "" 0 "" {TEXT 292 54 "j^2=-1, k^2=-1, \+
 ji=-k ,  jk=i,  kj=-i,  ki=j,  ik=-j." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 
0 "" {TEXT -1 22 "Quaternions are a Ring" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 70 "We have already shown that they are an abelian group unde
r addition.  " }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 60 "To show that t
he quaternions are closed under multiplication" }{TEXT 287 156 ", let \+
u,v be in H.  We will show that u*v is in H.  By definition of H, we k
now there exist real a,b,c,d, e,f,g,h, so that u=a+bi+cj+dk, v=e+fi+gj
+hk.  Now " }}{PARA 0 "" 0 "" {TEXT -1 4 "u*v=" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 22 "mult(a,b,c,d,e,f,g,h);" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 102 "Notice that each coefficient of 1,i,j,k is real sin
ce addition and multiplication of reals are real.  " }}{PARA 0 "" 0 "
" {TEXT -1 64 "Hence u*v is an element of H, and so H is closed under \+
addition." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 280 66 "To s
how that the quaternions are associative under multiplication," }
{TEXT -1 39 " let u,v,w be in H.  We will show that " }}{PARA 0 "" 0 "
" {TEXT -1 107 "u(vw)=(uv)w.  By definition of H, we know there exist \+
real a,b,c,d, e,f,g,h, q,r,s,t so that u=a+bi+cj+dk, " }}{PARA 0 "" 0 
"" {TEXT -1 32 "v=e+fi+gj+hk, and w=q+ri+sj+tk. " }}{PARA 0 "" 0 "" 
{TEXT -1 11 "Then u(vw)=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 
"leftside:=mult(a,b,c,d,mult(e,f,g,h,q,r,s,t));" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 10 "And (uv)w=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 47 "rightside:=mult(mult(a,b,c,d,e,f,g,h),q,r,s,t);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 31 "The real coefficient difference" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "leftside[1]-rightside[1];" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 31 "The coefficient of i difference" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "leftside[2]-rightside[2];" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 31 "The coefficient of j difference" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 37 "leftside[3]-rightside[3];simplify(%);" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "The coefficient of k difference" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "leftside[4]-rightside[4];
simplify(%);" }}}{PARA 0 "" 0 "" {TEXT -1 86 "Hence we have proven tha
t u(vw)=(uv)w, and so H is associative under multiplication.  " }}
{EXCHG {PARA 0 "" 0 "" {TEXT 278 79 "To show that the quaternions are \+
distributive under addition and multiplication" }{TEXT -1 201 ", let u
,v and w be in H.  We must show that u(v+w)=uv+uw and (u+v)w=uw+vw.  B
y definition of H, we know there exist real a,b,c,d, e,f,g,h, q,r,s,t \+
so that u=a+bi+cj+dk, v=e+fi+gj+hk, and w=q+ri+sj+tk. " }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 13 "Then, u(v+w)=" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 46 "leftside:=mult(a,b,c,d,plus(e,f,g,h,q,r,s,t));" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "rightside:=plus(mult(a,b,c,d
,e,f,g,h),mult(a,b,c,d,q,r,s,t));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
31 "The real coefficient difference" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "leftside[1]-rightside[1];" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 12 "simplify(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
31 "The coefficient of i difference" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "leftside[2]-rightside[2];" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "simplify(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "
The coefficient of j difference" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 37 "leftside[3]-rightside[3];simplify(%);" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 31 "The coefficient of k difference" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 37 "leftside[4]-rightside[4];simplify(%);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "Hence we have proven that u(v+w)=u
v+uw.  The proof of (u+v)w=uw+vw is similar.  " }}}{PARA 0 "" 0 "" 
{TEXT 279 22 "Therefore, H is a ring" }{TEXT -1 1 "." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 52 "Quaternions are
 Not a Field (but are Almost a Field)" }}}{EXCHG {PARA 260 "" 0 "" 
{TEXT -1 41 "To show that H has a multiplicative unit," }{TEXT 281 52 
"we must produce i in H so that i*u=u for all u in H." }}{PARA 0 "" 0 
"" {TEXT -1 111 "So, take i=1=1+0i+0j+0k.  Notice that 1 is in H since
 the coefficients of 1,i,j,k are real.  So, let u be in H." }}{PARA 0 
"" 0 "" {TEXT -1 50 "then u=a+bi+cj+dk for some real a,b,c,d.  Now 1*u
=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "mult(1,0,0,0,a,b,c,d);
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "=u, as desired.  Hence H has \+
a multiplicative unit." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 282 51 "To sho
w that non-zero elements of H have an inverse" }{TEXT -1 35 ", let u n
ot 0 be an element of H.  " }}{PARA 0 "" 0 "" {TEXT -1 86 "We must pro
duce an inverse q so that uq=1.     Take q=(a-bi-cj-dk)/(a^2+b^2+c^2+d
^2). " }}{PARA 0 "" 0 "" {TEXT -1 98 "Notice that (a^2+b^2+c^2+d^2) is
 not 0 since a,b,c, and d are not all 0.  Then the coefficients of" }}
{PARA 0 "" 0 "" {TEXT -1 51 "1,i,j,k are all real and so q is an eleme
nt of H.  " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "mult(a,b,c,d,
a/(a^2+b^2+c^2+d^2),-b/(a^2+b^2+c^2+d^2),-c/(a^2+b^2+c^2+d^2),-d/(a^2+
b^2+c^2+d^2));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "simplify(
%[1]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 133 "So, the real coefficie
nt is 1, and the other coefficients are 0.  Hence uq=1, as desired.  T
hus non-zero elements of H have inverses." }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }{TEXT 283 31 "To show that H is not abelian, " }{TEXT 
-1 55 "we must produce u,v in H so that uv is not equal to vu." }}
{PARA 266 "" 0 "" {TEXT -1 0 "" }{TEXT 284 40 "Finish this proof that \+
H is not abelian." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 262 "" 0 "" {TEXT 285 88 "So, H satisfies the structure o
f a field EXCEPT it is not abelian under multiplication. " }}}{SECT 0 
{PARA 3 "" 0 "" {TEXT -1 66 "Quaternions are Not an Integral Domain, b
ut have no zero-divisors." }}}{EXCHG {PARA 263 "" 0 "" {TEXT -1 51 "To
 show that the quaternions have no zero-divisors," }{TEXT 288 144 " as
sume for contradiction that the quaternions do have zero-divisors.  Th
en by definition, we know there exists u,v non-zero in H so that uv= 0
." }}{PARA 265 "" 0 "" {TEXT 291 62 "By definition of H, we know there
 exist real a,b,c,d, e,f,g,h," }}{PARA 0 "" 0 "" {TEXT -1 47 "so that \+
u=a+bi+cj+dk, and v=e+fi+gj+hk.  Then, " }}{PARA 0 "" 0 "" {TEXT -1 5 
"0=uv=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "mult(a,b,c,d,e,f,
g,h);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "Hence, each coefficient \+
(of 1,i,j,and k) must be 0.  Thus we have four conditions:" }}{PARA 0 
"" 0 "" {TEXT -1 18 "a*e-b*f-c*g-d*h=0," }}{PARA 0 "" 0 "" {TEXT -1 
18 "a*f+e*b+c*h-d*g=0," }}{PARA 0 "" 0 "" {TEXT -1 18 "a*g-b*h+c*e+d*f
=0," }}{PARA 0 "" 0 "" {TEXT -1 17 "a*h+b*g-c*f+d*e=0" }}{PARA 0 "" 0 
"" {TEXT -1 47 "We will solve these conditions for a,b,c and d:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "solve(\{a*e-b*f-c*g-d*h=0,a*
f+e*b+c*h-d*g=0,a*g-b*h+c*e+d*f=0,a*h+b*g-c*f+d*e=0\},\{a,b,c,d\});" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "Then, u=a+bi+cj+dk=0+0i+0j+0k, a
 contradiction to the fact that u was non-zero." }}{PARA 0 "" 0 "" 
{TEXT -1 64 "Therefore the quaternions do not have zero divisors, as d
esired." }}}}{MARK "0 0 1" 13 }{VIEWOPTS 1 1 0 1 1 1803 }