February14, 2016
Hiroshi Michiwaki
Orthogonality conditions for two straight lines, the structure of the Cartesian
coordinates, and division by zero
Two straight lines y = ax + b and y = a’x + b’ are said to be orthogonal to each other if
the following condition holds:
If the slopes of the lines arerespectively, then the lines are orthogonal if

Based on this, Equation (1) can be expressed as

Therefore,holds. Equation (4) can be transformed further to obtain

When the included angle α isEquation (5) is clearly －1, soOn the other hand, when the included angle α in Equation (5) isif we let k be an integer, α can be separated into the cases

andIn the case of Equation (9),and in the case of Equation (10),Note that 0/0 = 0 follows from the fundamental theorem of division by zero. Therefore,
if the included angle α satisfies Equation (8), thenholds. In other words, by combining Equation (7) and Equation (13), we obtain

Conversely, based on Equation (7) and Equation (13),holds. In particular, if we let α = π/2, then we obtain the relationshipIncidentally, if we let θ represent the angle formed by the two straight lines y = ax + b
and y = a’x + b’, this can be expressed as

The two straight lines are orthogonal, so θ = π/2. Therefore, when the included angle αisbased on Equation (4) and Equation (7),and when the included angle α isbased on Equation (4) and Equation (14), we obtain

Note that 1/0 = 0 follows from division by zero.
Based on the above result, it can be said that the slope between two straight lines thatare orthogonal to each other is 0, regardless of the Cartesian coordinate system.
Interestingly, in the situation where the two straight lines are parallel to each other, or inother words when θ = 0 and α = α’, Equation (17) becomes

and when α ≠ π/2, Equation (20) clearly becomes 0. On the other hand, when α = π/2,
Equation (20) becomes

As this is equal to 0, the slope with respect to all included angles α in two straight linesthat are parallel to each other is 0, and is equal to the slope of two straight lines that areparallel to each other.

The above result shows that when a straight line L’ is tilted with included angle of θwith respect to another straight line L, if θ = 0, then the slope a between the two straightlines is 0, and θ = π/2 ⇒ a = 0, θ = π ⇒ a = 0, θ = 3π/2 ⇒ a = 0, and θ = 2π ⇒ a = 0.
In other words, an orthogonal coordinate system is composed of units of half-open
angular intervals every π/2 (every quadrant) from the coordinate axis that serves as areference, such as 0 ≦ θ < π/2 (the first quadrant), π/2 ≦ θ < π (the second quadrant),π ≦ θ < 3π/2 (the third quadrant), and 3π/2 ≦ θ < 2π (the fourth quadrant). The borders
between neighboring quadrants are connected with discontinuities that are strong for theslope of the coordinate axis.
In other words, using the first quadrant as an example, 0 ≦ θ < π/2 ⇒ 0 ≦ a < ∞, and inparticular, θ → π/2 ⇒ a → ∞, and θ = π/2 ⇒ a = 0. The last relationship shows that thetarget range has transitioned to the range of the next unit (the second quadrant) as
viewed from the previous unit (the first quadrant).
Below, we decompose the four quadrants in the x-y Cartesian coordinate system, fromthe first quadrant to the fourth quadrant, into separate quadrants (hereafter referred to as
quadrant decomposition), and show the structure of the Cartesian coordinate system inFig. 1. As shown in Fig. 1, for the range of slopes of the axes, or in other words, theranges of the axes’ included angles, the side with the larger included angle is an open
angle (a concept that corresponds to the concept of an open point) in contrast to the sidewith the smaller included angle, which is a closed angle (a concept that corresponds tothe concept of a closed point). The range of the included angle of each quadrant is a
half-open angle.
Next, we synthesize the x-y Cartesian coordinates that were decomposed into quadrants
into the original x-y Cartesian coordinates formed from the first quadrant through thefourth quadrant. The relationship of the axis slope that corresponds to the minimum
value and the maximum value in the included angle range for each quadrant within theregion of each quadrant is shown in Fig. 2. The symbol 0 written in Fig. 2 refers to an
infinitesimally small number that is smaller than 0 and that is larger than any negative
constant.

Fig. 1: Structure of the x-y Cartesian coordinates decomposed into 4 quadrants and theirranges

Fig. 2: Relationship between the x-y Cartesian coordinates and the axis slope due to 4
quadrant synthesis