The ring of quaternions, discovered by W.R. Hamilton provides the most often cited example of a skew field or division ring. It consists of all possible sums of the form:

a1 + bi + cj + dk

where a,b,c and d are real numbers and 1,i,j and k are members of the quaternionic group. For concreteness we can take 1,i,j and k to be 4 x 4 matrices:

1 = 1  0  0  0     i = 0 -1  0  0     j = 0  0 -1  0     k = 0  0  0 -1
0  1  0  0          1  0  0  0           0  0  0  1            0  0 -1  0
0  0  1  0          0  0  0 -1          1  0  0  0            0  1  0   0
0  0  0  1          0  0  1  0           0 -1 0  0            1  0  0  0

--back to combinatorics--
Quaternions with unit norm are also useful in representing rotations around the origin in 3D. The unit norm restriction is necessary because rotations in 3D have only 3 degrees of freedom (whereas quaternions have 4).

Quaternions possess the following advantages over 3×3 rotation matricies in representing 3D rotations:

One can easily convert unit-norm quaternions to rotation matricies. Given that q = [ a b c d ] is a unit quaternion (i.e. a2 + b2 + c2 + d2 = 1), the rotation matrix representing that quaternion is denoted M(q) and is defined by

```       | a2+b2-c2-d2  2bc-2ad    2bd+2ac   |
M(q) = |  2bc+2ad   a2-b2+c2-d2  2cd-2ab   |
|  2bd-2ac    2cd+2ab   a2-b2-c2+d2 |
```
The composition of two rotations represented by the unit-norm quaternion p followed by q is represented by the quaternion product qp. In other words, the result of a 3-vector v rotated by a quaternion p and then rotated by quaternion q is
M(q)(M(p)v) = (M(q)M(p))v = M(qp)v.
This means that instead of multiplying two 3×3 matricies to get the composite rotation, you can just multiply the two quaternions.

Two interpolate between two quaternions, you must interpolate linearly over the surface of the unit hypersphere in four dimensions. This process is called spherical linear interpolation. The quaternion so interpolated t of the way from p to q is given by the vector r using the formula

r = (psin((1-t)φ) + qsin(tφ)) / sin(φ).
Here 0<t<1, and cos(φ) = p·q (regular old dot product between two 4-vectors). φ represents the angle between the 4-vectors p and q.

Finally, one can obtain the angle θ and axis u of rotation from the unit norm quaternion q using the following formulae:

cos(θ/2) = a

sin(θ/2) = (b2 + c2 + d2)1/2

u = [ b c d ] / (b2 + c2 + d2)1/2

The Quaternions denoted H are a division ring or skew field. This means that you can add,subtract,multiply and divide. Notice though that multiplication is non-commutative.

They were invented by Hamilton in 1843 who was so pleased that he scratched the defining relations on Brougham Bridge on the Royal Canal in Dublin.

As a real vector space the Quaternions have basis 1,i,j,k and the mutiplication can be deduced from the rules

```i2=j2=k2=-1,  ij=-ji=k, jk=-kj=i,  ki=-ik=j.
```
For each quaternion q=t+xi+yj+zk in H we can define b(q)=t-xi-yj-zk. It's easy to see that
```qb(q)=t2+x2+y2+z2
```

Note that if q is nonzero, so that one of t,x,y,z is nonzero, then qb(q) is a nonzero real number. It follows that such a q has inverse

```b(q)/(t2+x2+y2+z2)
```
The quaternions have a concrete description as a subalgebra of 2x2 complex matrices. The quaternion q=t+xi+yj+zk corresponds to the matrix
```  --         --
| t+xi   y+zi |
| -y+zi  t-xi |
--         --
```

The subgroup of the group of units of H consisting of {1,-1,i,-i,j,-j, k,-k} is called the Quaternion group (or Pauli spin group). In this group of order 8 all of the elements except for 1 and -1 have order 4, with -1 having order 2.

Multiplying Quaternions

No-one has really explained how to do this yet. We'll be working with the two quaternions:
A = a + bi + cj + dk
E = e + fi + gj + hk
As has already been mentioned, multiplying quaternions is non-commutative, and multiplying the units is actually anti-commutative. This means we have to be extra careful. So then:

AE = (a + bi + cj + dk)(e + fi + gj + hk)

Now we have to laboriously expand these brackets, being very careful not to change the order of the units as we go:

ae + afi + agj + ahk
+bei + bfi² + bgij + bhik
+cej + cfji + cgj² + chjk
+dek + dfki + dgkj + dhk²

Currently we have a lot of unresolved units all over the place, so let's tidy them up using Hamilton's Laws:

ae + afi + agj + ahk
+bei - bf + bgk - bhj
+cej - cfk - cg + chi
+dek + dfj - dgi - dh

Lovely, right right? Now we just need to group together the terms with the same unit:

ae - bf - cg - dh
+i( af + be + ch - dg )
+j( ag - bh + ce + df )
+k( ah + bg - cf + de )

And there you have it.

Qua*ter"ni*on (?), n. [L. quaternio, fr.quaterni four each. See Quaternary.]

1.

The number four.

[Poetic]

2.

A set of four parts, things, or person; four things taken collectively; a group of four words, phrases, circumstances, facts, or the like.

Delivered him to four quaternions of soldiers. Acts xii. 4.

Ye elements, the eldest birth Of Nature's womb, that in quaternion run. Milton.

The triads and quaternions with which he loaded his sentences. Sir W. Scott.

3.

A word of four syllables; a quadrisyllable.

4. Math.

The quotient of two vectors, or of two directed right lines in space, considered as depending on four geometrical elements, and as expressible by an algebraic symbol of quadrinomial form.

⇒ The science or calculus of quaternions is a new mathematical method, in which the conception of a quaternion is unfolded and symbolically expressed, and is applied to various classes of algebraical, geometrical, and physical questions, so as to discover theorems, and to arrive at the solution of problems.

Sir W. R. Hamilton.

Qua*ter"ni*on, v. t.

To divide into quaternions, files, or companies.

Milton.