A New Mathematical Trial on Division by Zero

A New Mathematical Trial on Division by Zero

— Through Formal Infinite Series Manipulation, Approximation Value Method, and a Mass–Energy Conversion Concept —

Abstract

This paper presents a trial attempt to examine 18÷018 \div 018÷0, which is usually regarded as undefined, by temporarily setting aside the conventional restriction.Three approaches are proposed:

1. Formal infinite series manipulation

2. Approximation value method (assuming Z=1Z=1Z=1)

3. A mass–energy conversion approach related to special relativity

This paper is not intended as a rigorous mathematical proof, but as a presentation of the basic concept of the idea. In particular, zero is considered not as a fixed nothingness, but as something that may vary depending on time and representation.

1. Introduction

18÷018 \div 018÷0 is undefined.Recently, however, this became a topic of discussion because it appeared in an elementary school mathematics problem.

Throughout life, the author has continued critical thinking, although without formal advanced mathematical lectures. Formal study of mathematics extended only to the first year of university, but there is confidence in originality.

Therefore, by excluding the condition that 18÷018 \div 018÷0 is undefined, the author attempted to solve the debated elementary school problem in three different patterns.

2. Solution 1: Formal Infinite Series Manipulation

2.1 Basic Form

First, let zero be considered as an infinite series:

0=(2−2)−(2−2)−(2−2)−⋯0 = (2 - 2) - (2 - 2) - (2 - 2) - \cdots0=(2−2)−(2−2)−(2−2)−⋯

Expanding formally:

0=+2−2−2+2−2+2−2+2−2+⋯0 = +2 - 2 - 2 + 2 - 2 + 2 - 2 + 2 - 2 + \cdots0=+2−2−2+2−2+2−2+2−2+⋯

2.2 Interpretation When Stopped Halfway

If one pair in the repetition ends halfway, namely at −(2−-(2-−(2−, then:

0=−20 = -20=−2

Therefore,

18÷0=18÷(−2)=−918 \div 0 = 18 \div (-2) = -918÷0=18÷(−2)=−9

This is a forced interpretation.

2.3 True Solution in Time

Now consider the same expression in time:

0=+2−2−2+2−2+2−2+2−2+⋯0 = +2 - 2 - 2 + 2 - 2 + 2 - 2 + 2 - 2 + \cdots0=+2−2−2+2−2+2−2+2−2+⋯

or equivalently,

0=(2−2)−(2−2)−(2−2)−⋯0 = (2 - 2) - (2 - 2) - (2 - 2) - \cdots0=(2−2)−(2−2)−(2−2)−⋯

continuing endlessly.

As a supplement, the endless repetition of (2−2)(2-2)(2−2) becomes 000 at one stopped time and −2-2−2 at another stopped time. Thus it oscillates like a wave indefinitely.

Therefore, the true answer to

18÷018 \div 018÷0

is considered to be an endless repetition of −9-9−9 and error.

2.4 General Solution

Generalizing the above, let

2−2=γ2 - 2 = \gamma2−2=γ

Then the general solution becomes

18÷γ18 \div \gamma18÷γ

This also appears as a repetition of 18÷γ18 \div \gamma18÷γ and error in the positive flow of time.

3. Solution 2: Approximation Value Method

3.1 Basic Setup

Consider:

18÷018 \div 018÷0

First, let

0=0.001×00 = 0.001 \times 00=0.001×0

Let:

• X=0X = 0X=0

• Y=0.001Y = 0.001Y=0.001

• Z=0Z = 0Z=0

so that

X=YZX = YZX=YZ

and also

Y=Z/XY = Z/XY=Z/X

3.2 Calculation

Calculating the 0.001 part first:

18÷0=18÷X=18÷(Y×Z)18 \div 0 = 18 \div X = 18 \div (Y \times Z)18÷0=18÷X=18÷(Y×Z)=18×(1/Y)×(1/Z)= 18 \times (1/Y) \times (1/Z)=18×(1/Y)×(1/Z)=18000×1/Z= 18000 \times 1/Z=18000×1/Z

At this point, an error occurs.

3.3 Retry

Try again:

18÷018 \div 018÷00=0.001×00 = 0.001 \times 00=0.001×0X=YZX = YZX=YZ18÷X=18÷(Y×Z)=18×(1/Y)×(1/Z)=18000×1/Z18 \div X = 18 \div (Y \times Z) = 18 \times (1/Y) \times (1/Z) = 18000 \times 1/Z18÷X=18÷(Y×Z)=18×(1/Y)×(1/Z)=18000×1/Z

Again, an error occurs.

However, if we assume

Z=1Z = 1Z=1

then:

18000×1/Z=1800018000 \times 1/Z = 1800018000×1/Z=18000

Finally, multiply by zero to return:

18000×0=018000 \times 0 = 018000×0=0

3.4 Meaning of This Method

This method is an attempt to confirm the error while also trying to create new mathematics.Instead of treating zero as fixed, it decomposes zero and handles it approximately, producing a temporarily large finite value.

4. Solution 3: Mass–Energy Conversion Approach

4.1 Basic Setup

First, consider zero as the combination of:

• a dark mass electron XXX: D+1D + 1D+1

• a white mass electron YYY: W−1W - 1W−1

Thus zero is regarded as the combination of these two.

4.2 Conversion into Energy

Using the special relativity formula:

E=mc2E = mc^2E=mc2

convert all mass of the dark and white particles into energy.

• Energy of dark mass electron XXX: Dc2Dc^2Dc2

• Energy of white mass electron YYY: Wc2Wc^2Wc2

Then:

18÷0=18÷(X+Y)18 \div 0 = 18 \div (X+Y)18÷0=18÷(X+Y)=18×1Dc2+Wc2= 18 \times \frac{1}{Dc^2 + Wc^2}=18×Dc2+Wc21​=Z= Z=Z

4.3 Conversion Back into Matter

The obtained answer ZZZ is energy only.Convert it back into matter:

α=Z/c2\alpha = Z/c^2α=Z/c2

Therefore, the answer is α\alphaα.

5. Additional Idea: On Expressing Light by an Equation

An additional idea arose: the infinite series in 1a becomes −2-2−2 at one waiting time and 0 at another, and this continues infinitely like a wave.

If 2 is replaced by 100 million, then:

100 million−100 million+100 million−100 million+⋯100\,million - 100\,million + 100\,million - 100\,million + \cdots100million−100million+100million−100million+⋯

and this may correspond to a strong light.

If time is accelerated, the amplitude may change.

It is still unclear what the rapid fluctuations of error, −9-9−9, or 18/γ18/\gamma18/γ in 1b and 1c truly mean.

Also, infinite series may arise from zero, and light may be formed when nothingness separates into two polarities.

Furthermore, it may be possible in simulation to assign one equation to another, create light faster than light, and possibly go into the past.

Although electromagnetic waves have already been expressed mathematically, this may become the basis for a completely new equation of light.

6. Conclusion

This paper proposed three approaches to 18÷018 \div 018÷0:

1. formal infinite series manipulation

2. approximation value method

3. mass–energy conversion approach

As a result, the following expressions were obtained:

• −9-9−9

• endless repetition of error

• 18÷γ18 \div \gamma18÷γ

• the intermediate finite value 18000

• α\alphaα, obtained by reconverting energy into matter

These are not rigorous mathematical answers, but they suggest the possibility that zero is not merely fixed nothingness, but something involving oscillation, decomposition, and transformation.