#### Defining the spherical numbers.

The spherical number is expressed in the Euler form as

where is the natural number, (the preceding subscript s reads ‘spherical’) is the spherical-imaginary number, a spherical angle. In the cartesian form the spherical number is expressed as

where defines the order of the -dimensional spherical number, with the coefficients , is the multiplicative identity and usually omitted in the expression,

and are imaginary unit-numbers orthogonally orientated to , and to each other.

The values of the coefficients are determined by the spherical angle

The angles can be interpreted as a series of ordered successive rotations of the number (thought as a vector) into the complex volume, starting with the rotation of by towards the -axis and ending with the rotation of by towards the -axis.

**Definition 1:** *Products of imaginary unit-numbers*: The -algebra defines three rules (the fourth is derived):

(1)

(2)

(3)

(4)

Using above rules, the Euler form of the spherical number can be expanded to a series of products

(5)

and represents a point on a -sphere of radius in the complex volume.

**Example:** The three-dimensional spherical number

**Definition 2:** *The binary operation of spherical-multiplication*, expressed with the symbol , is performed by addition of the rotation angles, and is defined as:

(6)

**Definition 3:** *The unitary operation of exponentiation* is performed by multiplying each rotation angle by the exponent , and is defined as

(7)

which includes inversion when , therefore

(8)

**Definition 4:** *The binary operation of spherical-addition*, is expressed with the symbol , is defined as:

(9)

where

(10)

and

(11)

where and is the largest value so that holds, and after taking into account which quadrant the point is in, and .

**Definition 5:** *The unitary operation of negation* is defined by adding to the highest order non zero rotation angle , defined as

(12)

where is the largest value so that holds, therefore

(13)

Without further ado:

**Theorem 1: ***Spherical numbers of any order , i.e. of dimension , for all populate the field to form a finite to infinite-dimensional associative, commutative and distributive algebra over the reals without divisors of zero. %The one-dimensional spherical numbers are the complex numbers .*

Using the trigonometric identity

(14)

and the above five definitions the distributivity is confirmed, i.e.

(15)

**Note:** The first order spherical numbers, i.e. two-dimensional numbers, are the complex numbers and first order spherical-addition is equivalent to vector-addition over the cartesian coordinates.

#### Spherical numbers and cartesian representation.

We note that in a multi-dimensional construct that has the solution

This does not imply the definition for by but is the result of the exponentiation by a half of a number , where and . If a spherical number is expressed in its cartesian form we note the cartesian-equivalences over when the pairs is replaced with

and we also note that for any the Euler form is indeterminate as all information regarding the rotation angles is lost.

Thus, the cartesian form is avoided to express spherical numbers, the only correct algebraic representation is in the Euler form.

#### Spherical numbers and calculus.

Spherical numbers can be used in calculus. Any function , which is smooth in is also smooth in . Differentiating and integrating

is performed with respect to the variable and not its individual components. The accustomed rules apply: \footnote{In fact, can be transformed by a series of axis or Euler rotations to a complex number , where the new complex plane that defines is any plane in the complex volume .}

(16)

If is a function of ,

then

(17)

where .% which is the length of the arc on a unit -sphere defined by the points and when .

### Spherical Waves

Link to Spherical Waves

### Concluding remarks

The numbers introduce the new study-fields of multi-dimensional algebra and multi-dimensional calculus. I can imagine that this algebra may be helpful in formulating new descriptions for the physical world. Even if, for whatever reason, the counter-intuitive spherical-addition is omitted an abelian group remains, as outlined in points (1) and (2) of the introductory paragraph. Spherical-addition would be omitted if the algebraic vector product is introduced into vector algebra.

Technically, the Frobenius 1877 theorem on division algebra, and many related theorems, are now falsified.

I thus conclude with the philosophical observation:

*The truth is*: If a theorem is interpreted (defined in Oxford dictionary) as “*a general proposition not self-evident but proved by a chain of reasoning; a truth established by means of accepted truths*” then there exist theorems that will be falsified in the future, if and only if our current knowledge of accepted truths is subordinated to a future superordinate knowledge of accepted truths.

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