# Quaternions

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## Quaternions

a number system proposed in 1843 by the British scientist W. Hamilton.

Quaternions were the result of attempts to generalize the complex numbers *x*+*iy,* where *x* and *y* are real numbers and *i* is a basis element satisfying the condition *i*^{2} = — 1. As is well know, complex numbers are depicted geometrically as points in a plane, and operations on them correspond to simple geometrical transformations of the plane (translation, rotation, dilation, or contraction, and their combinations). The search for a number system that would be geometrically realizable with the aid of points in three-dimensional space led to the discovery that it is impossible to “construct” from the points of a space of three or more dimensions a number system in which algebraic operations retain all the properties of addition and multiplication of real or complex numbers. However, if one of the properties— commutativity of multiplication—is dispensed with while the remaining properties of addition and multiplication are retained, then it is possible to construct a number system out of points of a space of four dimensions. (It is impossible to construct even such a number system out of points of a space of three, five, or more dimensions.) These numbers, realizable in four-dimensional space, are called quaternions.

Quaternions are linear combinations of the four basis elements 1, *i,j,* and *k:*

*X* − *x*_{0} · l + *x*_{1}*i* + *x*_{2}*j* + *x*_{3}*k*

where *x*_{0}, *x*_{1}, *x*_{2}, and *x*_{3} are real numbers. One operates on quaternions according to the rules of operation on polynomials in 1,*i, j,* and *k* except that one must not use the commutative law of multiplication and one multiplies the basis element as in Table 1. From Table 1 it is evident that 1 plays the role of ordinary unity and may therefore by omitted when writing down the quaternion:

(1) *X = x*_{0} + *x*_{1}*i* + *x*_{2}*j* + *x*_{3}*k*

The quaternion (1) may be separated into a scalar part *x*_{0} and a vector part

*V - x*_{1}*i* + *x*_{2}*j* + *x*_{3}*k*

so that

*X = x*_{0} + *V*

If *x*_{0} = 0, then the quaternion *V* is called a vector; it can be identified with ordinary three-dimensional vectors.

Table 1 | ||||
---|---|---|---|---|

1 | i | j | k | |

1 | 1 | i | j | k |

i | i | -1 | k | -i |

j | j | -k | -1 | i |

k | k | j | -i | -1 |

In the middle of the 19th century, quaternions were perceived as a generalization of the concept of number that was destined to play as important a role in science as the complex numbers. This viewpoint was reinforced by the fact that quaternions were applied in electrodynamics and mechanics. However, vector calculus in its modern form has displaced quaternions from these areas. It is clear that the role of quaternions can in no way be compared to the role of complex numbers, which have numerous and varied applications in different branches of science and technology.