September 3, 2015
Hiroshi Michiwaki
Mixed irreducible fractions and division by zero
1. Introduction
The author has recently presented a definition and a proof of the result that in thedivision problem a/b, if b = 0, the quotient becomes 0. The result, which states thatdivision by zero is possible, is revolutionary. However, it is also necessary to be awareof the existence of formulae in which it is not possible to correctly execute division byzero. In this paper, we discuss the rule that prohibits division by zero and present aspecific example below.
2. A mixed irreducible fraction that violates the rule that prohibits division by zeroLet a, b, c ∈ R and c ≠ 0 satisfy the following equation:

The right-hand side of the equation is referred to as a mixed irreducible fraction.
Transforming the left-hand side into the form on the right-hand side is referred to as
converting a fraction to a mixed irreducible fraction. Conversely, an expression thatsatisfies the left-hand side in Equation (1) is referred to as a mixed combined fraction.
Transforming the right-hand side into the form on the left-hand side is referred to as
converting to a mixed combined fraction. Note that in Equation (1), a > b, a = b, and a <b are all possible.
2. The rule that prohibits division by zero
Theorem: It is not possible to apply division by zero with a denominator of 0 to mixed
reduced fractions.
Proof: Let a, b, c ∈ R and c ≠ 0. Then, in the following equation,

let b ＝ 0. Then, the left-hand side of Equation (1) becomes:

Similarly, the right-hand side of Equation (1) becomesA comparison of Equation (2) and Equation (3) shows that the results contradict eachother. The reason that an illogical result such as this occurred is that Equation (3) is afraction that has been converted to a mixed reduced fraction, or in other words, because
it is a mixed reduced fraction. ■The reason that the result is illogical is that the right-hand side of Equation (1) isequivalent towhich is an irreducible binomial fractional expression, or an extension of the mixed
fractionIn other words, the process of conversion to an irreducible fraction and the process of
converting to a mixed fraction include operations in which the following is applied:

Therefore, the real reason is that a contradiction arises with division by zero

when b ＝ 0. Therefore, it can be said that it is not possible to apply division by zero to
mixed reduced fractions.
As shown in the discussion above, in division by zero, converting fractions to common
denominators and reducing fractions are both prohibited. The fundamental reason forthis is that both of these operations use the fact thatWhile, on the other hand, the denominator is 0. In other words, in division by zero,because

in cases in which 0 is included in the range of values for b, it is not possible to convert
fractions to a common denominator, nor is it possible to reduce fractions.

Although the above discussion may give the incorrect impression that division by zero
implies the rule that prohibits it, limiting the world in which division by zero is possible.
In actuality, this is not true. This implies that special solution methods that are valid inworlds in which division by zero is prohibited, such as inappropriately reducing
fractions or inappropriately converting fractions to common denominators, areprohibited. This shows that it is important to consider whether it is possible for thedenominator to have a value of zero, and whether it is allowable for the denominator tohave a value of zero. This can be expressed in the following phrases:
Axiomatic systems in which division by zero is impossible  conversion to a common
denominator is possible ˄ reduction is possible
Axiomatic systems in which division by zero is possible  conversion to a common
denominator is impossible ˄ reduction is impossible
Expressing this in terms of symbols, the above can be summarized as:

These relationships can be thought of as representing the symmetry of axiomaticsystems for the division by zero. This symmetry is referred to as the division by zero
symmetry axiom in the following.
According to the division by zero symmetry axiom, it can be said that it is not possible
to apply the division by zero to formulas and physical equations, etc. that were derived
by application of axiomatic system . This is because it is possible that conversion to acommon denominator or reduction was performed in the process of deriving the finalequation in axiomatic system . It is highly effective to compare forms that werederived without conversion of a fraction to a mixed irreducible fraction, and forms thatwere derived with conversion of a fraction to a mixed irreducible fraction in the processof deriving the final equation in a specific example. This is illustrated below using anexample that consists of a physics equation.
3. A comparison of forms that were derived with and without conversion of a fraction toa mixed irreducible fraction
A small wheel and a large wheel with radii r1 and r2(r1 < r2) were fixed on the same shaft
on the same axis a distance of d apart, and are structured in a way such that there is no
relative rotation. When this same-axis same-rotation two-wheeled device rolls, theradius of curvature R of the arc drawn by the trajectory of the large wheel can berepresented as:

(This equation is the principle formula of the needleless wheel compass invented by E.
Michiwaki to draw circles and arcs. The radius of curvature R in Equation (10) is
referred to as the EM radius). Interestingly, if the radius of the small wheel is set to r1 =0 in the EM radius R, thenIf d and r2 are considered to be the distance from the needle to the pivot and the distancefrom the tip of the pen to the pivot, then this becomes equivalent to the radius ofcurvature for the angle between the pen and the needle in a conventional compass with
the needle set to 90˚. Therefore, it can be seen that the needleless wheel-type compass inEquation (10) is a special case of the conventional needle compass.
In the EM radius R, it is common to set the radius of the small wheel r1 and the radius of
the large wheel r2 so that they satisfy (r1 < r2) . However, in the structure of theneedleless wheel-type compass, it is possible to set the radius of the small wheel r1 andthe radius of the large wheel r2 to be equal to each other. In other words, this implies
that (r1 = r2). This is similar to the back wheel in a passenger vehicle. This relationshipcan be substituted into the formula for the EM radius R and solved for the value of R,
which gives:

In other words, this implies that the solution 1/0 is the value of the EM radius R. The
physical meaning of division by zero is illustrated in a subtle manner here. Applying theinterpretation that 1/0 = infinity results in the following strange relationship. Thisimplies that the EM radius of the back wheels on a passenger vehicle is infinity, which
implies that the back wheels on a passenger vehicle traveling in a straight line have an
EM radius that is larger than the diameter of the universe, which has a finite radius. Thisimplies that the center point of the EM radius exists outside of the universe.
Next, we use the interpretation that 1/0 = undefined. This would imply that the EM
radius of the back wheels of a passenger vehicle is undefined. What does that mean?
This means that the EM radius of the back wheels on a passenger vehicle isundetermined and is not constant, which implies that the passenger vehicle wouldwobble while it travels. However, the back wheels of a passenger vehicle travel straight.
Therefore, this interpretation implies that the back wheels wobble too much and travel
straight (the wheels wobble an infinite and uncountable amount of times per instant andtravel straight as a result).
After introducing division by zero where a/0 = 0, the EM radius R becomes:

This equation implies that if the radii of the left and right wheels are equal, then the EM
radius is zero, and the wheels would travel straight without turning. This is the mostnatural and common-sense interpretation, and is compatible with reality.
It has been shown that introducing a/0 = 0 into the EM radius R results in a correct
answer that is compatible with reality. Transforming the EM radius R as shown belowgives:

Substituting r1 = r2 into Equation (14), from the equation in the center, we obtain:

From the equation on the right, we obtain:

The derived results contradict each other. The reason for this illogical result is thatdivision by zero was applied to an equation in which conversion of a fraction to a mixed
irreducible fraction has been applied. Therefore, it is obvious that it is not possible todivide by zero in equations in which conversion of a fraction to a mixed irreducible
fraction has been applied.
4. Appendix
Does the number infinity exist? Here, we will discuss the treatment of infinity using
Equation (1) under the assumption that the number infinity exists.
Let a, b, c ∈ R , and assume that the number infinity ∞ exists. Then, substituting b = ∞into the equation

the left-hand side becomes:=×=Conversely, the right-hand side becomes== 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑.

(17)which is the only value that has been determined. Therefore, this shows that acontradiction has arisen between Equation (17) and Equation (18). This is because

has been used in both conversion to a common denominator and reduction. In contrast,
becauseundefined (20)in cases in which ∞ is included in the range of values for b, it is not possible to convert
fractions to a common denominator, nor is it possible to reduce fractions. Of course, theconcept of limits and the concept of division by infinity and division by zero arecompletely separate, so it is necessary to exercise caution.

andThis implies that because the manner in which b changes is equal in the denominatorand in the numerator, it is possible to reduce the fraction even when b becomesmaximum or minimum.