Also found in: Dictionary, Thesaurus.



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 i2 = — 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:

Xx0 · l + x1i + x2j + x3k

where x0, x1, x2, and x3 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 = x0 + x1i + x2j + x3k

The quaternion (1) may be separated into a scalar part x0 and a vector part

V - x1i + x2j + x3k

so that

X = x0 + V

If x0 = 0, then the quaternion V is called a vector; it can be identified with ordinary three-dimensional vectors.

Table 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.

References in periodicals archive ?
The quaternions mathematics, or hypercomplex mathematics, were discovered by Hamilton in 1843 [20, 21].
This mathematical property of quaternions makes them ideal in the construction of algorithms for 3D computer graphics.
T] can be estimated using orientation represented in quaternion (8)
rel] are relative quaternion and angular rates written in LVLH frame.
The attitude kinematics are modeled using the quaternion differential equation
Moreover, setting the real parts of the quaternions to zero does not make full use of the nature of quaternions.
Next the joint positions and quaternions are transformed into 3x4 matrices.
Both the smoothing of markers trajectories (triangular filter kernel) and the filtering of quaternions (4th order Butterworth filter) were performed on the raw data including frame of impact.
Kamel, Finding the Position and Orientation of a Sensor on a Robot Manipulator Using Quaternions, The International Journal of Robotics Research 10, 240-254 (1991).
Glossing quaternions as an algebra of four dimensions, consisting of the three of space and the fourth of time, he goes on to declare that he formulated his best account of them in his 1846 poem "The Tetractys":
One might link this to the dispute between quaternions and vectorists which hinged upon the fact that quaternions were limited to three-dimensions, where vector analysis extends to four or more.