## What is the Continuous Nonlinear Numbers, What is ...

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What is the Continuous Nonlinear Numbers, What is the Nonlinear Zero? Ralph W. Lai, Ph.D. (04/14/2014, V.1) Abstract: This paper defines what the continuous nonlinear numbers is and discusses the existence of nonlinear zeroes. The variation of continuous nonlinear numbers is explored with examples. In general, we need multiple graphical expressions to explain the variation of continuous nonlinear numbers. The nonlinear numbers is best measured with nonlinear logarithmic scale, and its differential (or change) is always measured relative to its asymptotes. The nonlinear zero is an asymptote of the nonlinear numbers; it is an elusive number because its existence can only be implied but cannot be plotted on a graph.

1.0 Mathematical Background 1.1 Classification of Continuous Numbers From a viewpoint of classifying the collection of continuous numbers (Note: “numbers” refers to a series or a set of numbers. It is as singular or as plural in this paper), we classify the continuous numbers into linear numbers and nonlinear numbers. The key to the classification of numbers is asymptote. Liner numbers have no asymptote and are not associated with any asymptotes, such as ...-3, -2, -1, 0, 1, 2, 3, 4... , where the zero 0 is a linear zero that can be touched and cross over. In contrast, nonlinear numbers are associated with one or two asymptotes, such as …10-3, 10-2, 10-1, 100, 101, 102, 103, 104…, which has a nonlinear zero ϕ = (0) as its lower asymptote. This nonlinear zero ϕ = (0) can be approached but cannot be touched or crossed over. The latter numbers decrease in steps from (right to left) 10000, to 1000, to 100, to 10, to 1, to 0.1, to 0.01, and to 0.001 etc., These numbers are decreasing toward nonlinear zero but will never reach or touch the nonlinear zero. Nonlinear numbers always have continuity and always preserve the continuity - meaning it always has the next step or the next number. Asymptote is not part of the nonlinear numbers and can never be part of the nonlinear numbers. The nonlinear zero is an asymptote, which means nonlinear zero is not and never be the nonlinear numbers (see Axiom II at the end of this section) [1, 2, 11]. Table 1 gives examples of linear numbers and nonlinear numbers. Examples of linear numbers are X1, and X2. These numbers can increase or decrease in two directions, and can have a linear zero sandwiched between them - meaning the linear zero can be touched, or crossed over by the linear numbers. Examples of nonlinear numbers are Y’s, where Y1, Y2, and Y3 have a lower asymptote Ys = ϕ = (0); Y4 has an upper asymptote Yu = 1/9; Y5 has an upper asymptote Yu = 1; Y6 has an upper asymptote Yu = 50 and a lower asymptote Ys = 0.4. Y2, Y4, and Y5 are one-sided nonlinear numbers; these three nonlinear numbers can extend only in one direction. Y1, Y3, and Y6 are two sided nonlinear numbers that can extend in both directions. 1

2 Table 1 Examples of linear numbers and nonlinear numbers X1 -3 -2 -1 0 1 2 3 4 5 6 7

X2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

This is linear numbers

This is linear numbers

Y1 10000 1000 100 10 1 0.1 0.01 0.001 0.0001

Ys = ϕ = (0)

Y2 2 0.7 0.01 0.008 0.0002 0.00008 0.000001

Y3 64 32 16 8 4 2 1 0.5 0.25 0.125

Y4 0.1 0.11 0.111 0.1111 0.11111

Y5 0.9 0.99 0.999 0.9999

Y6 1.37 2.77 5.96 12.34 22.39 33.47 41.77 46.36 48.48 49.38 49.75

Ys = ϕ = (0) (this is e)

Ys = ϕ = (0) (this is 1/2n)

Yu = 1/9

Yu = 1

Ys = 0.4 Yu = 50

Sound mathematics need to base on simple, exact axioms. There are two (only two) essential axioms: Axiom I on continuity and Axiom II on asymptote. We will use these two axioms to address the existence of nonlinear zero: Axiom I:

Continuity exists for all collection of continuous numbers in relating to a physical phenomenon. The continuous numbers has continuity and always has a next step or a next number. Continuous numbers are dynamic, non-terminating, and can never be forced to stop (It is dishonest to use the uncertain word “infinity” as a disguise to stop the continuity of the numbers). Continuous numbers can be one-sided continuous numbers or can have continuity on both two sides. Axiom II: Continuous nonlinear numbers can approach the asymptote, but cannot touch or cross the asymptote. Asymptote is never a part of the continuous nonlinear numbers; Moreover, the continuous nonlinear numbers is dynamic and the asymptote is static; the former can move and the later cannot move (or change). 1.2 Axiom I and Axiom II versus Modern Definition of Limit The above two axioms are two common sense statements using plain English; they are simple and exact. The Axiom I is for the preservation of continuity for both continuous linear numbers and continuous nonlinear numbers. The Axiom II is on the relationship between the nonlinear numbers and their asymptotes. These two axioms can be restated with modern definition of “limit”, but they are not a preferred way to address the nonlinear numbers, because the limit theory is awkward and not easy to understand or memorize by common people and students. In his internet article “Funnels: A More Intuitive Definition of Limit”, Professor Eric Schechter of Vanderbilt University has the following interesting observations: “Many (perhaps most) calculus students have difficulty understanding and learning the epsilon-delta definition of a limit. I can state several reasons why the epsilon-delta definition is difficult to understand (although the student does not need to be aware of reasons): it has too many variables; it has too many nested clauses; it does not suggest anything about a rate of convergence; and it cannot be illustrated easily with a picture. I sometimes tell my students to memorize the epsilon-delta definition, word by word; understanding will come later (if at all). I caution the students to be careful with their memorizing; students who do not yet fully understand the definition may inadvertently change the wording slightly, in some fashion that sounds inconsequential 2

3 to the untrained ear but greatly changes the mathematical content.” (See Appendix A-1 for more discussion on limit vs. Axiom II). From the above discussions we may conclude that we do not need to teach the “limit”; all we need is to learn the simple fundamental Axiom I and Axiom II, just two simple straightforward English sentences. 1.3 The Most Important Linear Numbers and Nonlinear Numbers in this Universe Let us use two important continuous numbers as reference numbers: Universal linear numbers Ul and universal nonlinear numbers Un. It is believed that all human by age four, from Stone Age to modern man, can count 1, 2, 3, 4, 5, 6… at an increment of 1. It is also believed that every civilized human understands the existence of 0 and also understands how to count the negative numbers -1, -2, -3, -4, -5, -6 …in decreasing order at an increment of -1. Over all, the number series ...-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6...is a linear numbers with a difference of 1 between two adjacent numbers. This number series is the most important numbers in this universe, and is called the universal linear numbers Ul. In Ul the 0 exists between 1 and -1, and is said to cross over between positive numbers and negative numbers, or equivalently is said to be touched when counting from positive to negative or positive to negative numbers. The important characteristics of Ul are that: (1) The Ul has continuity in both positive and negative directions; (2) The Ul, no matter how large or how small, always has the next number, e.g., at 5, its next number is 6; at 1000000, its next number is 1000001 etc.; on negative side, at -1000, its next negative number is -1001 etc. The above characteristics remind us the concept of preserving the continuity, which will be used to discourage the use of “infinity”. Use of “infinity” is an ambiguous way to close a scientific argument, and thus discarded in the new math system. The most important nonlinear numbers is …10-3, 10-2, 10-1, 100, 101, 102, 103, 104…. This is called the universal nonlinear numbers Un. This Un has a nonlinear zero as its lower asymptote. No matter how large the negative of the power of 10, e.g., 10-100, 10 -10000, or 10-1000000, these numbers has continuity (Axiom I) and are approaching a nonlinear zero which can be approached but cannot be touched (Axiom II). Since the number 10 is extremely useful and will be used extensively, we introduce a symbol θ to represent 10, i.e., 10 = θ. Accordingly, the universal nonlinear numbers Un =…10-3, 10-2, 10-1, 100, 101, 102, 103, 104… is also written as Un =…θ-3, θ-2, θ-1, θ0, θ1, θ2, θ3, θ4… or Un = θ^Ul, or = . This is to say that the universal nonlinear numbers is θ raise to the power of universal linear numbers. Thus, the universal nonlinear numbers has the characteristic continuity of the universal linear numbers. As a special case, the numbers ….(-4)2, (-3)2, (-2)2, (-1)2, (0)2, (1)2, (2)2, (3)2, (4)2….is called the extended linear numbers, where the linear zero, (0)2, can be reached or touched. This set of numbers is useful in accounting for the symmetry of curves such as a symmetric bell curve. 1.4 Scales for Linear and Nonlinear Numbers, and Construction of two Dimensional Graphs A. Linear scale for linear numbers The standard scale for linear numbers is a linear scale, as shown in Fig. 1.4a below; its characteristic is the equal spacing between adjacent scale and the existence of a linear zero, which can be crossed 3

4 over or touched. The spacing between two adjacent numbers can be as small as you would like or as large as you like. Fig.1.4a linear scale Linear scale is a standard scale for linear numbers -6

-4

-2

0

2

4

6

8

10 12 14 16 18 20

Linear scale is a standard scale for linear numbers -300

-200

-100

0

100

200

300

400

B. Logarithmic scale is the standard scale for nonlinear numbers The standard scale for nonlinear numbers is a 10 based logarithmic scale, as shown in Fig. 1.4b; its characteristic is the existence of a nonlinear zero, which can be approached but cannot be reached or touched. The scale in Fig.1.4b is nonlinear and is approaching nonlinear zero as its lower asymptote but will never be able to touch this nonlinear zero. This is to say that the nonlinear zero is not part of the continuous nonlinear numbers: …0.01, 0.1, 1, 10, 100… These nonlinear numbers are measured from nonlinear zero, are continuous, and are able to be labeled on the logarithmic (nonlinear) scale. However, its nonlinear zero is its lower asymptote, is not part of the nonlinear numbers, and thus cannot be plotted on the nonlinear scale. Fig. 1.4b nonlinear scale Logarithmic scale is the standard scale for nonlinear numbers 0.01

0.1

1

10

100

1000

Although, there are possibilities of using other nonlinear numbers, such as e and 1/2 n, as standard scale for nonlinear numbers, however, they are inconvenience to use as discussed in references [1,2]. C. Construction of Two Dimensional Graphs Two-dimensional graphs are constructed from a combination of linear and nonlinear scales as their two axes. Most logically, the Cartesian coordinate graphs (such as Fig. 1.4c) with linear numbers as scales are used for initial expression of all data, including linear and nonlinear numbers. However, we need some other convenient scale for in depth expression of nonlinear numbers. To do so, the graphs with 10 based logarithmic scales in the forms of semi-log graph (such as Fig. 1.4d) and log-log graphs (such as Fig. 1.4e) are needed.

4

5 Fig. 1.4c Cartesian graph with Linear scale of Y by linear scale of X α0 or linear Y values

80

40

0 -40

-20

(0, 0) 0

-40

20

40

60

linear zero

-80

β0 or linear X values

Fig. 1.4e Log-log graph having nonlinear scales for both axes αi or nonlinear Y values

Fig. 1.4d Semi-log graph having nonlinear scale by linear scale αi or nonlinear Y values

1000 100 10 1 0.1 -5

0

5

β0 or linear X values Ys = ϕ = (0)

10

100 10 1 0.1 0.01 0.1

Ys = ϕ = (0) Xs = ϕ = (0)

1

10

βi or nonlinear X values

Let us pay attention to placing straight lines on the above graphs and also discuss the existing of zeros in the graphs. When placing straight lines in a Cartesian graph, the straight lines or their extension will intersect the vertical and horizontal axis at zeros, as shown in Fig. 1.4c. These zeros are linear zeros. They can be reached or crossed by the linear numbers. The existence of straight lines means that the linear numbers Y is proportional to the linear numbers X (See Sections 3.0 and 5.1). When placing a straight line on a semi-logarithmic graph, the straight line or its extension will intersect the vertical axis that is standing at X = 0 – meaning the horizontal numbers are linear numbers. The vertical numbers are continuous nonlinear numbers; this vertical numbers can decrease toward nonliner zero but will never reach the nonlinear zero ϕ. This nonlinear zero is a lower asymptote of continuous numbers Y. The lower asymptote zero is not part of the continuous nonlinear numbers and thus cannot be plotted on the graph, but can only be indicated separetely on the graph, such as indicated as Ys = ϕ = (0) on the graph in Fig. 1.4d. When placing a straight line on a double logarithmic (log-log) graph, the straight line or its extension will never intersect the axes at zero. Both the vertical and horizontal numbers are continuous nonlinear numbers; these continuous numbers can decrease toward nonliner zeros but will never reach the nonlinear zeros. These nonlinear zeros are the lower asymptotes of continuous nonlinear numbers Y and continuous nonlinear numbers X. The lower asymptotes, i.e., the nonlinear zeros, are not part of the continuous nonlinear numbers and thus cannot be plotted on the graph, but can only be indicated separately on the graph, such as indicated as Ys = ϕ = (0) and Xs = ϕ = (0) on the graph in 5

6 Fig. 1.4e. When trying to plot a zero value on a logarithmic graph using a Microsoft Excel, we will get a warning banner as shown in Fig. 1.4f bellow. Fig. 1.4f Microsoft Excel Spreadsheet Banner

Technically speaking, the above warning is correct, but it did not provide theoretical background and explain why. The theory behind it is explained in the previous paragraphs. In essence, the logarithmic scale is for the nonlinear numbers, and the nonlinear numbers has nonlinear zero as its lower asymptote. Furthermore, the nonlinear zero can be approached but cannot be touched, also, the nonlinear zero is not part of the continuous nonlinear numbers, and it cannot be plotted on the graph.

1.5 Differential of Linear and Nonlinear numbers A. Differential (or change ) of Linear Numbers The notation “d” stands for differential or change; sometimes, “∆” is used in lieu of “d”. Examples of linear change or differential of linear numbers are dY, dX, d0, and d0, where Y, X, 0, and 0 have no association with asymptotes. Subscript 0 in 0 and 0 is to indicate that α and β numbers are linear numbers that has no (0) asymptote and that it may include linear zero as parts of the linear numbers. Here dY, dX, d0, and d0 are read as “the change of linear numbers Y”, “the change of linear numbers X”, “the change of linear numbers 0 “ and “the change of linear numbers  0“. Integration of

the above linear changes are: dY  Y  C ,

 dX  X  C ,  d

0

  0  C , and  d 0   0  C ,

where C is an integral constant or a position constant. B. Differential (or change) of Nonlinear Numbers Nonlinear numbers are associated with their asymptotes; thus, their change or differential always needs to base on measurement relative to their asymptotes. Measurement of nonlinear numbers relative to their asymptote is called the face values of the nonlinear numbers. These face values can be a difference, or a ratio or a combination of both relative to the asymptotes. Since we need to account for asymptotes in all operation of nonlinear numbers, the traditional Y and X is insufficient for representing the face values. Accordingly, we introduce new symbols, i and i, as face values for corresponding Y and X nonlinear numbers. Because the face values can be a difference, or a ratio or a combination of both relative to the asymptotes, we need various sub symbols for representations. Overall, they can be represented by a series of sub symbols: such as 11 = (Y – Ys), 12 = (Yu – Y), and 21 = (Y – Ys)/(Yu – Y). The first number in the subscription, such as 1 in 11, or 2 in 21, refers to the number of asymptotes in the measurement of nonlinear face 6

7 values; there is one asymptote in 11 and 12, and there are two asymptotes in 21. The second number in the subscription refers to the forms of measurement relative to the asymptote. 11 takes the first form of measurement relative to the lower asymptote Ys, thus 11 = (Y – Ys); 12 takes the second form of measurement relative to the upper asymptote Yu, thus 12 = (Yu – Y). In 21 there are two asymptotes, upper and lower asymptotes Yu and Ys; the first form of nonlinear measurement relative to two asymptotes is (Y – Ys)/(Yu – Y); thus 21 = (Y – Ys)/(Yu – Y). In 22 there are two asymptotes, upper and lower asymptotes Yu and Ys; the second form of nonlinear measurement relative to two asymptotes is (Yu – Ys)/(Y – Ys); thus 22 = (Yu – Ys)/(Y – Ys). In 23 there are two asymptotes Yu and Ys; the third form of nonlinear measurement relative to two asymptotes is (Yu – Ys)/(Yu – Y); thus 23 = (Yu – Ys)/(Yu – Y). A special case exists for a higher order of nonlinearity where a third subscription number 1 is used to indicate the existence of an additional logarithmic q factor, such as 231 = q[(Yu – Ys) /(Yu – Y)]. Other sub symbols are listed in Appendix A - 2 Table A-2. The following four graphs illustrate the measurements of 11, 12, 21, and 221.

Fig. 1.5b Primary graph Y versus X in linear by linear scale

Fig. 1.5a Primary graph Y vs, X in linear by linear scale 20

80

Yu

15

40

5

α11 = Y - Ys

20

α12 = Yu - Y

10

Y

Y

60

0 0

0

Ys = 0

0

5

200

400

X (=β0)

10

X (=β0)

Fig. 1.5c Primary graph linear by linear scales 180

Fig. 1.5d Leading graph Y in log scale, X in linear scale 10000

Yu

qYu 1000

∝21 =

60

𝑌 − 𝑌𝑠 𝑌𝑢 − 𝑌

Y

Y

120

10

𝑌𝑠

0 0

10 X

20

∝221= 𝑞𝑌𝑢 − 𝑞𝑌 𝑌𝑠 = ϕ = (0)

100

1 0

2

4

6

8

X

7

8 The differential or the change of nonlinear face values is obtained by placing the nonlinear face values on nonlinear logarithmic scale and measure their changes. This is done by multiplying log to the face values, i.e.,(qi) and (qi), and followed by taking the differential “d” to give d(qi) and d(q i). There are two ways to read the two terms d(qi) and d(q i). First, we address the “d” first by reading as “the change of nonlinear face value (qi)” or ”the change of nonlinear face value (qi)”; Second, we address the “q” first by reading as “the nonlinear change of face-value (i)” or “the nonlinear change of face-value ( i)”. As shown in Table A-2 (Appendix A - 2), when there is only one asymptote, either lower asymptote Ys or upper asymptote Yu, there are four ways to measure nonlinear face-value relative to asymptotes: (Y – Ys), (Yu – Y), Ys/Y, and Yu/Y. The latter two forms have some use in physics but have not much use in life and biomedical experimental sciences, and we will omit them in this paper. Integral forms for the above nonlinear changes, d(qi) and d(q i) are:

 d (q )  q  1

i

i

 d (q )  q  1

i

i

 q1C , and

 q1C . [Note: q1 means: “please plot the following nonlinear numbers on the

nonlinear scale (i.e., log scale or q scale)”.] 1.6 Slope of a Right Triangle on the Graphs is the Proportionality Constant Slope of a right triangle is different from the slope of a curved line. Slope of curved lines is of interest in physical sciences and engineering but has little use in biomedical sciences, and we will omit them here. They are available in detail in reference books [1, 2]. We will discuss the slopes of right triangles and the slope of straight lines in graphs in this section.

Fig. 1.6a Linear change by linear change 10000 p2(X2,Y2)

Fig. 1.6b Nonlinear change by linear change

1000

p4(X4,qY4)

Y

∆Y or dY ∫dY = Y2 - Y1 p1(X1,Y1)

y = Yo

100 10 1

p3(X3,qY3)

0.1 ∆X or dX ∫dX = X2 - X1

∆X or dX ∫dX = X4 - X3

0.01 0.001 -1

X

∆(qYo)or d(qYo) ∫d(qYo) =(qY4 - qY3) = q(Y4/Y3)

1

3

5

7

X

Refer to Figure 1.6a, the slope of a right triangle in a Cartesian coordinate graph is define as ∆Y/∆X = K or dY/dX = K, where K is the slope of the line. ∆Y or dY is the incremental change in vertical distance, ∆X or dX is the incremental change in horizontal distance. In this paper, we write the above equation of the slope into ∆Y = K∆X or dY= KdX and read this equation as “the change of linear numbers Y, ∆Y or 8

9 dY, is proportional to the change of liner numbers X, ∆X or dX”. Accordingly, the K is called the proportionality constant. In general, we can represent ∆Y or dY with dα0; and ∆X or dX with dβ0 on a Cartesian graph. Total change from point P1 to point P2 in Fig. 1.6a is the integral of dY and dX between two points, i.e., ∫dY = Y2 – Y1, and ∫dX = X2 – X1. Refer to Figure 1.6b, the nonlinear face-value Yo is plotted on vertical logarithmic scale, while the linear face-value X is plotted on horizontal linear scale; the slope of the right triangle in the semi-log graph is define as ∆(qYo)/∆X = K or d(qYo)/dX = K, where K is the slope of the right triangle. Yo indicates that the nonlinear number Yo as well as logarithmic scale is measured from its lower asymptote nonlinear zero, which cannot be shown in the graph. In this paper, we write the above slope equation as ∆(qYo) = K∆X, or d(qYo) = KdX, meaning the change of nonlinear true-values is proportional to the change of linear true-values; the K is the proportionality constant. In general term, Figure 1.6b illustrates the change of nonlinear true-values qαi versus the change of linear true-values β0 on a semi-logarithmic graph. Total vertical change from point p3 to point p4 is ∫d(qYo) = qY4 – qY3 = q(Y4/Y3) and total horizontal change is ∫dX = X4 – X3.

Fig. 1.6c Nonlinear change by nonlinear change 1000.000 100.000 p6(qX6, qY6)

y = Yo

10.000

∆(qYo ) or d(qYo ) ∫d(qYo) = (qY6 - qY5) = q(Y6/Y5)

1.000 p5(qX5, qY5)

0.100

∆(qXo) or d(qXo) ∫d(qXo) = (qX6 - qX5) = q(X6/X5)

0.010 0.001 0.1

1.0 X = Xo

10.0

Refer to Figure 1.6c, the two axes are plotted with nonlinear face-value Yo and Xo; the slope of the right triangle in the log-log graph is define as ∆(qYo)/∆(qXo) = K or d(qYo)/d(qXo) = K, where K is the slope of the right triangle. In this paper, we write it as ∆(qYo) = K∆(qXo), or d(qYo) = Kd(qXo), meaning the change of one nonlinear true-value is proportional to the change of another nonlinear true-value; the K is the proportionality constant. In general term, Figure 1.6c illustrates the change of nonlinear true-value qαi versus the change of nonlinear true-value qβi on a double-logarithmic graph. The changes of nonlinear true-values qαi and qβi are d(qYo) and d(qXo). Total changes from point p5 to point p6 are ∫d(qYo) = qY6 – qY5 = q(Y6/Y5) and ∫d(qXo). = qX6 – qX5 = q(X6/X5).

2.0 Simple Examples for Comparison of Nonlinear Numbers versus Linear Numbers In the following examples, we will compare the change of nonlinear numbers having a single asymptote Ys or Yu, versus change of linear numbers X. Where the nonlinear change of Y is 9

10

1

Nonlinear numbers 0.9, 0.99, 0.999, 0.9999… has an upper asymptote 1 (Yu = 1). Let us compare the change of this nonlinear numbers with the change of universal linear numbers Ul and check whether the number 1 is the unique asymptote of the nonlinear numbers. Let us input the universal linear numbers in the Column A as X and the nonlinear numbers in the Column B as Y, as shown in Microsoft Excel Screen Table 2.1A. As a reminder, the nonlinear change is always measured relative to the asymptote. In our case, we need to measure the nonlinear change of (Yu – Y).

10

11 Table 2.1A

In the Excel Screen, we reserve Cell E1 for imputing an upper asymptote Yu. This asymptote is needed for calculating (Yu – Y) in Column C. By plotting Column A vs. Column B, we obtain Fig. 2.1a for Y vs. X in a linear scale, where the data line is approaching an upper asymptote. Next, let us input “1” into Cell E1. The next step is to calculate (Yu – Y) in Column C. In Cell C2, we input “=\$E\$1 – B2”, then copy Cell C2 to Cell C3 through Cell C10, as shown in Table 2.1B. By plotting (Yu – Y) vs. X (Column A vs. Column C), we obtain Fig. 2.1b in linear scale; this is a pre-proportionality graph. We can copy Fig. 2.1b into Fig. 2.1c and convert the vertical scale into nonlinear logarithmic scale, as shown in Fig. 2.1c. Notice in Fig. 2.1b we are comparing (Yu – Y) vs. X in linear by linear scale; in Fig. 2.1c we are comparing q(Yu – Y) vs. X in log-linear scale. The next step is to display proportionality equation (trendline equation) and coefficient of determination for the proportionality plot. Fig. 2.1a Primary graph Y vs. X in linear scale

Fig. 2.1b Pre-proportionality graph (Yu - Y) vs. X in linear scale

1.04

0.12

Yu - Y

Y

1.00 0.96 0.92 0.88

0.08

0.04

0.00 0

2

4

6

8

10

X

0

2

4

6

8

10

X

Table 2.1B

11

12

Fig. 2.1d Proportionality graph q(Yu - Y) vs. X in log-linear scale

1.E-01

1.E-01

1.E-03

1.E-03

y = (Yu - Y)

Yu - Y

Fig. 2.1c Proportionality graph q(Yu - Y) vs. X in log-linear scale

1.E-05 1.E-07 1.E-09

y = 1θ-X

1.E-05 y = 1e-2.303x R² = 1

1.E-07 1.E-09

0

2

4

6

8

10

X

0

2

4

6

8

10

X

Let us right clicking on data series in Fig. 2.1c followed by selecting “Add Trendline”, and then selecting “Exponential” from Trendline Options, also selecting “Display Trendline” and “Display R-squared”, then click Close. We obtain Fig. 2.1d. The coefficient of determination is R2 = 1, indicating that Yu = 1 is the perfect choose as the upper asymptote. How do we know this is the perfect asymptote? Let us try out with some other numbers. [*Special Note: This paper relies on Microsoft Excel for graphing and calculation. However, there is a drawback in using Excel when we need to plot semi-log graph and issue trendline equation on it. Ideally, the trend line equation for a straight line in a semi-log graph should be written as y = Cθ-Kx, where C is the position constant (intercept of the straight line at X = 0) or integral constant, θ is a notation for 10, and K is the proportionality constant or the slope of the straight line. In Fig. 2.1d, the original equation for y (= Yu – Y) vs. x from Excel is y = 1e-2.303x, it should be written as y = 1θ-x, where C = 1 and K = -1(note: 2.303/2.303 = 1; 2.303 is a conversion factor between natural logarithm and 10 based logarithm)].

Table 2.1C

Let us pick a number slightly larger than 1, say 1.0000001, and input 1.0000001 into Cell E1, as shown in Table 2.1C. Figure 2.1d will turn into Fig. 2.1d-1, where the data line strays from the straight line and R2 reduced to 0.9603. When we change the number into 1.00001 and input 1.00001 into Cell E1, we obtain Fig. 3.3d-2 and R2 reduced to 0.8244. We can improve R2 by increasing the number of zero after 1 in the numerator, such as 1.000000001. In doing this, we get Fig. 2.1d-3 and the R2 improves to 0.9991; it is still less than 1. The overall trend is that all the numbers are eventually approaching 1 as an upper asymptote.

12

13

Fig. 2.1d-1 Proportionality graph q(Yu - Y) vs. X in log-linear scale Yu = 1.0000001 y = 0.2483e-1.85x R² = 0.9603

y = 0.0452e-1.151x R² = 0.8244

1.E-02 y = (Yu - Y)

1.E-02 y = (Yu - Y)

Fig. 2.1d-2 Proportionality graph q(Yu - Y) vs. X in log-linear scale Yu = 1.00001

1.E-04 1.E-06

1.E-04

1.E-08

1.E-06 0

2

4

6

8

10

0

2

X

4

6

8

10

X

Fig. 2.1d-3 Proportionality graph q(Yu - Y) vs. X in log-linear scale Yu = 1.000000001

y = (Yu - Y)

1.E-01 1.E-03

y = 0.8455e-2.251x R² = 0.9991

1.E-05 1.E-07 1.E-09 0

2

4

6

8

10

X

2.2 Nonlinear Numbers with one Upper Asymptote – 0.1, 0.11, 0.111, 0.1111…

1/9

Nonlinear numbers 0.1, 0.11, 0.111, 0.1111… has an upper asymptote 1/9 (Yu = 1/9). Let us compare the change of this nonlinear numbers with the change of universal linear numbers Ul and check whether the number 1/9 is the unique asymptote of the nonlinear numbers. Let us input the universal linear numbers in the Column A as X and the nonlinear numbers in the Column B as Y, as shown in Excel Screen Table 2.2A. As a reminder, the nonlinear change is always measured relative to the asymptote. In our case, we need to measure the nonlinear change of (Yu – Y). In the Screen, we reserve Cell E1 for imputing an upper asymptote Yu. This asymptote is needed for calculating (Yu – Y) in Column C. By plotting Column A vs. Column B, we obtain Fig. 2.2a for Y vs. X in linear scale, where the data line is approaching an upper asymptote. This is a primary graph. Next, let us input “=1/9” into Cell E1. The Excel will give certain build-in number of digit but will only show limited number of digit as shown in Table 2.2B. The next step is to calculate (Yu – Y) in Column C. In Cell C2, we input “=\$E\$1 – B2”, then copy Cell C2 to Cell C3 through Cell C10, as shown in Table 2.2C.

13

14 Table 2.2A

Table 2.2B

Table 2.2C

By plotting (Yu – Y) vs. X (Column A vs. Column C), we obtain Fig. 2.2b in linear scale; this is a preproportionality graph. We can copy Fig. 2.2b into Fig. 2.2c and convert the vertical scale into nonlinear logarithmic scale, as shown in Fig. 2.2c. The next step is to display proportionality equation (trendline equation) and coefficient of determination for the proportionality plot. Let us right clicking on data series in Fig. 2.2c followed by selecting “Add Trendline”, and then selecting “Exponential” from Trendline Options, also selecting “Display Trendline” and “Display R-squared”, then click Close. We obtain Fig. 2.2d. The coefficient of determination is R2 = 1, indicating that Yu = 1/9 is the perfect choose as the upper asymptote. How do we know this is the perfect asymptote? Let us try out with other numbers.

14

15

Fig.2.2a Primary graph Y vs. X in linear scale

Fig.2.2b Pre-proportionality graph (Yu - Y) vs. X in linear scale

0.12 0.012 Yu - Y

Y

Yu - Y

0.11

0.10

0.008 0.004 0

0.09 0

4

8

0

12

4

X

Fig.2.2c Proportionality graph q(Yu - Y) vs. X in linear scale

y = (Yu - Y)

Yu - Y

Yu - Y

1E-07 1E-09 1E-11

y = 0.1111θ-x

0.001 1E-05 1E-07 1E-09

y = 0.1111e-2.303x R² = 1

1E-11 0

4

8

12

Fig.2.2d Proportionality graph q(Yu - Y) vs. X in linear scale Yu = 1/9

0.001 1E-05

8 X

12

X

0

4

8

12

X

Table 2.2D

Let us pick a numerator slightly larger than 1, say 1.0000001, and input 1.0000001/9 into Cell E1, as shown in Table 2.2D. Figure 2.2d will turn into Fig. 2.2d-1, where the data line strays from the straight line and R2 reduced to 0.9603. When we change the numerator into 1.00001 and input 1.00001/9 into Cell E1, we obtain Fig. 2.2d-2 and R2 reduced to 0.8244. We can improve R2 by increasing the number of zero after 1 in the numerator, such as 1.000000001. In doing so, we get Fig. 2.2d-3 and the R2 improves to 0.9991; it is still less than 1. The overall trend is that all the numbers are eventually approaching 1/9 as an upper asymptote. 15

16

Fig.2.2d -2Proportionality graph q(Yu - Y) vs. X in linear scale Yu = 1.00001/9

y = 0.0276e-1.85x R² = 0.9603

0.001 1E-05 1E-07

y = 0.005e-1.151x R² = 0.8244

0.001

y = (Yu - Y)

y = (Yu - Y)

Fig.2.2d -1Proportionality graph q(Yu - Y) vs. X in linear scale Yu = 1.0000001/9

1E-05

1E-09

1E-07 0

4

8

12

0

4

X

8

12

X

y = (Yu - Y)

Fig.2.2d -3Proportionality graph q(Yu - Y) vs. X in linear scale Yu = (1.000000001)/9 y = 0.0939e-2.251x R² = 0.9991

0.001 1E-05 1E-07 1E-09 1E-11 0

4

8

12

X

3.0 Paradox or Not Paradox - Historical Cases of Dichotomy Let us examine two ancient wisdoms of dichotomy and explain them with modern concepts of nonlinear numbers. First, let us review the Eastern wisdom of Lie Tzu (Lie Zi, 450 – 375 BCE). Lie Tzu stated 百 尺之竿, 日折其半, 永世不休, it states that “giving you a 100-foot pole, halving it every day, day by day, continue from you to your offspring, generations after generations, yet the task cannot be finished in a million generations (direct translation is 10 thousand generations – meaning forever many generations).” Lie Tzu’s wisdom of dichotomy explained the conservation of continuity for the nonlinear numbers. This explained the parts of Axiom I; however, his statement is short of addressing the lower asymptote of the nonlinear numbers and the relationship between the lower asymptote and the nonlinear numbers, as required in Axiom II. To complete his statement, he should add, “The process of halving the pole will give the size of the pole approaching nonlinear zero, which is a lower asymptote of the continuous process and which can never be reached”. Now, let us turn to a Western wisdom. In the fifth century B.C., Greek philosopher Zeno of Eleatic argued that a person could never cross a room and bumps his nose into the opposite wall. Zeno pointed 16

17 out that in order to cross the room one would first need to cross half the distance of the room. Then, half the remaining distance. Then half of that distance. And so forth. However, everyone knows one can cross a room and bump one’s nose into the opposite wall (certainly if one is not paying attention). This is often called a Zeno’s paradox. However, is it really such a paradox? No, it is only a simple nonlinear mathematical joke. A complete argument is that a person can have two types of strides: A linear walk and a nonlinear walk. When a person is allowed to walk freely across the room without any restriction, he will cross the room and bump his nose into the wall in a single stage. This casual stride is a linear walk that can be represented by a single straight line where one starts from one end and reaches other end in one stage regardless of the size and number of steps. To illustrate the concept more concretely, simply use numerical values and graph the results. For example, suppose the wall-to-wall distance in a room is 64 meters. When a person walks from one wall at 64 meters toward the other wall at 0 meters, he may walk continuously in a single stage and reach the other wall at 0 meters in 100 steps, as shown in the demulative (opposite of cumulative) distance 2 in Fig. 3.0a. When a person walks from one wall at 0 meters toward the other wall at 64 meters, he may walk continuously in a single stage and reach the other wall at 0 meters in 100 steps, as shown in the cumulative distance 1 in Fig. 3.0a. Table 3.0 gives the markers of steps and distances for each walk. The linear walk can proceed in any way regardless of the markers. In the graph, the distance Y is plotted versus number of steps X. Both linear walks are completed in a single stage of 100 steps to reach 0 or 64 meters and are represented by two straight lines with two linear equations. Table 3.0 Linear Walk from Either Side of the Wall (Walk in straight line in a single stage)

X, Steps (markers)

From 64 m to 0 m Incremental distance

Cumulative Distance 1

X, Steps (markers)

From 0 m. to 64 m. Incremental Demulative distance Distance 2

0

0

0

0

0

64

25

16

16

25

16

48

50

16

32

50

16

32

75

16

48

75

16

16

100

16

64

100

16

0

DistanceY, in meters

Fig. 3.0a Distance in Linear Walk 80

Cumulative Distance 1

60

Demulative Distance 2

y = 0.64x R² = 1

40 y = -0.64x + 64 R² = 1

20 0 0

20

40

60

80

100

120

X, Steps

17

18 Now consider Zeno’s mathematical joke abstracting in nonlinear fashion: that as a person walks across a room, he walks in stages, each stage being half the distance remaining to be walked. The counting of stages is 0, 1, 2, 3, 4… forever. These are linear numbers obeying the accounting rule of universal linear numbers by adding 1 in sequence. Imposing “stages” along with imposing corresponding “half the distance” is imposing nonlinear restriction and nonlinear rule. This nonlinear “stride” can no longer be described by a single dimensional straight line, but needs to be represented by a two dimensional graph. This nonlinear “stride” is a nonlinear (of distance) by linear (of stages) phenomenon, where the nonlinear change in distance is negatively proportional to the linear change in the number of stages (the number of steps per stage doesn’t matter). In the nonlinear change of nonlinear numbers (the distance), the nonlinear distance is measured from asymptote, either an upper asymptote or a lower asymptote. Table 3.1 Nonlinear Walk from Either Side of the Wall (Walking in multiple stages by nonlinear rule and with nonlinear restriction) X, Stages

y1, Incremental distance

Y1, Cumulative Distance 1

X, Stages

y2, Incremental distance

Y2, Demulative Distance 2

∝11 = Y - Ys

0

0

0

64

0

0

64

64

1

32

32

32

1

32

32

32

2 3

16

48

16

2

16

16

16

8

56

8

3

8

8

8

4

4

60

4

4

4

4

4

5

2

62

2

5

2

2

2

6

1

63

1

6

1

1

1

7

0.5

63.5

0.5

7

0.5

0.5

0.5

8

0.25

63.75

0.25

8

0.25

0.25

0.25

∝12 = Yu - Y

9

0.125

63.875

0.125

9

0.125

0.125

0.125

10

0.0625

63.9375

0.0625

10

0.0625

0.0625

0.0625

11

0.03125

63.96875

0.03125

11

0.03125

0.03125

0.03125

12

0.015625

63.984375

0.015625

12

0.015625

0.015625

0.015625

Yu =

64

Ys = (0) = Ф

0

Now, let us use numerical values for illustration: starting with 64 meter at stage 0; halving the distance to the opposite wall for each stage. The first stage is 32 meters. The second stage is 16 meters, and so on. Plotted in Series 2 in Figure 3.1a, it is a nonlinear phenomenon showing a curve. The Y numbers, 64, 32, 16, 8, 4, 2, 1, 0.5, 0.25, 0.125… are nonlinear numbers having continuity, and are associated with lower asymptote Ys = (0) = Ф, which cannot be touched or reached; their corresponding X numbers, 0, 1, 2, 3, 4, 5… are linear numbers, also having continuity, but its 0 is reachable linear zero. For every X there is a corresponding Y. X is forever continuously increasing, so does the Y which is forever continuously decreasing. The change in distance is a nonlinear change measured relative to the lower asymptote Ys = (0) = Ф. The distance one walks approaches ϕ = (0), but never reaches this nonlinear zero ϕ = (0).

18

19 Series 1 in Figure 3.1a is a case where the person walks from 0 meters toward the other wall at 64 meters. In this case, the upper asymptote Yu is 64. The line in series 1 is cumulative numbers Y1 versus cumulative numbers X; the line in series 2 is demulative (opposite of cumulative) numbers Y2 versus cumulative numbers X. A graph with plot of cumulative and/or demulative numbers is called a primary graph. For a nonlinear phenomenon of higher order of nonlinearity, there may have more associated graphs, including primitive graph, primary graph, leading graph, and proportionality graph. Fig. 3.1a Nonlinear Walk Y1 vs. X and Y2 vs. X

Distance, Y meters

80

60

Y1, Cumulative Distance 1

40

Y2, Demulative Distance 2

20

0 0

4

8

12

16

X, Stages

Fig. 3.1b Pre-proportionality plot (Yu - Y) vs, X and (Y - Ys) vs. X

Fig. 3.1c Proportionality plot q(Yu - Y) vs, X and q(Y - Ys) vs. X 100 y = α12 = (Yu - Y), Yu = 64 y = α11 = (Y - Ys), Ys = Ф

(Yu - Y) or (Y - Ys)

80 60 Yu - Y 40

Y - Ys

20 0

Yu - Y Y - Ys

10

y = 64θ-0.3X

1 y = 64e-0.693x R² = 1

0.1

0.01 0

5

10 X, stages

15

0 Ys = Ф = 0

5

10

15

X, stages

In series 2, the nonlinear change in distance, denoting as α11 or (Y – Ys), is negatively proportional to the linear change in stage X. For comparing nonlinear change of nonlinear numbers with linear change of linear numbers, we need to use nonlinear scale for nonlinear numbers and linear scale for linear numbers. To do this we use the logarithmic scale as standard for nonlinear scale, and common linear scale as standard for linear scale. Thus, we can first plot (Y – Ys) versus X on a Cartesian coordinate graph to show a nonlinear line for series 2 in Fig. 3.1b. Then, we need to convert the y-axis from linear into nonlinear logarithmic scale to obtain a plot of q(Y – Ys) versus X on a semi-log graph, which yields a straight line with negative slope, as shown in Fig. 3.1c. This is a proportionality graph.

19

20 On the other hand, in series 1, we can first plot (Yu – Y) versus X on a Cartesian coordinate graph to show a nonlinear line; then, to convert the y axis from linear into nonlinear logarithmic scale to obtain a plot of q(Yu – Y) versus X on a semi-log graph, which yields the same straight line as series 2. The nonlinear numbers measured relative to their asymptote, such as (Y – Ys) and (Yu – Y), are called nonlinear face values. They can be calculated and placed on the axis of linear scale or nonlinear scale of the graph. When the nonlinear face-values are placed on nonlinear scale, their nonlinear true-values are q(Y – Ys) and q(Yu – Y). The straight lines in Fig. 3.1c represent that the change of nonlinear true-value q(Y – Ys) or q(Yu – Y) is negatively proportional to the change of linear true-values X, i.e., d(q(Y – Ys)) = -K(dX) or d(q(Yu – Y)) = -K(dX)

(3.1a)

K is the proportionality constant. Their integral forms are: q(Y – Ys) = -KX + qC and q(Yu – Y) = -KX + qC

(3.1b)

C is an integral constant or position constant (position of a straight-line moving up/down in a graph). The above two equations are also written as (Y – Ys) = Cθ-KX and (Yu – Y) = Cθ-KX, where θ is a notation of 10. Figure 3.1c is a proportionality graph due to having two straight lines in a semi-log graph, where one nonlinear true-value is proportional to the other linear true-value. The requirement for expression with two-dimensional graphs for nonlinear walk, along with the requirement of Figure 3.0a for linear walk, inspires the establishment of the first letter G (graph-based) in GVP math system. Moreover, Fig. 3.1c inspires the establishment of the second letter V (true-value compared) and the third letter P (proportionality-oriented) in GVP math system. By now, we have started to encounter more variety of comparing true-values through proportionality. Later on, we shall demonstrate some more complicated series of comparing nonlinear true-values versus nonlinear true-values. In Fig. 3.1c, the straight line decreases as X increases. Ideally, the obtained straight line equation on the graph should be written as y = 64θ-0.3x, where y is (Y – Ys) or (Yu – Y) , 64 is C, θ is 10 and K is 0.3. The R2 is coefficient of determination for trendline. Unfortunately, when using Microsoft Excel for drawing trend line we can only get the exponential equation as y = 64e-0.693x. (Note there is a factor of 2.30285 between 10 based logarithm and natural logarithm). I hope that someday in the near future, Microsoft will modify the Excel program to provide direct plot of log-linear equation such as for the equations (Y – Ys) = Cθ-KX and (Yu – Y) = Cθ-KX on the graph. For series 2, at X = 0, (Y – Ys) = Y – 0 = Cθ-KX = C, i.e., C = Y = 64; and for series 1, at X = 0, Yu – Y = Yu – 0 = Cθ-KX = C, i.e., C = Yu = 64. Figure 3.1d shows the change in Y from stages 60 to 75, where Y decreases from 2.8x10 -17 to 1.7x10-21. The proportionality plot for this range is given in Fig. 3.1e. In Figures 3.1c and 3.1e, the slope of the lines is the same, and the proportionality constant K is 0.6931/2.30285 = 0.3. Meanwhile, we also recognize that the nonlinear plots in Figs. 3.1b and 3.1c are always continuous. In essence, both continuity and the nonlinear restriction must be preserved. The line in Fig. 3.1d is an extension of series 2. The nonlinear line is approaching nonlinear zero, but will not reach this nonlinear zero. Figure 3.1e is a proportionality plot for the same range from stage 60 to 75. The straight line in the semi-log graph decreases continuously. The semi-log graph can continue from this graph to extend outside the room to go into space to reach the moon and return. We will still get a similar

20

21 semi-log graph. It will never change into a linear graph. The straight line is decreasing toward nonlinear zero but will never reach nonlinear zero, and the nonlinear line will never cross zero. Fig. 3.1e Proportionality plot II

Fig. 3.1d Nonlinear walk II 6E-17

1.E-16 y = Y - Ys, Ys = (0)

5E-17 Y, meters

4E-17 3E-17 2E-17 1E-17 0 60

65

70

75

y = 64θ-0.3X

1.E-17 1.E-18 1.E-19 1.E-20

1.E-21 Ys = ϕ = (0) 60

X, stages

y = 64e-0.693x R² = 1 65

70

75

x = X, stages

What do we learn from the above inspiration? We learned eight new concepts: First, there are two types of continuous numbers (walks): linear and nonlinear numbers (walks). A linear numbers have no asymptote, and a nonlinear numbers have an asymptote. Second, there are two types of zeroes: linear zero and nonlinear zero. A linear zero is attainable and can be crossed, whereas a nonlinear zero is an asymptote, which is not attainable. Additionally, we learned: Third, a nonlinear change of nonlinear numbers is best expressed by a change of nonlinear true-values. Fourth, asymptotes can be any real numbers including zero (e.g., 64 and 0), and asymptotes are not attainable by the nonlinear numbers. Fifth, a graph with a logarithmic scale (semi-log graphs in this section) is needed to express the nonlinear change in nonlinear face values (Y – Ys) and (Yu – Y), and to indicate the existence of asymptotes (e.g., Ys = (0) = ϕ and Yu = 64). Here (Y – Ys) is the difference between Y and the lower asymptote Ys, and (Yu – Y) is the difference between the upper asymptote Yu and Y. Sixth, in the logarithmic scale, when Y decreases it continuously decreases toward the nonlinear zero as its asymptote, but will never attain nonlinear zero. Y has continuity and the logarithmic scale will never disappear or (magically) turn into a linear scale. Seventh, GVP mathematics (i.e., αβ math) has continuity everywhere, whereas traditional math has a problem of continuity (e.g., y is undefined at x = 0 for y = 1/x, to be discussed in section 5.1). Eighth, use of logarithmic paper, either a semi-log or a log-log, is important in scientific analyses, especially in analyses of any nonlinear phenomena. (Note 1: nonlinear numbers Y will never reach zero on a logarithmic scale. Note 2: 2.303 is a conversion factor between the natural logarithm and 10-based logarithm. log x 

ln x . Note 3: The linear walk of ln 10

Fig. 3.1a is a linear by linear phenomenon where the change of distance is proportional the change in number of steps in a single stage. This linear by linear phenomenon is a phenomenon obeying the proportionality law of the first kind (PL1) [1, 2]. The nonlinear walk described by Fig3.1c is a nonlinear by linear phenomenon where the straight line on a semi-log plot shows the proportionality. This nonlinear by linear phenomenon is a phenomenon obeying the proportionality law of the second kind (PL2) [1, 2]. In PL2, the straight line on the semi-log graph will always intercept at X = 0. There always exists a C at X = 0. C may be any real number including 1. When C is 1, log 1 is zero (log 1 = 0). It is a special case. Note 4: Readers are encouraged to read the arguments on Zeno’s Room Walk (or The Dichotomy), and 21

22 the Zeno’s Achilles in references [1, 2]. You may find out the drawback of those arguments without the knowledge of nonlinear zeroes and with no discipline of preserving the continuity.) Reason for Confusion in Dichotomy – The GVP Math System Comes to Rescue The basic reasons for confusion in dichotomy are that, up to now, there is no clear-cut definition of linear zero and nonlinear zero, there is no clear-cut classification for linear numbers and nonlinear numbers, and that there is no clear-cut definition for “what a linear line is, and what a nonlinear line (a curve) is”. This confusion put many teachers in difficult situation to explain the dichotomy to the pupils. As an example, recently a 7th – 8th grade math teacher explained to the author how she tried to explain to her students that the dichotomy series, ½, ¼, 1/8, 1/16…, is forever decreasing and won’t reach zero. She said that many students think that the dichotomy series will reach zero. Her explanation is that the dichotomy series will reach “infinity”, and at “infinity” there is still a small number but it is not zero. I told her that her explanation sounds correct but confusion, and asked her whether her explanation is in the textbook. Her answer is “no, it is not explained in the text book”. Her explanation is a little bit awkward, because of using the term “infinity” and without explanation of the nonlinear zero. The fact is that the readers should realize that it is time for every student to learn the GVP math system, such that even the simple dichotomy can be explained correctly and precisely. The math teacher can simply explain to the students based on Axiom II and I. From Axiom I, we can say that the dichotomy series ½, ¼, 1/8, 1/16…, is forever decreasing and forever has continuity. From Axiom II, we can say that the dichotomy series, ½, ¼, 1/8, 1/16…, is a nonlinear numbers having a lower asymptote Ys. This lower asymptote is a nonlinear zero, which forever cannot be reached; the nonlinear zero is never parts of the nonlinear numbers. Nonlinear zero is static and the nonlinear numbers is dynamic. The dichotomy series is a nonlinear numbers associated with an asymptote, and the asymptote can be approached but cannot be reached. The asymptote in the dichotomy series happens to be a nonlinear zero, it cannot be reached forever. The use of “infinity” is not desirable because it implies that at certain point the continuity is terminated; it is as ambiguous as the use of “approximation”, “discount” and “limit”. 3.2 Arguments in Dichotomy – let us detect whether there is a violation of the law of nature Let us detect the violation of the law or the change of the rule of a game by citing three arguments in the followings. Argument I: A beautiful woman is sitting at one corner of the room; a handsome honest young man is standing at the other corner of the room. Although the young man is allowed to kiss the young woman, he can never be able to kiss the woman, because he first has to walk half the distance between them, then half the remaining distance, and so forth forever. The man has never kissed the woman. This is a dichotomy joke similar to Zeno’s nonlinear walk or Lie Tzu’s halving pole, there is no violation of the rule of the game. Argument II: A man states that the difference between a scientist and an engineer is that the engineer knows how to approximate and the scientist does not. The game goes like this: a woman sits on a chair in the middle of the field holding a gun in her hand; a scientist stands at one end of the field and an engineer stands at the other end of the field. When the woman fires the gun, the men can walk toward the woman. Whoever reaches the woman can kiss the woman. However, the rule of the game is that each time the man can walk only half of the remaining distance. Therefore, the 22

23 woman fired and fired and fired the gun, the men getting closer and closer to the woman. By the time the engineer gets to the distance two steps away from the woman, in the name of “approximation” he runs over to kiss the woman. The scientist keeps on walking stage by stage, he has never reach the woman. This is a dichotomy joke with engineer violated the rule of the game. When he was two steps away, he removed the nonlinear rule of the game and changed the game into a linear walk. Without the nonlinear restriction, with changing the rule of the game, and in the name of “approximation”, he reached the woman by linear walk. Argument III: In a carnival, an activity director asked three female volunteers - a three years old girl, a department store manager, and a math teacher, to participate in a game. In the game, the three volunteers lined up in one line. At a 100-meter distance from the line, there stand the girl’s mother with a toy, manager’s husband, and teacher’s boyfriend. The director announced that when he blows the whistle, everybody could walk half the distance toward his or her targets. Upon hearing the whistle, the girl walks straight toward her mother to get the toy. The other two women stop at 50 meters spot. After second whistle, the two women reached at 75-meter spot. Upon third whistle, the manager walk straight toward her husband to hug her husband because she thinks in her business a 25% discount is a common practice to close a deal, so in the name of “discount” she can ignore the rest of the rule and march through the rest of the distance in one stage. Meanwhile, the teacher walks to the 87.5 meters spot. In the fourth whistle, the teacher walks straight toward her boyfriend to kiss him. She thinks she can apply the limit theory and round up the rest of 12.5 meters to reach 100 meters. In this game, there is one winner and two game violators. The three years old girl is a winner because she does not understand the rule of the nonlinear game, but she is allowed to walk as she like, meaning she can take a linear walk without any restriction. Accordingly, she walks a linear walk and finished the walk without violating the rule of the game. The store manager violated the rule of the game for the last 25 meters of the walk where she walked the linear walk. She has changed the rule of the game from nonlinear to linear. The teacher is also a violator of the game. She changed the rule of the game from nonlinear to linear walk for the last 12.5 meters in the name of “limit theory”. According to the nonlinear rule, the two women can never reach their asymptotes if they walk a nonlinear walk. Unless the nonlinear rules are violated, they cannot reach their target. However, people can change the rule of the game in the name of “discount” “approximation” or “limit theory” to satisfy their desire. Nevertheless, everyone should recognize that they are violators of the rule and law, and should know how to tell the difference between the true from the disguising results. More discussion on the inappropriate use of “limit” is given in Appendix A-1.

4.0 Nonlinear Numbers Y with Two Asymptotes α(Y, Yu, Ys) 4.1 Nonlinear Numbers Y (with two asymptotes) versus Linear Numbers X α(Y, Yu, Ys) vs. β(X) Three ways to measure the Y relative to its lower asymptote Ys and upper asymptote Yu simultaneously 𝑠 𝑢 𝑠 𝑢 are the following three nonlinear face values: ∝21 = 𝑢 ; ∝22 = ; and ∝2 = 𝑢 𝑠 as illustrated in 𝑠 Fig. 1.5c. In comparing nonlinear numbers vs. linear numbers, we need to compare the nonlinear logarithmic value of nonlinear face values versus linear value of linear numbers, i.e., q∝21 vs. ; q∝22 vs. ; and q∝2 vs. 23

24 𝑠

𝑢

. [Note: 𝑞 ∝21 = 𝑞 ) = −𝑞 ) ]. In graphical expression, we plot q∝21 , q∝22 , and q∝2 on 𝑢 𝑠 nonlinear logarithmic scale, and on linear scale. When the above nonlinear face values, such as ∝21, are proportional to the linear face value , we can write their differential and integral equations as Eq. (4.1a) and Eq. (4.1b). It states that the nonlinear change of nonlinear face value ∝21 is proportional to the linear change of linear face value X. We will obtain a straight line when we plot the values of ∝21 on a logarithmic scale and the values of X on a linear scale. d[q((Y – Ys)/(Yu – Y))] = KdX

(4.1a)

q((Y – Ys)/(Yu – Y)) = KX + qC

(4.1b)

Taking anti-log on both sides of Eq. (4.1b), we get Eq. (4.1c) and Eq. (4.1d). (Y – Ys)/(Yu – Y) = 10(KX + qC) Y = [(Yu*10(KX + qC) + Ys]/(1 + 10(KX + qC) )

(4.1c) (4.1d)

An example for illustration with logistic equation In the previous sections, we have discussed several one-sided nonlinear numbers with single asymptote. In this section, we will explore the nature of two-sided nonlinear numbers Y having two asymptotes, an upper asymptote Yu and a lower asymptote Ys. To explore two-sided nonlinear numbers, it is best to either compare nonlinear numbers Y with linear numbers X or compare nonlinear numbers Y with nonlinear numbers X. Table 4.1 gives one simulated data for comparing nonlinear numbers Y with linear numbers X. Table 4.1 Comparing Nonlinear Numbers Y (with Two Asymptotes) versus linear numbers X α(Y, Yu, Ys) vs. β(X)

In Table 4.1, Column A gives the elementary x, Column B gives the cumulative X for succession of x. Column D gives the calculated Y with formula shown in formula bar: Y = ((Yu*10^(KX + qC) + Ys)/(1 + 10^(KX + qC)). This Y is from equation Eq. (4.1d). The K, C, Yu, and Ys are parameters in the table. 24

25 Column C gives y, which is the difference of adjacent Y in column D. In other words, Y is the cumulative of successive y. Fig. 4.1a Primitive graph y vs. X linear by linear scale

Fig. 4.1b Primary graph Y versus X Linear by linear scale

12 60 50 8

30

Y

y

40

4

20 10

0 0

20

40

0

60

0

20

X

Fig. 4.1c Leading graph also pre-proportionality graph (Y - Ys)/(Yu - Y) vs. X 10000 y = (Y - Ys)/(Yu - Y)

(Y - Ys)/(Yu - Y)

1000

500

0 20

60

Fig. 4.1d Proportionality plot q[(Y - Ys)/(Yu - Y)] vs. X

1500

0

40 X

40

60

X

80

y = 0.02θ 0.08X

100 y = 0.02e0.1842x R² = 1

1

0.01 0

20

40

60

80

X

Figure 4.1a is the plot of Column C versus Column B for y versus X; this is the primitive graph. It is a plot of elementary numbers y versus cumulative numbers X. Figure 4.1b is the plot of Column D versus Column B for Y versus X; this is the primary graph. It gives a solid comparison of one cumulative numbers versus another cumulative numbers. Figure 4.1c is the plot of Column E versus Column B for ∝21 =

𝑠 𝑢

versus X; this is the leading graph, it is also a pre-proportionality graph. It shows that the

larger the X the larger the ∝21 =

𝑠 𝑢

. By converting the vertical axis from linear to nonlinear logarithmic

scale, we get a straight line in Fig. 4.1d; this is the proportionality graph. In axis converting, we right clicking on vertical axis and selecting “Format Axis”. Then, in “Axis Option” selects “Logarithmic Scale”, we get Fig. 4.1d with log by linear scale. After converting the axis from linear to logarithmic scale, we click on data series followed by right clicking on the mouse and selecting “Add Trendline”. Next, in Trendline Options, select “Exponential”, select “Display Equation on Chart”, and select “Display R-squared value on Chart”. Then, select Close, we get Fig. 4.1d. Where we have an exponential equation y = 0.02e0.1842X with R2 = 1. 25

26 Placing an exponential equation on a log-linear graph is an awkward expression. We need to convert the equation to a 10 based proportionality equation by replacing e with θ, and convert 0.1842 to 0.1842/2.303 = 0.08 [note: the converting factor for e to 10 is 2.30285]. The final equation is y = 0.02θ0.08X. By taking logarithm on both sides, this equation is qy = q(

𝑠 𝑢

) = 0.08X + q(0.02) = KX + qC; K is the slope of the

straight line, or the proportionality constant; C is the position constant, which gives the intercept of the line at X = 0. This example with a lower asymptote Ys not equal to zero is to demonstrate that the lower asymptote not necessarily being zero. Figure 4.1a is a primitive elementary graph, it does not provide any physical meaning; however, many researchers in biological and radiation fields look at their curve and call it as a biphasic hormesis phenomenon and interpret it as “ low dosage (low X) beneficial and high dose (high X) inhibiting” phenomenon. The fact is that the true meaning should base on interpretation with primary graph and proportionality graphs but not the primitive graphs. The primary graph and proportionality graphs state that the nonlinear change of nonlinear numbers Y is proportional to the linear change of linear numbers X. 4.2 Nonlinear Numbers Y (with two asymptotes and at higher order of nonlinearity) versus Linear Numbers X – α(q(Y, Yu, Ys)) vs. β(X) In section 4.1 we have discussed the first case among q∝21 vs. , q∝22 vs. , and q∝2 vs. . In this section, let us discuss a case with a higher order of nonlinearity of Y numbers using the second case, i.e., we will discuss the case for q∝22 vs. , but with an additional nonlinearity by adding one q to q∝22 and designate as qq∝22 = q∝221 . In other words, we will discuss the cases with q221, where 221 = q((Yu – Ys)/(Y – Ys). In the subscript 221, the first number 2 stands for the total number of asymptote in the phenomenon, there are two asymptotes Yu and Ys in this case. The second number 2 refers to the form of measurement relative to the asymptotes — the form ((Yu –Ys)/(Y – Ys)) is the second form of measurements relative to two asymptotes ; the third number 1 indicates that there is an additional nonlinearity added to the above form to reach a higher order of nonlinearity, i.e., qq((Yu –Ys)/(Y – Ys)). When proportionality exists between the both sides of q∝221 vs. (4.2a) or Eq. (4.2b) and Eq. (4.2c).

(X), the differential equation is Eq.

d(qα221) = Kdβ0(X)

(4.2a)

d(q(q(Yu – Ys) – q(Y – Ys))) = KdX d(q(qYu – qY)) = KdX

when Ys ≠ 0

when Ys = 0

(4.2b) (4.2c)

The above equation states that the nonlinear change of the nonlinear face-values q((Yu –Ys)/(Y – Ys)) is proportional to the linear change of linear face-values X; it can also be read as the change of the nonlinear true-values q(q((Yu –Ys)/(Y – Ys))) is proportional to the change of linear true-values X. When the lower asymptote Ys is non-zero, the differential is Eq. (4.2b); when the lower asymptote Ys is zero, the differential equation is Eq. (4.2c).

26

27 Integration of Eq. (4.2b) and Eq. (4.2c) yields Eq. (4.2d) and Eq. (4.2e). q1[q(Yu – Ys) – q(Y – Ys)] = KX + q1C q1(qYu – qY)) = KX + q1C

when Ys ≠ 0

when Ys = 0

(4.2d) (4.2e)

The first log with superscript 1, q1, emphasizes “please plot the following face-values on the nonlinear logarithmic scale”, i.e., plotting the face-values (qYu – qY) or [q(Yu – Ys) – q(Y - Ys)] on logarithmic scale. Once the nonlinear face-values are plotted on logarithmic scale, their nonlinear true-values are q1(qYu – qY) or q1[(q(Yu – Ys) – q(Y - Ys))]. These true-values are what we need to compare with the linear true-values (X) on a semi-log graph. Ultimately, we are interested in obtaining proportionality between the two continuous numbers. Equations (4.2a) to (4.2e) describe sigmoid curves between the upper and lower asymptotes Yu and Ys. Data showing the above phenomenon possess two characteristics: (1) Y data (Note: Y is cumulative of (y) for succession of X, or (y) is individual increment of Y between two X) can be shown as a sigmoid curve with an upper asymptote Yu and a lower asymptote Ys, (2) the (y) data can be shown as an asymmetric bell-shaped curve. We can plot equations (4.2d) and (4.2e) on a graph of log-linear (semi-log) scales. In the above equation, (q(Yu - Ys) – q(Y - Ys)) is the nonlinear face-values, which always assumes positive values. The existence of proportionality in the equation implies that, when plotting one face value versus another face value on appropriate scales, we obtain a linear proportionality relationship. By stripping off the notion of GVP, We can write equation (4.2e) as a conventional math form Eq. (4.2f) when Ys = 0. This is one form of Gompertz’s Law [8]. K

y  Yu

x) 1 (  (  2.303 ) C

or

y  Yue

(

2.303  Kx e ) C

Ys  0

(4.2f)

5.0 Let’s Leap to Nonlinear by Nonlinear Phenomena In the previous section, we used Zeno’s mathematical joke to discuss the nonlinear (of Y) by linear (of X) phenomena. In this section, let us extend the discussion to the nonlinear (of Y) by nonlinear (of X) phenomena. 5.1 Anything Wrong with Y = 1/X? In algebra, two of the simplest ways of writing the relationship between Y and X are Y = X and Y = 1/X. We can consider the former equation as a special case of linear equation Y = KX + C with proportionality constant K, K = 1 and integral constant C, C = 0. The differential form of this linear equation is dY = KdX. It says that linear change of Y is proportional to the linear change of X, where K is proportionality constant. This linear phenomenon is simple and straightforward. The linear equation is expressed with straight lines on a Cartesian graph in Fig. 5.1a. Depending on whether K is positive or negative the straight line can have different slope and orientation; and depending on whether C is positive, negative, or zero, the line can pass through zero or intercept the axes at different locations.

27

28 In the graph, the straight-line y = 3x + 30 has the slope K = 3, and the line intercepts the vertical axis at C = 30. The straight line y = -3x passes the linear zero at (0,0); with the negative slope K = -3. Fig. 5.1a Linear (of Y) by linear (of X ) phenomena 80

y = 3x + 30 40

C = 30 linear zero (0, 0)

Y

0 -40

-20

0

20

40

-40

y = -3x -80 X

Now, let us look at a new arrangement of Y and X in the form of Y = 1/X, this is called an inverse equation. In traditional math, students are taught that when X = 0, Y is undefined or infinity. This teaching is flawed and is incorrect. A sound math should have nothing that is undefined; all sound mathematics should have everything defined. Let us examine the deficiency of the traditional math through graphical explanation.

6

10

4 2 0 0

2

4 x

6

y = (Y - Ys), Ys = (0) = ϕ (Y - Ys) = 1/(X - Xs)

y, y = 1/x

Fig. 5.1c proportionality plot q(Y - Ys) = q[1/(X - Xs)] = -q(X - Xs)

Fig. 5.1b Plot of nonlinear numbers in linear graph y vs. x with y = 1/x

y = x-1 R² = 1

1

0.1 0.1

1

10

x = (X - Xs), Xs = (0) = ϕ

The equation y = 1/x, when written in αβ (GVP) math form (Y – Ys) = 1/(X – Xs), is a nonlinear by nonlinear phenomenon. Figure 5.1b shows a plot of y versus x on a linear graph. In contrast to Fig. 5.1a, obviously, it shows not a linear line but a nonlinear line. Two different ways of arranging Y and X in an equation, apparently, have resulted in significant different meaning. The writing of Y = X is for linear case, with both Y and X as linear numbers. The writing of Y = 1/X is for nonlinear case; if it is a nonlinear case, then we need to treat both Y and X as nonlinear numbers. From previous section, we have learned the difference between a linear numbers and a 28

29 nonlinear numbers. A nonlinear numbers is associated with an asymptote. Now, where are the asymptotes? There are two lower asymptotes; they are Ys = (0) = ϕ and Xs = (0) = ϕ, both Ys and Xs have nonlinear zero as their lower asymptote. In Figure 5.1b, the line of y and x can extend forever, but will never touch their asymptotes. Asymptotes are not parts of the data line. Figure 5.1c gives the proportionality plot, where (Y – Ys) versus (X – Xs) is plotted on a log-log graph to give a straight line. Note that the straight line can extend in both directions. As X increases, Y decreases toward asymptotic nonlinear zero, and as Y increases, X decreases toward asymptotic nonlinear zero. The graph shows the continuity of the line in logarithmic scale and shows an implication of the existence of nonlinear asymptotes Ys = (0) and Xs = (0); these two asymptotes are not parts of the nonlinear numbers and cannot be touched or reached, and thus cannot be put on the graph but can only be implicated. Figure 5.1c is a plot of face values (Y –Ys) versus face values (X – Xs) on a loglog graph. By accounting for the nonlinear logarithmic scale for the nonlinear face values, the equation of the straight line is Eq. (5.1a). q(Y – Ys) = -Kq(X – Xs) + qC

(5.1a)

It states that the nonlinear true-values q(Y – Ys) is negatively proportional to the nonlinear true-values q(X – Xs), where K = -1, and C is the position constant or integral constant, which is 1, and q1 = 0 [q(Y – Ys) = - q(X – Xs) + q1 = q1 - q(X – Xs) = q(1/(X – Xs)), and (Y – Ys) = 1/(X – Xs)]. Fig. 5.1d Nonzero lower asymptote Series 2 has Ys = 1.5, Xs = 1.5

Fig. 5.1e Proportionality plot for series 2

10

8

Y

6

series 2

4 2

Ys = 1.5

y = (Y - Ys), Ys = 1.5 (Y - Ys) = 1/(X - Xs)

Xs = 1.5

1

series 2

0 0

2

4 X

6

8

0.1 0.1

1

10

x = (X - Xs), Xs = 1.5

The lower asymptotes Ys and Xs are not necessarily being zeroes. Figure 5.1d illustrates that the data in series 2 exhibit non-zero lower asymptotes, Ys = 1.5 and Xs = 1.5. Figure 5.1e shows the proportionality plot of this nonlinear data set. When Y and X assume both positive and negative values, one curve exists in the first quadrant and the other in the third quadrant in a Cartesian graph, as shown in Figure 5.1f. Their common asymptotes are the pivot asymptotes Yp = 0 and Xp = 0. In the first quadrant, the nonlinear change in (Y – Ys) is negatively proportional to the nonlinear change in (X – Xs). Eq. (5.1b) describes their equation. In the third quadrant, the nonlinear change in (Yu – Y) is proportional to the nonlinear change in (Xu – X), their equation is described by Eq. (5.1c). A nonlinear by nonlinear phenomenon as described by the equation 29

30 Eq. (5.1a), Eq. (5.1b), or Eq. (5.1c), is a phenomenon that obeys the proportionality law as power law. This is the proportionality law of the fourth kind (PL4) [1, 2]. d(q(Y - Ys)) = Kd(q(X – Xs))

or

d(q(Y – Yp) = Kd(q(X – Xp)

(5.1b)

d(q(Yu – Y)) = Kd(q(Xu – X))

or

d(q(Yp – Y) = Kd(q(Xp – X)

(5.1c)

Fig. 5.1g

8

y = 1/x

4

0 -6

-3

0

3 Series1

-4

Series2 -8 x

6

y = (Y - Yp) or (Yp - Y), Yp = 0 (Y - Yp) = 1/(X - Xp) or (Yp - Y) = 1/(Xp - X)

Fig. 5.1f 10

Series1 Series2

1

0.1 0.1

1

10

x = (X - Xp) or (Xp - X) , Xp = 0

In the first quadrant, Yp = (0) and Xp = (0) are the lower asymptotes of Y and X. In the third quadrant, Yp = (0) and Xp = (0) are the upper asymptotes of Y and X. In any case, measurements of difference relative to the asymptotes are the upper values minus the lower values, e.g., (Y – Yp) for the first quadrant and (Yp – Y) for the third quadrant. In the third quadrant, the negative X value makes (0 – X) positive and the negative Y value makes (0 – Y) positive. The plot of (Y – Yp) versus (X – Xp) and (Yp – Y) vs. (Xp – X) on a log-log graph yields a straight line with slope K = -1, as shown in Figure 5.1g. Thus, Y and X are continuous everywhere, and Y and X will never attain the pivot asymptotes. The above illustration leads to a new concept that, when dealing with two collections of continuous numbers, we are comparing paired numbers side by side in the GVP equation with proportionality; there is no need for tagging “dependent” and “independent” for α and β in the equation dα = Kdβ. Equations Eq. (5.1a), Eq. (5.1b), and Eq. (5.1c) are the power law equations. 5.2 Nonlinear Numbers (with two asymptotes) versus Nonlinear Numbers (with one asymptote) α(q(Y, Yu, Ys)) vs. β(q(X, Xs)) Let us discuss a nonlinear by nonlinear phenomenon having three asymptotes. In this phenomenon, the nonlinear change of the first nonlinear numbers is proportional to the nonlinear change of the second nonlinear numbers. The first nonlinear numbers have both upper and lower asymptotes, and have a higher order of nonlinearity in the form of ratio relative to the asymptotes; while the second nonlinear numbers have a single lower asymptote. Nonlinear phenomena of this group are abundance in natural sciences. Usually, we need a combination of four types of graphs to fully expressing this nonlinear phenomenon: a primitive graph, a primary graph, a leading graph, and a proportionality graph. 30

31

In this example, we will have a lower asymptote Ys. However, for a concise expression, we shall omit some of the Ys when we assume Ys = 0. In this phenomenon, we have a nonlinear face-value 221, 221 = q((Yu– Ys )/(Y – Ys)) = qYu – qY and a nonlinear face-value β11, β11= X – Xs. The proportionality equation is Eq. (5.2a): d(qα221) = Kd(qβ11)

(5.2a)

The above equation states that the nonlinear change of nonlinear face-value α221 is proportional to the nonlinear change of nonlinear face-value β11. It is also read as the change of nonlinear true-values qα221 is proportional to the change of nonlinear true-values qβ11. The subscript in α221 indicates that the α nonlinear numbers has two (2) asymptotes and has the second (2) form of measurement relative to the asymptotes, (Yu– Ys )/(Y – Ys), and there is an additional (1) enforcement of nonlinearity to the above second form, i.e., q[(Yu – Ys)/(Yu - Y)]. The subscript in β11 indicates that the β nonlinear numbers has one (1) asymptote and has the first (1) form of measurement relative to the asymptote (X - Xs). Substituting q(Yu/Y) = (qYu – qY) for 221 and (X – Xs) for 11, we obtain Eq. (5.2b) or Eq. (5.2c), where nonlinear numbers Y exist between upper asymptote Yu and lower asymptote Ys. d(q(qYu – qY)) = Kd(q(X – Xs))

Ys = 0

d(q(q(Yu –Ys) – q(Y - Ys))) = Kd(q(X – Xs))

(5.2b)

Ys ≠ 0

(5.2c)

when Ys = 0

(5.2d)

Integration of Eq. (5.2b) yields Eq. (5.2d). q1(qYu – qY) = Kq1(X – Xs) + q1C

The log with superscript 1, q1, means “please plot the following face-values on the nonlinear scale, e.g., plotting the face-values (qYu – qY) and (X – Xs) on logarithmic scale. q1 also means that this first q is a differentiable q. Once the nonlinear face-values are plotted on logarithmic scale, their nonlinear truevalues are q1(qYu – qY) and q1(X – Xs). These true-values are what we need to compare on a log-log graph to obtain proportionality between two nonlinear numbers in terms of two true-values. When the equation Eq. (5.2d) is plotted on nonlinear graphs of log-log scales, we obtain proportionality graphs having the slope of the straight line K and the intercept of the line at X = 1 is C. By stripping down the GVP notion, Eq. (5.2d) can be written in a conventional math form as Eq. (5.2e) when (qYu – qY) and (X - Xs) exhibit an inverse proportionality relationship or negative K (–K).

y  Yu  Cx

K

or

y  Yu

1 K x C

(5.2e)

Let us use a set of simulated data to illustrate the relationship of one nonlinear numbers (with two asymptotes) versus another nonlinear numbers (with one asymptote), e.g., q(Y, Yu, Ys) vs. q(X, Xs). Table 5.2 gives the simulated data for q(Y, Yu, Ys) vs. q(X, Xs). Column B is the calculated data for

31

32 cumulative Y based on the parameters Yu, Ys, C, and K listed in the table. The formula for Y is given in the formula bar, it is Y = (Ys + Yu*C*(X^K))/(1 + C*(X)^K). Table 5.2 Simulated data for q(Y, Yu, Ys) vs. q(X, Xs)

Fig. 5.2b Primary graph Y and X in linear by log scale

Fig. 5.2a Primary graph Y vs. X in linear by linear scale

120

120

Y

Y, inlinear scale

80

40

Yu

80

40

Ys = 0

1.E-06

0 0.00

0.05 X

0.10

1.E-04

1.E-02

1.E+00

X, in log scale

Fig. 5.2c Primitive graph y vs. X in linear by linear scale

Fig. 5.2d Primitive graph y vs. X in linear by log scale 30

20

20

y

y

30

10

10

0 0.00

0.05 X

0.10

0 1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 X

32

33

Fig. 5.2d Proportionality plot (Yu = 110) 1000.00

y = (Y - Ys)/(Yu - Y)

100.00

y = 800x0.85 R² = 1

10.00 1.00 qy = 0.85qX + q800 qy = KqX + qC K = 0.85

0.10 0.01 1.E-06

1.E-04

1.E-02

1.E+00

x = X - Xs, Xs = (0)

References [1]. [2]. [3]. [4]. [5]. [6]. [7].

[8]. [9]. [10]. [11].

Ralph W. Lai, A text book on Nonlinearity in Life and Biomedical Sciences, the Cornerstone Company, pp.314, 2011. Ralph W. Lai, A text book on Nonlinearity and Proportionality in Science, Medicine, and Engineering, the Cornerstone Company, pp.406, 2012. G. J. Diaz, E. Calabrese, and R. Blain, Aflatoxicosis in Chickens (Gallus Gallus): An Example of Hormesis? Poultry Science, 87:727-732, 2008. A. R. D. Stebbing, Adaptive Response Account for the β-Curve – Hormesis is Linked to Acquired Tolerance, Nonlinearity Biol Toxicol Med, 1(4), 493 -511, October, 2003. Stephen H. Curry, Clinical Pharmacokinetics- The MCQ Approach, The Telford Press, Caldwell, New Jersey, pp. 39-40, 1987. Ralph W. Lai, Get More Information from Flotation-Rate Data, Chemical Engineering, October 19, 1981, pp.181-182. Ralph W. Lai, Melisa W. Lai, and Alec G. Richardson, Unified Proportionality Equation for Modeling Biological and Pharmacological Data, Proceedings, 11th IEEE Symposium on Computer-Based Medical Systems, June 12-14, 1998, Lubbock, Texas, USA, pp.104-109. B. Gompertz, “On the Nature of the Function Expressive of the Law of Human Mortality”, Philosophical Transactions, Royal Society of London, 1825. Bernard L. Cohen, Test of the Linear-No Threshold Theory of Radiation Carcinogenesis for Inhaled Radon Decay Products, Health Physics, February, Vol. 68, No. 2, 1995, pp. 157-174. Bernard L. Cohen, Data Set “SHORT92”, University of Pittsburgh, Pittsburgh PA 15260, 1992. Ralph W. Lai, “The Myth of Biological a d Radiatio Hormesis,” Appendix A, www. Researchgate.net, pp.25-26, 2013.

33

34 Appendix A - 1 The formal definition of limit is the following: lim f ( x)  L . This equation with equal sign “=” is x c

incorrect. The correct definition should be lim f ( x)  L but not lim f ( x)  L . Use of equal sign is an x c

x c

abuse of mathematical Axioms I and II. Other than wrongfully using the equal sign, the intended meaning of limit is correct. It means for every ε > 0, there exists a δ > 0 such that if x differs from c by less than δ, then f(x) differs from L by less than ε. i.e., ε > 0, Ǝ δ > 0, such that if | x – c | < δ then |f(x) – L | < ε . The above definition is nothing more than intended to state that f(x) is a collection of continuous numbers that forever has continuity. For collection of continuous linear numbers, there is no need for the definition of “limit”. The definition of “limit” is applicable to nonlinear numbers. In internet, some people have posted examples of linear numbers and trying to use the definition of “limit” to prove the continuity of linear numbers. It is a wasteful exercise in math. Axiom I is an Axiom, there is no need to go around a circle to prove the existence of continuity (Axiom) for the linear numbers. For collection of nonlinear numbers, it is correct to say that |f(x) – L | < ε, since the nonlinear f(x) has continuity and never equal to L. It does not say that nonlinear f(x) = L. Traditional math does not realize that L is the asymptote of nonlinear f(x), and that nonlinear f(x) can never be able to touch the L. There is “” but no “=”. However, this definition of “limit” is nothing more than Axiom II. It means the nonlinear numbers f(x) has continuity and has an asymptote, where the nonlinear numbers can approach the asymptote but cannot touch the asymptote. The |f(x) – L | can be as small as you would like, but the f(x) will never touch the asymptote L. L is static and f(x) is dynamic; a static can never equal to the dynamic. 

With the above clarification, we can recognize that

n 1

1

( 2) n 1

n

1

( 2)

n

1 1 1    ...  1 is wrong. 2 4 8

1 1 1    ...  1 is correct. The later says 1 is the asymptote of the collections of nonlinear 2 4 8

numbers to the left. One is dynamic and the other is static. They are never equal. [Note 1: When a series is a nonlinear numbers and has continuity, it can be equated to another nonlinear number set. For example, Euler’s power series and its solution of Basel problem are equating the nonlinear numbers on both sides (Leonhard Euler, 1707 – 1783). It is a correct and perfect way to use an equal sign as long as the continuity is preserved on both sides. (Both “e” and “π” are nonlinear numbers). Note 2: In his classic math book, The Theory of Probability, B. V. Gnedenko was able to use limit in his proof by using a lot of unequal sign “”, and a lot of “→”. He really understood his math. On the contrary, many people get lost by using “=” and “is” to equate nonlinear numbers to its asymptote. Note 3: It is highly probable that the original definition of limit has used “~” or “→” or other non-equal symbol but did not use the equal sign “=”; but some absent minded people intentionally or unintentionally put the “=” sign to the definition of limit, resulting in misleading almost everybody].

34

35 Appendix A -2 Extension of XY Math into αβ Math (Fundamentals of αβ Math)

   

There are two classes of continuous numbers: linear numbers and nonlinear numbers. The key to the classification of the numbers is the asymptote [1, 2]: liner numbers have no asymptote and are not associated with any asymptotes, such as ...-3, -2, -1, 0, 1, 2, 3, 4... ; In contrast, nonlinear numbers are associated with one or two asymptotes, such as …10-3, 10-2, 10-1, 100, 101, 102, 103, 104…, which has a nonlinear zero as its lower asymptote. The latter numbers decrease in steps from (right to left) 10000, to 1000, to 100, to 10, to 1, to 0.1, to 0.01, and to 0.001 etc., These numbers are decreasing toward nonlinear zero but will never reach or touch the nonlinear zero. Nonlinear numbers always have continuity and always preserve the continuity - meaning it always has the next step or the next number. Asymptote is not part of the nonlinear numbers and can never be part of the nonlinear numbers [1, 2]. [*Note: “numbers” refers to a series or a set of numbers. We use it either as singular or as plural]. Not all the zeroes are the same. There are two types of zero, linear zero and nonlinear zero. Liner numbers has no association with asymptote, as shown for Y in Fig. 3.0a and Fig.5.1a; while nonlinear numbers is associated with one or two asymptotes, as shown for Y in Fig. 1.5a, 2.1a, 2.2a, and 5.1d. The change (or differential) of linear numbers is dY, dX, as demonstrated in Fig. 3.0a for the linear phenomena in the text. The change (or differential) of nonlinear numbers is the change of nonlinear numbers with measurement relative to their asymptotes and then place them on nonlinear logarithmic scale (i.e., multiply by q, q = log). For example, when there is only one upper asymptote Yu or lower asymptote Ys associated with the nonlinear numbers Y, the nonlinear change of Y is d(q(Yu – Y)) or d(q(Y – Ys)). The measurement of (Y – Ys) is shown as α11 = Y – Ys in Fig. 1.5a. The measurement of (Yu – Y) is shown as α12 = Yu – Y in Fig. 1.5b. When there are two asymptotes associated with the nonlinear numbers Y, one upper and one lower asymptote, the nonlinear change of Y can be in three forms: ∝21 = . Subscript 2 in α21 indicates there is two asymptotes associated with nonlinear numbers Y, and the second subscript 1 indicates is the first form of measurement relative to Y. The second and third forms are ∝22 =

and

∝2 = 

There are two mathematical Axioms in the αβ Math: Axiom I on continuity and Axiom II on asymptote. 35

36 Axiom I: Continuity exists for all collection of continuous numbers in relating to a physical phenomenon. The continuous numbers has continuity and always has a next step or a next number. Continuous numbers are dynamic, non-terminating, and can never be forced to stop (It is dishonest to use the uncertain word “infinity” as a disguise to stop the continuity of the numbers).

Axiom II: Asymptote can be approached, but cannot be touched or crossed by the continuous nonlinear numbers. In another words, asymptote is never a part of the continuous numbers. Asymptotes are static and cannot move. 

The standard scale for nonlinear numbers is a 10 based logarithmic scale, as shown in Fig. 1.4b; its characteristic is the existence of a nonlinear zero, which can be approached but cannot be reached or touched. The scale in Fig.1.4b is nonlinear and is approaching nonlinear zero as its lower asymptote but will never be able to touch this nonlinear zero. This is to say that the nonlinear zero is not part of the continuous nonlinear numbers: …0.01, 0.1, 1, 10, 100… These nonlinear numbers are measured from nonlinear zero, are continuous, and are able to be labeled on the logarithmic (nonlinear) scale. However, its nonlinear zero is its lower asymptote, is not part of the nonlinear numbers, and thus cannot be plotted on the nonlinear scale. When trying to plot a zero value on a logarithmic graph using a Microsoft Excel, we will get a warning banner as shown in Fig.1.4f. Technically speaking, the listed warning is correct, but it did not provide theoretical background and explain why. The theory behind it is that the logarithmic scale is for the nonlinear numbers, and the nonlinear numbers has nonlinear zero as its lower asymptote. The nonlinear zero can be approached but cannot be touched. Meanwhile, because the nonlinear zero is not part of the continuous nonlinear numbers, it cannot be plotted on the graph. One of the conventions adopted in the αβ math is that when assigning the nonlinear numbers αi or βi onto the logarithmic scale, their true-values are logαi, logβi, or qαi, qβi (q is a notation of log, q = log). Other convention used in this math system is the equivalency of scale: Nonlinear scale = logarithmic scale = q scale. To distinguish αβ math from the traditional math, several terms such as “dependent variable”, “independent variable”, “functions”, “limit”, “infinity”, and “polynomial” are discarded in this extended math. When either the linear numbers or the cluster of nonlinear numbers are assigned or plotted on the axes of graphs, these numbers are called face values of the numbers. Truevalues of numbers are defined as the face values embedded with the linear or nonlinear scale. For linear numbers, the face values are always the same as the true-values. For nonlinear numbers, the face values of nonlinear numbers are not the same as the truevalues of nonlinear numbers. The true-values of nonlinear numbers are obtained by multiplying q to the nonlinear face values. For example, when we assign a nonlinear numbers αi to the nonlinear scale, its face-vales is αi; however, its true-vales are qαi. True-values, but not the face values, are what we need to account for when evaluating nonlinear changes. Face values of nonlinear numbers that can be assigned to the nonlinear scales may include a difference, a ratio, or a combination of both of nonlinear numbers, all having nonlinear numbers measured relative to the asymptotes. Examples of useful face values in α and β, and their corresponding X, Y expressions are listed in the following table. 36

37 Table A-2 Examples of useful face-values in ,  and their corresponding X, Y expressions In αijk and βijk, i = number of asymptote, j = forms of measurement, k = extra higher order of nonlinearity  numbers and corresponding Y’s 0 asymptote

 numbers and corresponding X’s

0

Y

0

X, X2

11

(Y – Ys)

11

(X – Xs)

12

(Yu – Y)

12

(Xu – X)

13

Ys/Y

13

X/Xs

14

Yu/Y

14

Xu/X

1 asymptote

21 2 asymptotes

22 23

Y − Ys Yu − Y

211

q(

Yu − Ys Y − Ys

221

q(

Yu − Y Yu − Ys

231

Y Ys

Yu Y

)

Yu Ys Y Ys

Yu Ys

q(

Yu Y

β21

) β22 )

X − Xs Xu − X

β211

q(

Xu − Xs X − Xs β221

q(

X Xs

)

Xu X

Xu Xs X Xs

)

1.

Yu = upper asymptote, Ys = lower asymptote.

2.

We can interchange the ratio terms (Y – Ys)/(Yu – Y) for (Yu – Y)/(Y – Ys) etc., because the nonlinear true-values of these nonlinear face-values are q multiply these ratio terms, and q((Y – Ys)/(Yu – Y)) = -q((Yu – Y)/(Y – Ys)). We only need to remember to change the sign in the equation from positive to negative or vise visa when needed.

37