#### Abstract

In this article, we obtain the polar forms for two types of split octonions. We calculate De Moivre’s formulas for all polar forms of split octonions. Thus, we give the powers and roots of split octonions and the matrix representation of split octonions. In addition, we present an illustrative example with Matlab codes.

#### 1. Introduction

De Moivre’s formula,is used to calculate the powers and roots of a complex number in polar form. Therefore, it has been one of the important subjects studied after some fundamental operations in mathematics. Many studies have been carried out on the adaptation of the formula to the lots of number systems, which are the expansion of the complex number system (references [17] and therein).

Split octonions have been worked on physics and mathematics [820]. Split octonions represent Minkowski space with (4 + 4)-signature. However, split octonions have also been studied with different orders of base elements in the articles in different fields. So split octonions have been studied as split octonions, countercomplex octonions, hyperbolic octonions, or split type octonions. But, it should be emphasized that there are direct relationships between all octonions in the literature. The geometrical fundamental properties for split octonions are located in the article [812].

In this study, we first give polar forms for two types of split octonions whose polar forms have not been given before. After that, we obtain De Moivre’s formulas of the split octonions and the matrices related with the split octonions in all polar forms. Finally, we present the roots of the split octonions and the matrix representation of split octonions in all polar forms.

#### 2. Preliminaries

A split octonion [8, 9] is introduced aswhere the number is called scalar part of the split octonion andis called vector parts of split octonion.

In (2), , , and denote the speed of the light, time, and space coordinates; and are considered as the phase (classical action) and the wavelengths associated with octonionic signals in geometric application. The set of all split octonions is denoted by . The fundamental algebraic properties for split octonions are located in Table 1. The conjugate of a split octonion, denoted by , is introduced by the following equation:

The inner product over split octonions is defined by the following equation [12]:where (where and ). So split octonions are divided into three classes having positive, negative, or zero value with respect to the inner product. In this case, similar to the literature of the split quaternions, if , , and , can be also called a spacelike split octonion, a timelike split octonion, and a lightlike split octonion, respectively. Also, the split octonions can be divided into three classes with respect to the vector part: if , , and , then it is called a split octonion with spacelike, timelike, and lightlike vector parts, respectively. The norm of the split octonion, denoted by or for brevity , is defined by [12].

The polar representation of a split octonion depends on its inner product and its vector part. So, it can be summarized in 5-case as follows:(i)Every split octonion with the inner product having a negative value iswhereand is a unit timelike vector [12].(ii)Every split octonion with the inner product having a positive value and timelike vector part iswhereand again is a unit timelike vector [12].(iii)Every split octonion with the inner product having a positive value and spacelike vector part iswhereand now is a unit spacelike vector [12].Now, we will define the polar forms for two types of split octonions which is not given before as follows.(iv)Every split octonion with the inner product having zero value, , iswhereand now is a unit vector and . Indeed, the following decomposition for the split octonion with the inner product having zero value, , can be written as follows:and since ,is satisfied.(v)Every split octonion with the lightlike vector part, , is written aswhereand . Because the split octonion with the lightlike vector part, , can be determined asand since ,is provided.

#### 3. De Moivre’s Formulas of Split Octonions

In this section, we express De Moivre’s formulas when split octonions are given in polar forms.

Theorem 1. Given the polar form for a unit split octonion with the inner product having a negative value. Then, we have the following equation:

If is an odd integer, then the equation (20) is provided.

If is an even integer, then the equation (21) is provided.

Proof. For , it is trivial and for and , with the hyperbolic angle sum formulas and considering that is timelike, we have the following equation:Now, let us assume that the formula is valid for that is,Then,or by using the hyperbolic angle sum formulas and , we obtain the following equation:Similarly, suppose that the formula is provided for that is,Then,or by using the hyperbolic angle sum formulas and , we have the following equation:Also, we can calculate for negative integer power by a similar method. For , by the properties of the multiplicative inverse of the split octonion , we have and . If the calculation process is repeated as mentioned above, the formula can be obtained similarly.

Theorem 2. Let be a unit split octonion with the inner product having positive value and time-like vector part. Then, we have the following equation:for all

Proof. Like the proof of Theorem 1, by using the induction and the hyperbolic angle sum formulas, the proof can be seen clearly.

Theorem 3. Given the polar form for a unit split octonion with the inner product having a positive value and spacelike vector part. Then, we have the following equation:for all

Proof. Like the steps of the proof of Theorem 1, by applying the induction, the poof can be obtained.

Theorem 4. Let us consider the polar form for a split octonion with the inner product having zero. Then, we have the following equation:for all

Proof. Let us apply the induction method. Suppose that is a split octonion with the inner product having zero and holds. Then, and

Theorem 5. Assume that is a unit split octonion with the light-like vector part. Then, we have the following equation:for all

Proof. By applying the induction method and considering , the proof can be seen directly.

#### 4. Roots of Split Octonions

In this section, the roots of a split octonion or in other words the solutions of the equation will be examined.

Theorem 6. Suppose that is a unit split octonion with the inner product having a negative value. Let us consider the equation .(i) If is an even number, then there is no root(ii) If is an odd number, then there exist only root as follows:where .

Proof. Let be a unit split octonion with the inner product having a negative value and is an odd number. Then, and using Theorem 1, we can writeThe other case is clear from Theorem 1.

Theorem 7. Let be a unit split octonion with the inner product having a positive value and timelike vector part. Let us consider the equation of .(i)If is an even number, then the 4-distinct roots are given in the following forms:where .(ii)If is an odd number, then the only root iswhere

Proof. Let be a unit split octonion with the inner product having a positive value and timelike vector part:(i)Assume that is an even number. And, letbe roots of the equation , where . Then, using Theorem 1, we can obtain the following equation:Thus, and hold the equation . The other cases can be obtained by using Theorems 1 and 2.

Theorem 8. Let be a unit split octonion with the inner product having a positive value and spacelike vector part. Then, there exist rootswhere .

Proof. One can directly verify this result with a similar way of the proof of Theorem 6.

Theorem 9. Let be a light-like split octonion with the inner product having zero. Then,(i)If is an even number, then roots of the equation are in the following forms:where .(ii)If is an odd number, then only root of the equation iswhere .

Proof. The proof can be easily obtained by using the same way of Theorem 6.

Theorem 10. Let be a unit split octonion with the light-like vector part.(i)If is an even number, there are roots of the equation for . The roots are in the following forms:But there exist no roots of the equation for .(ii)If is an odd number, the only root of the equation of is in the form as follows:

Proof. One can prove the formula by using the same way of Theorem 6.

#### 5. De Moivre’s Formulas of the Matrices of Split Octonions

Let and be any two split octonions in . Given the following two linear transformations in ,

Then, using the transformation and basis vectors , for , the left matrix representation of the transformation is determined by

Similarly, using the transformation and basis vectors , for , the right matrix representation of the transformation is given by

In addition to this, we can define an isomorphism where

Similarly, we can define a bijective and surjective transformation where

But note that, and , for are hold.

Lemma 1. Let be a split octonion. Then,where N is the norm of .

Proof. One can easily prove with some algebraic operations.
From now on, we consider only the right matrix representation of any split octonion through all our calculations. But all results also can be easily obtained for the left matrix representations of any split octonion by the same methods:(i)If is a unit split octonion with the inner product having a negative value and a unit timelike vector , the right representation of the split octonion can be written as follows:(ii)If is a split octonion with the inner product having a positive value and a unit timelike vector part , then the right representation of the split octonion can be written as follows:(iii)If is a split octonion with the inner product having a positive value and spacelike vector part, then the right representation of the split octonion can be obtained as follows:(iv)If is a split octonion with the inner product having zero. Then, the right representation of the split octonion can be given as follows:(v)If is a unit split octonion with the lightlike vector part  = 0. Then, the right representation of the split octonion is found as

Theorem 11. Let us consider the matrix defined byfor a unit split octonion with the inner product having a negative value and a unit timelike 7-vector . Then, if is an odd number, the matrix isand if is an even number, the matrix is

Proof. By applying matrix multiplication and using the hyperbolic angle sum formulas and the factThe and power of the given matrix is calculated as