Definitions and basic properties of antiquaternions

It is known [1, 2], the four-dimensional hypercomplex numerical system called a system of quaternions with basis the multiplication table of elements which is;

 

 

.

Quaternions are the result of anticommutative doubling of complex numbers by same system of numbers. Or, using a system of markings, introduced in [3], we can write:

. (1)

If we redouble the system of complex numbers by system of binary numbers with multiplication table

 

,

 

we obtain a system of antiquaternions , or through operator of doubling:

. (2)

Indeed, if we take the composition of bases and , we obtain a basis

.

Multiplication table of the obtained hypercomplex numerical system is constructed using multiplying of elements of this basis. However, we believe that the basic elements of the same name are multiplied by the rules of systems and . When multiplying them together remains commutative only, when at least one multiplier is or . Basic elements and are multiplied anticommutatative.

 

.

We will give some examples of multiplication of basic elements, taking into account these rules:

.

If to rename two-symbolical names of basic elements in one-symbolical:

, , , ,

то that we will receive the multiplication table of basic elements of system of antiquaternions:

 

 

 

(3)

 

 

The principle of multiplication of basic elements is represented in fig. 1, on which basic elements of antiquaternion’s system are represented by triangle tops. Product of any two elements from this three is equal to the third if movement from the first to the second multiplier coincides with the arrow direction, if the movement is opposite to the direction of the arrow - the third with a minus sign.

Fig. 1 Schematic image of the multiplication table of basic elements of antiquaternion’s system.

 

Thus, anti-quaternions are numbers of a look

, (4)

where: .

 

Addition and multiplication of antiquaternions

In system of anti-quaternions is entered addition and multiplication operations thus:

The antiquaternion :

is called the sum of two antiquaternions and .

The antiquaternion :

(5)

is called the product of two antiquaternions and .

 

According to rules of addition and multiplication of antiquaternions it is possible to mark out their main properties:

1) operation of addition is commutative: ;

2) operation of addition is associative: ;

3) operation of multiplication is noncommutative: . (6)

Really:

,

but opposite order is such as:

 

That is carried out (6).

4) operation of multiplication is associative: .

It can be proved directly, using (5).

5) In the same way it is possible to prove the distributivity of antiquaternions: ;

6) for antiquaternions is determined action of multiplication by a scalar: ,

;

7) for is performed .

 

Definition of norm of antiquaternions

 

In the work [2] the norm of hypercomplex number generally is determined by a formula

, (7)

where - structural constants of hypercomplex numerical system of antiquaternions , which are defined from (3). On this basis the norm of matrix is constructed [2].

. (8)

Having calculated a determinant of a matrix (8) we will receive norm of hypercomplex numbers :

(9)

By analogy to the theory of quaternions we will call a root of the norm a pseudonorm of anti-quaternions (9), which will be denoted as :

. (10)

Apparently from (10), the pseudonorm can be negative. It is possible to show that the pseudonorm entered by such method is multiplicative:

. (11)

 

Definition and characteristics of conjugate antiquaternions

We introduce the definition of conjugate antiquaternion

(12)

on the basis of equality

, (13)

as it is offered in [2]. If (13) substitute (5) and (10), and to equate coefficients at identical basic elements that we will receive linear algebraic system concerning variables :

, (14)

which solutions have the form:

. (15)

Therefore, if the original antiquaternion , that the conjugate antiquaternion to it has view:

. (16)

We will define some properties of conjugate antiquaternions.

1) the sum and the product of conjugate antiquaternions is a real numbers;

2)the conjugate of the sum is the sum of conjugated ;

3) the conjugate of the product is the product of conjugated , which can be verified directly.