December 14, 2014
Hiroshi Michiwaki
Reducible set theoretical division of complex numbers
Theorem 1: In the division of two complex numbers, z1 and z2, z1/z2, there is a complex
quotient z3 and a complex remainder z4 that satisfy

where “…” means that the value immediately after “…” is the remainder term.
Theorem 2: In the division of two complex numbers, z1 and z2, z1/z2, if we define thequotient of reducible set theoretical division of real numbers |{r2}| using the complex radius
r2 of z2 as the smallest non-negative real number when |{r2}| is maximized,is established. However, 𝜃1 and 𝜃2 are real numbers, and are the complex azimuth of z1 and z2,respectively. “…” means that the value immediately after “…” is the remainder term.
Theorem 3: When 𝑧 ∈ C,holds.
Proof: Proofs for Theorems 1, 2, and 3 are shown below without separation.
If we express the complex number 𝑧 = a + b𝑖 (a, b ∈ R, i: imaginary unit) using Euler’s
formula as a polar form,𝑧 = 𝑟𝑒𝑖𝜃 = (cos𝜃 − 𝑖sin𝜃) (1)where r is the distance from the origin on the complex plane (complex radius), and θ is theslope angle from the positive side of the real axis. Both r and θ are real numbers. Using thesevalues, two complex numbers z1 and z2 are expressed as𝑧1 = r1𝑒𝑖𝜃1𝑧2 = r2𝑒𝑖𝜃2
If z1 is divided by z2, it is expressed as(2)(3)

The result of Equation (4) is a complex number with a complex radius of r1/r2 that occurs as aresult of division z1/z2, and the term inside of { } is the unit radial azimuth component.
Definition of reducible set theoretical division of real numbersWhen A and B are real numbers, the calculation of division B/A is as follows. In|B| – (|{A}|・|A| + a) = 0 ( 0 ≤ a) (A, B, a ∈ R) (5)we assume B0 = B and Aj =A (j = 0, 1, 2, …), and define a set based on Aj+1 (this Aj+1 is called
the j+1th reducible number) in the reducible recurrence relation Bj – Aj+1 = Bj+1 as the j+1threducible number set {Aj+1}, and when Bj (this Bj is called the jth reduced number) satisfies Bj> Bj+1 ≥ 0, {Aj+1} ≠ ∅, and when this condition it not satisfied, {Aj+1} = ∅. The set whoseelements are all within the reducible number set {Aj+1} is defined as the reducible set {A}.
Here, B is the dividend, A is the divisor, and |{A}| is the number of elements of the reducible
set {A}, which is the quotient of B/A, where a is the remainder, which is the smallest nonnegative real number when |{A}| is maximized.
Therefore, if r1/r2 is treated based on reducible set theory, when the remainder is s, it isexpressed as|r1| − (|{r2}| ∙ | r2| + s) = 0 (0 ≤ s) (r1, r2, s∈R) (6)Therefore Equation (4) becomes

Here, the quotient and remainder terms of Equation (7) arerespectively, and can be expressed as

Here, we can see that the quotient of Equation (10) is the complex quotient z3, which is acomplex number, and the remainder is the complex remainder z4, which is a complex number.
In other words, division of a complex number by a complex number can be defined by thereducible set theory as shown above. It questions the ratio of the complex radius of z1 (r1) tothe complex radius of z2 (r2); i.e., the complex radius ratio. At the same time, it refers to thenumber of times the complex radius r2 can be reduced from the complex radius r1 of 𝑧1, and it
asks the size of the difference in the complex azimuth. The true nature of asking these twoquestions is that the amount of information in the original complex number z consists of twopoints; in other words, two sets of information on the real axis and the imaginary axis.

In Equation (7), let us suppose a case where 𝜃1 = 𝜃2 = 0. In this case, Equation (7) can be
reduced as follows:

Therefore, we can see that the reducible set theoretical complex division expressed inEquation (7) is a natural expansion of the reducible set theoretical division of real numbers.
Also, in Equation (7), if r2 = 0, it is clear that z2 = 0 based on Equation (3), and can be
expressed as 𝜃2 = |∅|; therefore,

is obtained.