Division by Zero Calculus in Trigonometric Functions

Transcription

1 Division by Zero Calculus in Trigonometric Functions Saburou Saitoh Institute of Reproducing Kernels, Kawauchi-cho, , Kiryu , JAPAN April 1, 019 Abstract: In this paper, we will introduce the division by zero calculus in triangles trigonometric functions as the first stage in order to see the elementary properties. Key Words: Zero, division by zero, division by zero calculus, 0/0 = 1/0 = z/0 = 0, triangle, trigonometric function, Laurent expansion. 010 Mathematics Subject Classification: 30E0, 30C0, 00A05, 00A09. 1 Division by zero calculus The essence of the division by zero is given by the simple division by zero calculus. For this conclusion, see the papers in the references. Therefore, we will simply introduce the division by zero calculus. For any Laurent expansion around z = a, f(z) = 1 n= C n (z a) n + C 0 + C n (z a) n, (1.1) n=1 we define the identity f(a) = C 0. (1.) 1

2 By considering derivatives in (1.1), we define any order derivatives of the function f at the singular point a; that is, f (n) (a) = n!c n. In addition, we will refer to the naturality of this division by zero calculus. Recall the Cauchy integral formula for an analytic function f(z); for an analytic function f(z) around z = a for a smooth simple Jordan closed curve γ enclosing one time the point a, we have f(a) = 1 πi γ f(z) z a dz. Even when the function f(z) has any singularity at the point a, we assume that this formula is valid as the division by zero calculus. We define the value of the function f(z) at the singular point z = a with the Cauchy integral. The basic idea of the above may be considered that we can consider the value of a function by some mean value of the function. The division by zero calculus opens a new world since Aristotele-Euclid. See, in particular, [] also the references for recent related results. On February 16, 019 Professor H. Okumura introduced the surprising news in Research Gate: José Manuel Rodríguez Caballero Added an answer In the proof assistant Isabelle/HOL we have x/0 = 0 for each number x. This is advantageous in order to simplify the proofs. You can download this proof assistant here: J.M.R. Caballero kindly showed surprisingly several examples by the system that tan π = 0, log 0 = 0, exp 1 (x = 0) = 1, x others. Furthermore, for the presentation at the annual meeting of the Japanese Mathematical Society at the Tokyo Institute of Technology:

3 March 17, 019; 9:45-10:00 in Complex Analysis Session, Horn torus models for the Riemann sphere from the viewpoint of division by zero with [], he kindly sent the message: It is nice to know that you will present your result at the Tokyo Institute of Technology. Please remember to mention Isabelle/HOL, which is a software in which x/0 = 0. This software is the result of many years of research a millions of dollars were invested in it. If x/0 = 0 was false, all these money was for nothing. Right now, there is a team of mathematicians formalizing all the mathematics in Isabelle/HOL, where x/0 = 0 for all x, so this mathematical relation is the future of mathematics. lp15/grants/alexria/ Surprisingly enough, he sent his at :4 as follows: Nevertheless, you can use that x/0 = 0, following the rules from Isabelle/HOL you will obtain no contradiction. Indeed, you can check this fact just downloading Isabelle/HOL: copying the following code theory DivByZeroSatoih imports Complex Main begin theorem T: x/ = 000 for x :: complex by simp end he referred to horn torus models [] in the system. Meanwhile, on ZERO, S. K. Sen R. P. Agarwal [3] published its long history many important properties of zero. See also R. Kaplan [3] E. Sondheimer A. Rogerson [5] on the very interesting books on zero infinity. In particular, for the fundamental relation of zero infinity, we stated the simple fundamental relation in [] that The point at infinity is represented by zero; zero is the definite complex number the point at infinity is considered by the limiting idea that is represented geometrically with the horn torus model []. S. K. Sen R. P. Agarwal [3] referred to the paper [4] in connection with division by zero, however, their understings on the paper seem to be not suitable (not right) their ideas on the division by zero seem to be 3

4 traditional, indeed, they stated as the conclusion of the introduction of the book that: Thou shalt not divide by zero remains valid eternally. However, in [1] we stated simply based on the division by zero calculus that We Can Divide the Numbers Analytic Functions by Zero with a Natural Sense. They stated in the book many meanings of zero over mathematics, deeply. In this paper, we will introduce the division by zero calculus in triangles trigonometric functions as the first stage in order to see the elementary properties. Trigonometric functions In order to see how elementary of the division by zero, we will see the division by zero in trigonometric functions as the fundamental object. Even the cases of triangles trigonometric functions, we can derive new concepts results. Even the case tan x = sin x cos x, we have the identity, for x = π/ 0 = 1 0. Of course, for identities for analytic functions, they are still valid even at isolated singular points with the division by zero calculus. Here, we will see many more direct applications of the division by zero 1/0 = 0/0 = 0 as in the above with some meanings. Note that from the inversion of the both sides cot x = cos x sin x, for example, we have, for x = 0, 0 =

5 By this general method, we can consider many problems. We will consider a triangle ABC with BC = a, CA = b, AB = c. Let θ be the angle of the side BC the bisector line of A. Then, we have the identity tan θ = c + b c b tan A, b < c. For c = b, we have tan θ = b 0 tan A. Of course, θ = π/; that is, tan π = 0. Here, we used b 0 = 0 we did not consider that by the division by zero calculus for c = b c + b c b = 1 + b c b c + b c b = 1. In the Napier s formula a + b a b = tan(a + B)/ tan(a B)/, there is no problem for a = b A = B. Masakazu Nihei derived the result (H. Okumura sent his result at :06): For the bisector line of A in the above formula, if we consider a general line with a common point with BC, then we have tan θ = bc sin A (b c)(b + c). Here, for b = c, of course, we have θ = π/ tan π = 0. 5

6 Similarly, in the formula b c 1 b + a tan A + b + c b c tan A = sin(b C), for b = c, B = C, we have that is right. We have the formula 0 + c 0 tan A = 0, a + b c a b + c = tan B tan C. If a + b c = 0, then by the Pythagorean theorem C = π/. Then, 0 = tan B tan π = tan B. 0 Meanwhile, for the case a b + c = 0, B = π/, we have In the formula a + b + c abc a + b c 0 = cos A a = tan π tan C = 0. + cos B b + cos C, c for the case a = 0, with b = c B = C = π/ the identity holds. Meanwhile, the lengths f f of the bisector lines of A in the out of the triangle ABC are given by f = bc cos A b + c respectively. f = bc sin A b c, 6

7 If b = c, then we have f = 0, by the division by zero. However, note that, from f = sin A ) (c + c, b c by the division by zero calculus, for b = c, we have f = b sin A = a. The result f = 0 is a popular property, but the result f = a is also an interesting popular property. See [8]. Let H be the perpendicular leg of A to the side BC let E M be the mid points of AH BC, respectively. Let θ be the angle of EMB (b > c). Then, we have 1 tan θ = 1 tan C 1 tan B. If B = C, then θ = π/ tan(π/) = 0. In the formula cos A = b + c a, bc if b or c is zero, then, by the division by zero, we have cos A = 0. Therefore, then we should underst as A = π/. This result may be derived from the formulas sin A = (a b + c)(a + b c) 4bc cos A (a + b + c)( a + b + c) =, 4bc by applying the division by zero calculus. This result is also valid in the Mollweide s equation for a = 0 as sin B C = (b c) cos A a 0 = (b c) cos A 0., 7

8 We have the identities, for the radius R of the circumscribed circle of the triangle ABC, S = ar A = 1 bc sin A = 1 sin B sin C a sin A = abc 4R = R sin A sin B sin C = rs, s = 1 (a + b + c). If A is the point at infinity, then, S = s = r A = b = c = 0 the above identities all valid. For the identity tan A = r s a, if the point A is the point at infinity, A = 0, s a = 0 the identity holds as 0 = r/0. Meanwhile, if A = π, then the identity holds as tan(π/) = 0 = 0/s. In the identities cot A + cot B + cot C = cot A cot B cot C cot A + cot B + cot C = cot A cot B cot C + csc A csc B csc C, we see that they are valid for A = π B = C = 0. For the identity cot(z 1 + z ) = cot z 1 cot z 1 cot z 1 + cot z, for z 1 = z = π/4, the identity holds. For a triangle, we have the identity cot A + cot(b + C) = 0. For the case A = π/, the identity is valid. For the identity tan A + tan B + tan C = tan A tan B tan C, for A = π/, the identity is valid. 8

9 In the sine theorem: a sin A = b sin B = c sin C = R, for A = π, B = C = 0 then we have In the formula cos A cos B ab + a 0 = b 0 = c 0 = 0. cos B cos C bc + cos C cos A ca = sin A a for a = 0, A = 0, b = c, B = C = π/, the identity is valid. In the formula R = abc 4S, for S = 0, we have R = 0 (H. Okumura: :40). In the formula cos A + cos B = (a + b) c sin C, for c = 0, we have b = c A = B = π/ the identity is valid. In a triangle ABC, let H be the orthocenter J be the common point of the three perpendicular bisectors. Then, we have AH = d a = a tan A the distance of J to the line BC = h a = For A = π/, we have that d a = h = 0 (V. V. Puha: :10). a tan A. For the tangential function, note the following identities. In the formula tan θ = sin θ 1 cos θ 1 + cos θ = ± 1 + cos θ, 9

10 for θ = π, we have that 0=0/0. For the inversions from x = 0, we have 1 0 = 0 = ± 0 = 0. In the formula tan z 1 ± tan z = sin(z 1 + z ) sin z 1 sin z, for z 1 = π/, z = 0, we have that 0=1/0. In the formula tan x = 1 ± 1 sin x, sin x for x = π, we have In the elementary identity tan(α + β) = for the case α = β = π/4, we have 0 = 0 0. tan α + tan β 1 tan α tan β, tan π = = 0 = 0. Further note that it is valid for α + β = 0 α + β = π/ (H. Okumura: :03). Furthermore, the formula tan(α + β) = tan α tan β 1 1, tan α tan β is valid for α = π/ or β = π/ for α + β = π/ (H. Okumura: :05). In the identity 1 sin α 1 + sin α = 1 tan α, cos α for α = π/, we have 0 =

11 For the double angle formula for α = π/, we have that In the identity for α = π/6, we have that is right. In the identities tan α = tan α 1 tan α, 0 = tan 3α = tan α tan3 α 1 3 tan α, 1 + cos x sin x tan π = 0, = sin x 1 cos x 1 cos x = tan x sin x for x = 0, we have the identity In the identities tan(x + iy) = cot(x + iy) = for x = π/, y = 0, they are valid. In the identity arctan 0 0 = 0. sin x + i sinh y cos x + cosh y sin x i sinh y cosh x cos x b a = arcsin b a + b, for a = 0, the identity is valid. We can find similar interesting identities in the following identities: 11

12 sin 3x + sin x = tan x, cos 3x + cos x sin 3x + sin x cos 3x cos x = cot x sin 3x sin x cos 3x + cos x = tan x (H. Okumura: the first one is obtained by M. Nihei). We consider a triangle ABP with A( a, 0), B(a, 0), P (x, y) with a > 0. Then we have QP A = tan 1 x + a y For y = 0, we obtain QP B = tan 1 y a. y QP A = tan 1 x + a = tan 1 0 = π 0 QP B = tan 1 x a = tan 1 0 = π 0. In addition, M. Nihei remarked the following identities through H. Okumura ( :1; ): in a triangle sin x 1 + cos x = 1 cos x = tan x, sin x 1 + sin x + cos x = cot x, 1 + sin x cos x sin x 1 + cos x = = cot x, 1 cos x sin x cot A + cot B + cot C = a + b + c. 4S Consider the case x = π/ in the above two formulas in the triangle, consider the case A = π/. 1

13 3 Triangles For three points a, b, c on a circle with its center at the origin on the complex z-plane with radius R, we have a + b + c = ab + bc + ca. R If R = 0, then a, b, c = 0 we have 0 = 0/0. For a circle with radius R for an inscribed triangle with side lengths a, b, c, further for the inscribed circle with radius r for the triangle, the area S of the triangle is given by If R = 0, then we have S = r abc (a + b + c) = 4R. S = 0 = 0 0 (H. Michiwaki: ). We have the identity If a + b + c = 0, then we have r = S a + b + c. 0 = 0 0. For the distance d of the centers of the inscribed circle circumscribed circle, we have the Euler formula If R = 0, then we have d = 0 r = 1 R d R. 0 = Let r be the radius of the inscribed circle of the triangle ABC, r A, r B, r C be the distances from A, B, C to the lines BC, CA, AB, respectively. Then we have 1 r = r A r B r C 13

14 When A is the point at infinity, then, r A = 0 r B = r C = r the identity still holds. Thales theorem We consider a triangle BAC with A( 1, 0), C(1, 0), BOC = θ; O(0, 0) on the unit circle. Then, the gradients of the lines AB CB are given by sin θ cos θ + 1 sin θ cos θ 1, respectively. We see that for θ = π θ = 0, they are zero, respectively. For a triangle OAB (O(0, 0), A(1, 0), B(0, tan θ), θ = OAB), we consider the escribed circle (x + r) + y = r of A. Then, from we have r r + 1 = tan θ, r = tan θ. 1 tan θ For θ = π/, we have the reasonable result r = = 0. On the point (p, q)(0 p, q 1) on the unit circle, we consider the tangential line L p,q of the unit circle. Then, the common points of the line L p,q with x-axis y-axis are (1/p, 0) (0, 1/q), respectively. Then, the area S p of the triangle formed by three points (0, 0), (1/p, 0) (0, 1/q) is given by S p = 1 pq. Then, p 0; S p +, 14

15 however, S 0 = 0 (H. Michiwaki: ). We denote the point on the unit circle on the (x, y) plane with (cos θ, sin θ) for the angle θ with the positive real line. Then, the tangential line of the unit circle at the point meets at the point (R θ, 0) for R θ = [cos θ] 1 with the x-axis for the case θ π/. Then, ( θ θ < π ) π = R θ +, ( θ θ > π ) π = R θ, however, R π/ = [ ( π )] 1 cos = 0, by the division by zero. We can see the strong discontinuity of the point (R θ, 0) at θ = π/ (H. Michiwaki: ). The line through the points (0, 1) (cos θ, sin θ) meets the x axis with the point (R θ, 0) for the case θ π/ by Then, ( θ θ < π ) ( θ θ > π ) R θ = cos θ 1 sin θ. π = R θ +, π = R θ, however, R π/ = 0, by the division by zero. We can see the strong discontinuity of the point (R θ, 0) at θ = π/. Note also that [ ( π )] 1 1 sin = 0. In a triangle ABC, let X be the leg of the perpendicular line from A to the line BC let Y be the common point of the bisector line of A the line BC. Let P Q be the tangential points on the line BC with the 15

16 incircle of the triangle the inscribed circle in the sector with the angle A, respectively. Then, we know that XP P Y = XQ QY. If AB = AC, then, of course, X=Y=P=Q. Then, we have 0 0 = 0 0 = 0. Let X, Y, Q be the common points with a line three lines AC, BC AB, respectively. Let P be the common point with the line AB the line through the point C the common point of the lines AY BX. Then, we know the identity AP AQ = BP BQ. If two lines XY AB are parallel, then the point Q may be considered as the point at infinity. Then, by the interpretation AQ = BQ = 0, the identity is valid as We write lines by AP 0 = BP 0 = 0. L k : a k x + b k y + c k = 0, k = 1,, 3. The area S of the triangle surrounded by these lines is given by where is S = ± 1 D 1 D D 3, a 1 b 1 c 1 a b c a 3 b 3 c 3 D k is the co-factor of with respect to c k. D k = 0 if only if the corresponding lines are parallel. = 0 if only if the three lines are parallel or they have a common point. We can see that the degeneracy 16

17 (broken) of the triangle may be stated by S = 0 beautifully, by the division by zero. Similarly we write lines by M k : a k1 x + a k y + a 3k = 0, k = 1,, 3. The area S of the triangle surrounded by these lines is given by 1 A 11 A 1 A 13 S = A 11 A A 33 A 1 A A 3 A 31 A 3 A 33 where A kj is the co-factor of a kj with respect to the matrix [a kj ]. We can see that the degeneracy (broken) of the triangle may be stated by S = 0 beautifully, by the division by zero. Acknowledgements The author expresses his thanks to Professors Masakazu Nihei Hiroshi Okumura Mr. Hiroshi Michiwaki V. V. Puha for their kind sendings of their results. Further, the author wishes to express his deep thanks to Professor Hiroshi Okumura Mr. José Manuel Rodríguez Caballero for their very exciting informations. References [1] M. Abramowitz I. Stengun, HANDBOOK OF MATHEMATICAL FUNCTIONS WITH FORMULAS, GRAPHS, AND MATHEMATI- CAL TABLES, Dover Publishings, Inc. (197). [] W. W. Däumler, H. Okumura, V. V. Puha S. Saitoh, Horn Torus Models for the Riemann Sphere Division by Zero, vixra: submitted on :39:18. [3] R. Kaplan, THE NOTHING THAT IS A Natural History of Zero, OX- FORD UNIVERSITY PRESS (1999). [4] M. Kuroda, H. Michiwaki, S. Saitoh M. Yamane, New meanings of the division by zero interpretations on 100/0 = 0 on 0/0 = 0, Int. J. Appl. Math. 7 (014), no, pp , DOI: /ijam.v7i.9. 17

18 [5] T. Matsuura S. Saitoh, Matrices division by zero z/0 = 0, Advances in Linear Algebra & Matrix Theory, 6(016), Published Online June 016 in SciRes. [6] T. Matsuura, H. Michiwaki S. Saitoh, log 0 = log = 0 applications, Differential Difference Equations with Applications, Springer Proceedings in Mathematics & Statistics, 30 (018), [7] H. Michiwaki, S. Saitoh M.Yamada, Reality of the division by zero z/0 = 0, IJAPM International J. of Applied Physics Math. 6(015), [8] H. Michiwaki, H. Okumura S. Saitoh, Division by Zero z/0 = 0 in Euclidean Spaces, International Journal of Mathematics Computation, 8(017); Issue 1, [9] H. Okumura, S. Saitoh T. Matsuura, Relations of 0, Journal of Technology Social Science (JTSS), 1(017), [10] H. Okumura S. Saitoh, The Descartes circles theorem division by zero calculus, ( ). [11] H. Okumura, Wasan geometry with the division by 0, International Journal of Geometry, 7(018), No. 1, [1] H. Okumura S. Saitoh, Harmonic Mean Division by Zero, Dedicated to Professor Josip Pečarić on the occasion of his 70th birthday, Forum Geometricorum, 18 (018), [13] H. Okumura S. Saitoh, Remarks for The Twin Circles of Archimedes in a Skewed Arbelos by H. Okumura M. Watanabe, Forum Geometricorum, 18(018), [14] H. Okumura S. Saitoh, Applications of the division by zero calculus to Wasan geometry, GLOBAL JOURNAL OF ADVANCED RE- SEARCH ON CLASSICAL AND MODERN GEOMETRIES (GJAR- CMG), 7(018),,

19 [15] H. Okumura S. Saitoh, Wasan Geometry Division by Zero Calculus, Sangaku Journal of Mathematics (SJM), (018), [16] H. Okumura, To Divide by Zero is to Multiply by Zero, vixra: [17] S. Pinelas S. Saitoh, Division by zero calculus differential equations, Differential Difference Equations with Applications, Springer Proceedings in Mathematics & Statistics, 30 (018), [18] S. Saitoh, Generalized inversions of Hadamard tensor products for matrices, Advances in Linear Algebra & Matrix Theory, 4 (014), no., [19] S. Saitoh, A reproducing kernel theory with some general applications, Qian,T./Rodino,L.(eds.): Mathematical Analysis, Probability Applications - Plenary Lectures: Isaac 015, Macau, China, Springer Proceedings in Mathematics Statistics, 177(016), [0] S. Saitoh, Mysterious Properties of the Point at Infinity, arxiv: [math.gm]( ). [1] S. Saitoh, We Can Divide the Numbers Analytic Functions by Zero with a Natural Sense, vixra: submitted on :47:53. [] S. Saitoh, Zero Infinity; Their Interrelation by Means of Division by Zero, vixra: submitted on :57:5. [3] S.K.S. Sen R. P. Agarwal, ZERO A Lmark Discovery, the Dreadful Volid, the Unitimate Mind, ELSEVIER (016). [4] F. Soddy, The Kiss Precise, Nature 137(1936), 101. doi: /137101a0. [5] E. Sondheimer A. Rogerson, NUMBERS AND INFINITY A Historical Account of Mathematical Concepts, Dover (006) unabridged republication of the published by Cambridge University Press, Cambridge (1981). [6] S.-E. Takahasi, M. Tsukada Y. Kobayashi, Classification of continuous fractional binary operations on the real complex fields, Tokyo Journal of Mathematics, 38(015), no.,