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Apparently from the last formula, the scalar part of work is equal to a scalar product of vectors and with the return sign. The vector part is our old acquaintance – the vector work which is written down in coordinates.

Quaternions are the fours of real numbers (x; y; u; v), which are convenient for writing down in the form of q = x + yi + uj + vk, where i, j, k – the new numbers which are analog of imaginary unit in complex numbers. It is required that numbers i, j, k satisfied to the following ratios:

Thus, each quaternion of q is presented in q sum form = x + by r, where x – scalar part of a quaternion of q, and r – vector part. If r = 0, q = x and a quaternion of q is called as a scalar quaternion. If x = 0, q = r and q is called as a vector quaternion.

At multiplication the situation is more difficult. If – scalar quaternions, their work too a scalar quaternion. In a case when = x – a scalar quaternion, and = r – the vector quaternion, work is a vector quaternion, and operation of multiplication coincides with multiplication of a vector of r in space on a real number x.

Just as complex numbers decay in the sum of the valid and imaginary parts, the quaternion too can be spread out in q sum = x + (yi + uj + vk). First composed in this decomposition the second is called as scalar part of a quaternion, and – vector part. The scalar part x is simply real number, and the vector part can be represented by r vector = yi + uj + vk in three-dimensional space where i, j, k we consider as single a vector of rectangular system of coordinates now.

Optimism was replaced by scepticism. At the beginning of our century of mathematics ceased to be interested in quaternions. But time went, and physicists persistently looked for a mathematical formalism for some effects connected with so-called backs of elementary particles. Quaternions again gained recognition when their role in creation of various geometrical transformations of space used in quantum physics was understood. Geometrical properties of quaternions is a special big subject.

But there is a problem of transformation of points of space into numbers. Here again we will enter system of coordinates and we will write down points in the form of a set already of three coordinates (x; y; z). These so-called triplets too develop coordinate-wise:

If it is about points on a straight line it is simple. Having chosen a reference mark and scale with the direction, it is possible to receive from a straight line a numerical axis and by that to turn each point into a real number – its coordinate.

The task first seemed simple. It was necessary to put vectors on a formula (. It was necessary to find the multiplication formula similar to a formula (. But Hamilton unsuccessfully tried to select formulas for multiplication of triplets.

With points on the plane it is more difficult. We choose two axes and a reference mark. For each point of the plane it is compared its coordinates (x; y). This couple will be called as a doublet. To make a doublet number, it is necessary to learn "to put" and "multiply" them according to properties of addition and multiplication.