March 20, 2017
Hiroshi Michiwaki
Density Principle and Division by Zero with the Resulting Remainder
Let us assume in a space with a uniform gravitational acceleration 𝑔, there is a cylinder with radius 𝑅,
height ℎ, uniform density ρ, and total mass 𝑀, positioned on a level, circular pressure receptor A₀ withtotal surface radius 𝑟₀ ≧ 𝑅.
The total force of gravity 𝐹 acting on the total mass 𝑀 in this spatial system is then expressed as

Inside the receptor A₀, a pressure receptor A ≦ A₀ is considered, which will be the target of thediscussion. This surface A, which has radius 𝑟, is flat and concentric to A₀. If now 𝑀out is used torepresent the mass of the cylinder that is not on the target surface A, the total force of gravity 𝐹, basedon Equation (1), is

where 𝑝 is the pressure acting on the target surface A and 𝑠 is its area, expressed as 𝑠(𝑟)=π𝑟².
Consequently, if 𝑀in is used to represent the mass of the cylinder that is on the target surface A, thepressure 𝑝 can be obtained by

Based on this, eliminating 𝑔 by substituting (3) into (2) yields

which can also be expressed as

Alternatively, if σ(𝑟) is used to represent the mass surface density on the target surface A, (4) can beexpressed asIf (5) and (6) are considered from the perspective of division, the total mass 𝑀 can be regarded as thedividend, the mass surface density σ as the quotient, the area 𝑠 as the divisor, and the mass outside thetarget surface, 𝑀out, as the remainder. That is, (6) can be expressed asHere, ··· indicates that the 𝑀out written after the ··· is the remainder of the division operation.
The effect of a reduction in radius 𝑟 to the target surface A if 𝑟₀ is fixed will now be examined. Thatis, the conditions of the system remain unchanged except for the size of surface A, and then Equations(4)–(7) are considered.
The mass surface density σ(𝑟) then isand as 𝑟 gradually decreases, this causes the part of the cylinder that is not on the target surface A toexpand. Consequently, the mass outside the target surface, 𝑀out, is expressed as

Combining (6), (8), and (9) yields

Equation (10) represents the transitional mechanism of the mass corresponding to the radius 𝑟 of thearea 𝑠 of the surface A. If now 𝑟 = 0, the surface area 𝑠 of A will also be 0, and thus the surface
becomes a dimensionless point. Hence, the volume of the cylinder on the target surface A, 𝑣 = π𝑟²ℎ, isalso zero. Consequently, the mass atop a point with an area of 0 is necessarily 𝑀in(𝑟) = ρ𝑣(𝑟) = ρ𝑣(0)= 0. It can also be claimed that a surface with area 𝑎=0 is a point and thus a nonexistent area (null φ =0). The mass 𝑚 atop the point—that is, the nonexistent area (null φ)—is also 𝑚=0, which makes area𝑎 and mass 𝑚 both 0 (null φ) and nonexistent. This implies that the mass surface density σ is also
nonexistent (null φ) and thus σ = 0. Therefore, it is clear that

The following will be referred to as the mass surface density principle (or simply the “densityprinciple,” as it can also be applied to other densities as well):This shows that the first term in (6) would then be 0. Moreover, as clearly seen from (9), the secondterm in (6) would become

That is, (6) implies

If, as in (7) above, the total mass 𝑀 is regarded as the dividend, the mass surface density σ as thequotient, the area 𝑠 as the divisor, and the mass outside the target surface, 𝑀out, as the remainder, then(14) yields

where ··· denotes that 𝑀out is the remainder of the division operation. Consequently, this arguably
demonstrates that this physical system can be expressed mathematically by using division by zerowith a remainder. Conversely, it arguably shows that such a method is mathematically correct for thisphysical system as well. Below, Equation (15) will be referred to as the First Law of Mass Surface
Density.
The case where 𝑟₀ (the radius of the entire level pressure receptor A₀ in this system) atop which thecylinder is placed becomes 𝑟₀ < 𝑅 with respect to 𝑅 (the radius of the cylinder) is now considered.
That is, when A = A₀ (𝑟 = 𝑟₀). Naturally, the total force of gravity 𝐹 acting on the total mass 𝑀 in thisspatial system is expressed as

Thus, if A=A₀, then the level circular pressure receptor A₀ with radius 𝑟₀ is the now target surface Aand its radius is 𝑟. As the mass in this system is concentrated on target surface A, if 𝑀off denotes themass not sitting on target surface A, the total force of gravity 𝐹 is expressed asby (16), where 𝑝 is the pressure acting on target surface A and 𝑠 is its area, expressed as 𝑠(𝑟) = π𝑟².
Consequently, if 𝑀on denotes the mass on target surface A, 𝑝 can be determined by

Based on this, eliminating 𝑔 by substituting (18) into (17) yields

which can also be expressed as

Alternatively, if σ(𝑟) represents the mass surface density on the target surface A, (19) can beexpressed as

Considering (20) and (21) from the perspective of division, the total mass 𝑀 can be regarded as thedividend, the mass surface density σ as the quotient, the area 𝑠 as the divisor, and the mass outside thetarget surface, 𝑀out, as the remainder. That is, (21) can be expressed asHere, ··· indicates that 𝑀out written after the ··· is the remainder of the division operation.
The effect of a reduction in the area of both the target surface A and the surface A₀ will now be
examined so that that A₀ becomes smaller than the bottom surface of the cylinder with radius 𝑅. Thatis, the size and shape of the cylinder remain unchanged, and only the size of surface A (= A₀) isvaried, and then (19)–(22) are considered. Of course, there will be some changes in the system, as thetarget surface A, which is the entire pressure receptor, is changed, but the total mass 𝑀 will evidently
remain unchanged.
The mass surface density σ(𝑟) then isand as 𝑟 gradually decreases, the mass surface density σ(𝑟) on the target surface A increases rapidly,
in inverse proportion to 𝑟². Furthermore, based on (21) and (23), the total mass 𝑀 can be expressed asEquation (24) represents the transitional mechanism of the mass corresponding to the area (𝑟) of thesurface A of this system. Moreover, as A occupies a finite area, the entire mass 𝑀 is sitting on A,
which implies that 𝑀off, the mass not sitting on A, isHere, if 𝑟 = 0 for the target surface A, its area will then be (0) = 0, and the surface will become adimensionless point. Then, according to the mass surface density principle, 𝑀on(𝑟), the mass sitting on
the area 𝑠(0) of this dimensionless point (target surface A), is nonexistent. (This fact is self-evident, as
it is not possible to place a mass on a nonexistent area.) Accordingly,and(It can be stated that the pressure acting on a surface with no area is 𝑝 = σ𝑔 = σ(0)𝑔 = 0x𝑔 = 0.)
Consequently, the first term in (21) then becomesMoreover, based on the definition of 𝑀off as the mass not sitting on the target surface A, the second
term in (21) isThus, (21) is

where, as above, the total mass 𝑀 can be regarded as the dividend, the mass surface density σ as thequotient, the area 𝑠 as the divisor, and the mass outside of the target surface, 𝑀off, as the remainder.
Therefore,(where ··· denotes that 𝑀off is the remainder of the division operation), which implies that this system
can be described mathematically by using division by zero with a remainder.
Of course, multiplying both sides of (30) by gravitational acceleration 𝑔 yields

which implies that the force of gravity acting on the zero-area (point) of the target surface is zero andthat the force exerted by the mass outside the target surface area is equal to the total force 𝐹;
furthermore, without any part of the mass supported by the surface, the force of gravity will cause thetotal mass to drop.
Thus, it can be claimed that
1. Phenomena such as mass concentrating on a point whose size is zero will not occur.
2. The surface of a table upon which a cup is placed has an uncountably infinite number of points,
and the surface pressure at each of these points is zero.
That is, it is clear that the values, for instance, for mass surface density, surface pressure, and force ofgravity on a zero-area surface (a point) are constantly zero, without becoming infinitely large,
diverging, becoming undefined, and taking any value. Moreover, the mass atop the point neverbecomes infinitely large, the total mass never becomes zero, and the force of gravity acting on thepoint never becomes infinitely large. Furthermore, the mass surface densities σ(𝑟) shown in (8) and(23) arerespectively, which in turn yield