Dr. Sarah's Quaternion Demo

Quaternions are an extension of complex numbers.

Instead of just i, we have three different numbers that are all square roots of -1 labelled i, j, and k. The 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.

Applications of Quaternions

Beyond complex Numbers By Girish Joshi

I N NATURE, there is a deep connection between exceptional mathematical structures and the laws of micro- and macro-physics --- Quaternions
and Octonions have played an important role in the recent development of pure and applied physics.

Discovered by Hamilton in 1843 , Quaternions' main use in the 19th century
consisted in expressing physical theories in "Quaternionic notation". An important work
where this was done was
Maxwell's treatise on electricity and magnetism. Towards the end
of the century, the value of their use in electromagnetic theories led to a heated debate
dubbed "The Great Quaternionic War".

In a 1936 paper, Birkhoff and von Neumann presented a propositional calculus for
Quantum Mechanics and showed that a concrete realisation leads to the general result that a
Quantum Mechanical system may be represented as a vector space over the Real, Complex,
and Quaternionic fields. Since then this area has remained active, aiming to extend
Complex Quantum Mechanics (CQM) by generalising the complex unit in CQM to
Quaternions and to find observable effects of QQM. Jordan Algebras were proposed by
Jordan, Neumann and Wigner in formulating non-associative Quantum Mechanics, where
quantisation is achieved through associators rather than commutators. This formulation
allows mixing of space-time and internal symmetries. Another attractive feature of Jordan
Algebra is that critical dimensions of 10 and 26 arise naturally, suggesting a connection to
string theory .

Away from physics, Quaternions have recently been used for
robotic control , computer
graphics, vision theory, spacecraft orientation and geophysics
. The space shuttle's flight
software uses Quaternions in its
guidance navigation and flight control computations.

Quaternions are an Abelian Group Under Addition

To show that the quaternions H are closed under addition, let u,v be quaternions.

We will show that u+v is a quaternion.

By definition of H, we know there exist real a,b,c,d, e,f,g,h

so that u=a+bi+cj+dk, v=e+fi+gj+hk. Then,

Then, u+v= (I'm defining a procedure to add two quaternions)

> plus:=proc(a,b,c,d,e,f,g,h)

> a+e,b+f,c+g,d+h

> end;

Now plus is a procedure that inputs the coefficients of two quaternions and outputs the new coefficients of the sum of the two quaternions, with commas separating them:

> plus(a,b,c,d,e,f,g,h);

So u+v=(a+e) + (b+f)i + (c+g)j + (d+h)k. These coefficients are real since the sum of real numbers is still real. Thus u+v is a quaternion, and so the quaternions are closed under addition.

To show that the quaternions (H) are associative under addition,

let u,v, and w be in H be arbitrary. We must show that

(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

so that u=a+bi+cj+dk, v=e+fi+gj+hk, and w=q+ri+sj+tk. Then,

Then, (u+v)+w=

> plus(plus(a,b,c,d,e,f,g,h),q,r,s,t);

Hence (u+v)+w= (a + e + q) + ( b + f + r)i + ( c + g + s)j + (d + h + t)k

Also, u+(v+w)=

> plus(a,b,c,d,plus(e,f,g,h,q,r,s,t));

>

u+(v+w)= (a + e + q) + ( b + f + r)i + ( c + g + s)j + (d + h + t)k.

Since these are the same, we see that (u+v)+w=u+(v+w), as desired. Hence H satisfies the additive associative property.

To show the additive identity property , we must produce 0 in H so that for all u in H, 0+u=u.

So, take 0=0+0i+0j+0k, which is an element of H since 0 is real. Then, let u be in H.

By definition of H, we know there exist real a,b,c,d so that u=a+bi+cj+dk.

Then, 0+u=

> plus(0,0,0,0,a,b,c,b);

=a+bi+cj+dk, as desired. Hence H satisfies the additive identity property.

To show that the quaternions have additive inverse s, let u be in H. We must produce z in H so that

u+z=0.

By 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.

We see that z is an element of H since -a,-b,-c,-d are also real. Now,

u+z=

> plus(a,b,c,d,-a,-b,-c,-d);

=0+0i+0j+0k, as desired. Hence H satisfies the additive inverse property.

To show that H is abelian under addition , let u,v be in H. We must show that u+v=v+w.

By definition of H, we know there exist real a,b,c,d, e,f,g,h,

so that u=a+bi+cj+dk, and v=e+fi+gj+hk. Then,

Then, u+v =

> plus(a,b,c,d,e,f,g,h);

And v+u=

> plus(e,f,g,h,a,b,c,d);

Since these are equal, then we know that the quaternions are abelian under addition.

Hence the quaternions are an abelian group under addition

Multiplication of Quaternions

Mult is a procedure that inputs the coefficients of two quaternions and outputs the new coefficients of the product of the two quaternions, with commas separating them:

> mult:=proc(a,b,c,d,e,f,g,h)

> 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

> end;

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 product is k=(0,0,0,1)

> mult(0,1,0,0,0,0,1,0);

To show that i^2=-1, we'll show that i*i=-1=(-1,0,0,0)

> mult(0,1,0,0,0,1,0,0);

Write down similar Maple commands to show that the other 7 commands hold:

j^2=-1, k^2=-1, ji=-k , jk=i, kj=-i, ki=j, ik=-j.

>

>

>

>

Quaternions are a Ring

We have already shown that they are an abelian group under addition.

To show that the quaternions are closed under multiplication , let u,v be in H. We will show that u*v is in H. By definition of H, we know there exist real a,b,c,d, e,f,g,h, so that u=a+bi+cj+dk, v=e+fi+gj+hk. Now

u*v=

> mult(a,b,c,d,e,f,g,h);

Notice that each coefficient of 1,i,j,k is real since addition and multiplication of reals are real.

Hence u*v is an element of H, and so H is closed under addition.

To show that the quaternions are associative under multiplication, let u,v,w be in H. We will show that

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,

v=e+fi+gj+hk, and w=q+ri+sj+tk.

Then u(vw)=

> leftside:=mult(a,b,c,d,mult(e,f,g,h,q,r,s,t));

And (uv)w=

> rightside:=mult(mult(a,b,c,d,e,f,g,h),q,r,s,t);

The real coefficient difference

> leftside[1]-rightside[1];

> simplify(%);

The coefficient of i difference

> leftside[2]-rightside[2];

> simplify(%);

The coefficient of j difference

> leftside[3]-rightside[3];simplify(%);

The coefficient of k difference

> leftside[4]-rightside[4];simplify(%);

Hence we have proven that u(vw)=(uv)w, and so H is associative under multiplication.

To show that the quaternions are distributive under addition and multiplication , let u,v and w be in H. We must show that u(v+w)=uv+uw and (u+v)w=uw+vw. 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, v=e+fi+gj+hk, and w=q+ri+sj+tk.

Then, u(v+w)=

> leftside:=mult(a,b,c,d,plus(e,f,g,h,q,r,s,t));

> rightside:=plus(mult(a,b,c,d,e,f,g,h),mult(a,b,c,d,q,r,s,t));

The real coefficient difference

> leftside[1]-rightside[1];

> simplify(%);

The coefficient of i difference

> leftside[2]-rightside[2];

> simplify(%);

The coefficient of j difference

> leftside[3]-rightside[3];simplify(%);

The coefficient of k difference

> leftside[4]-rightside[4];simplify(%);

Hence we have proven that u(v+w)=uv+uw. The proof of (u+v)w=uw+vw is similar.

Therefore, H is a ring .

Quaternions are Not a Field (but are Almost a Field)

To show that H has a multiplicative unit, we must produce i in H so that i*u=u for all u in H.

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.

then u=a+bi+cj+dk for some real a,b,c,d. Now 1*u=

> mult(1,0,0,0,a,b,c,d);

=u, as desired. Hence H has a multiplicative unit.

To show that non-zero elements of H have an inverse , let u not 0 be an element of H.

We must produce an inverse q so that uq=1. Take q=(a-bi-cj-dk)/(a^2+b^2+c^2+d^2).

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

1,i,j,k are all real and so q is an element of H.

> 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));

> simplify(%[1]);

So, the real coefficient is 1, and the other coefficients are 0. Hence uq=1, as desired. Thus non-zero elements of H have inverses.

To show that H is not abelian, we must produce u,v in H so that uv is not equal to vu.

Finish this proof that H is not abelian.

>

>

>

>

So, H satisfies the structure of a field EXCEPT it is not abelian under multiplication.

Quaternions are Not an Integral Domain, but have no zero-divisors.

To show that the quaternions have no zero-divisors, assume for contradiction that the quaternions do have zero-divisors. Then by definition, we know there exists u,v non-zero in H so that uv= 0.

By definition of H, we know there exist real a,b,c,d, e,f,g,h,

so that u=a+bi+cj+dk, and v=e+fi+gj+hk. Then,

0=uv=

> mult(a,b,c,d,e,f,g,h);

Hence, each coefficient (of 1,i,j,and k) must be 0. Thus we have four conditions:

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

We will solve these conditions for a,b,c and d:

> 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});

Then, u=a+bi+cj+dk=0+0i+0j+0k, a contradiction to the fact that u was non-zero.

Therefore the quaternions do not have zero divisors, as desired.