Quaternion

Graphical representation of quaternion units product as 90°-rotation in 4D-space, ij = k, ji = −k, ij = −ji

Quaternions, in mathematics, are a non-commutative extension of complex numbers. They were first described by the Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. They find uses in both theoretical and applied mathematics, in particular for calculations involving three-dimensional rotations, such as in 3D computer graphics, although they have been superseded in many applications by vectors and matrices.

In modern language, quaternions form a 4-dimensional normed division algebra over the real numbers. The algebra of quaternions is often denoted by H (for Hamilton), or in blackboard bold by (Unicode ℍ). It can also be given by the Clifford algebra classifications Cℓ_0,2(R) = Cℓ⁰_3,0(R). The algebra H holds a special place in analysis since, according to the Frobenius theorem, it is one of only three finite-dimensional division rings containing the real numbers as a subring.

Hamilton product

For two elements a₁ + b₁i + c₁j + d₁k and a₂ + b₂i + c₂j + d₂k, their Hamilton product (a₁ + b₁i + c₁j + d₁k)(a₂ + b₂i + c₂j + d₂k) is determined by the products of the basis elements and the distributive law. The distributive law makes it possible to expand the product so that it is a sum of products of basis elements. This gives the following expression:

a₁a₂ + a₁b₂i + a₁c₂j + a₁d₂k + b₁a₂i + b₁b₂i² + b₁c₂ij + b₁d₂ik + c₁a₁j + c₁b₂ji + c₁c₂j² + c₁d₂jk + d₁a₁k + d₁b₂ki + d₁c₂kj + d₁d₂k².

Now the basis elements can be multiplied using the rules given above to get:

(a₁a₂ − b₁b₂ − c₁c₂ − d₁d₂) + (a₁b₂ + b₁a₂ + c₁d₂ − d₁c₂)i + (a₁c₂ − b₁d₂ + c₁a₂ + d₁b₂)j + (a₁d₂ + b₁c₂ − c₁b₂ + d₁a₂)k.

Ordered list form

Using the basis 1, i, j, k of H makes it possible to write H as a set of quadruples:

Then the basis elements are:

1 = (1,0,0,0),

i = (0,1,0,0),

j = (0,0,1,0),

k = (0,0,0,1),

and the formulas for addition and multiplication are:

(a₁,b₁,c₁,d₁) + (a₂,b₂,c₂,d₂) = (a₁ + a₂,b₁ + b₂,c₁ + c₂,d₁ + d₂).

(a₁,b₁,c₁,d₁)(a₂,b₂,c₂,d₂) = (a₁a₂ − b₁b₂ − c₁c₂ − d₁d₂,a₁b₂ + b₁a₂ + c₁d₂ − d₁c₂,a₁c₂ − b₁d₂ + c₁a₂ + d₁b₂,a₁d₂ + b₁c₂ − c₁b₂ + d₁a₂).

 

 

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