International J. of Math. Sci. & Engg. Appls. (IJMSEA) ISSN 0973-9424, Vol. 3 No. II (2009), pp.
INTRODUCTION OF A CIRCULAR NUMBER LINE D. K. YADAV, SARITA RANI AND SIPRA MOHANTY
Abstract The present paper introduces a circular number line, the superset of imaginary number line previously given by Yadav [1] and an imaginary circular plane, the superset of circular complex plane given by Yadav [2]. It also introduces the new concepts of imaginary circles and imaginary spheres. Taking different values of n, the natural numbers in in , we find that it takes all the values on the imaginary number line. Giving in the geometrical meaning as the sum of arithmetical distances, we find that the values of in lie on a circle and the imaginary number line lies on this circle, which gives the concept of circular number line. This circular number line is not a straight line but is a circle of imaginary radius. At last some axioms of Elliptical geometry and Euclidean geometry have been observed true in the paper. These axioms have been observed on the imaginary sphere and imaginary circle. The circular number line, imaginary circle and imaginary sphere will play a major role in explaining the concepts of Elliptical geometry and Hyperbolic geometry as well as they will be very helpfull in explaining the universe geometrically.
1. Introduction The real number system consists of rational and irrational numbers is playing
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Key Words : Real numbers, Real number line, Imaginary unit ‘i’, Imaginary numbers, Imaginary number line, Imaginary circle, Imaginary sphere, Axioms of elliptical geometry, Hyperbolic geometry and Euclidean geometry etc.
AMS Subject Classification 2000 : 14H50, 14H45, 30A99, 97B60, 51M10, 51M09, 51M30. @2007 Ascent Publishing House
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a major role in every sciences specially in Mathematics. To denote the real numbers geometrically the real number line has been introduced. This line can be visualized as a horizontal line on infinite plane that extends from a special point called the origin in both directions towards infinity. Thereafter complex numbers came into existence after the introduction of imaginary unit ‘iota i’ with the property i2 = −1 or −1 = i by Swiss Mathematician Leonhard Euler (1707-1783) in 1748. But no line could be drawn or introduced to represent the imaginary numbers or complex numbers. Recently Yadav [1] proved that the complex number a+ib; a, b real numbers are also imaginary numbers, if we take a+ib as the arithmetical sum of a and ib. He introduced imaginary number line to represent the imaginary/complex numbers on it by extending the real number line ]−∞, +∞[ upto imaginary length. Thereafter Yadav [2] also defined D-law of trichotomy to order the complex numbers and gave many order properties for the complex numbers.
2. Preliminary Ideas We shall discuss the present paper under the following terms: 2.1 Real Numbers : The set of rational and irrational numbers. It can be represented by points on a line called the real number line as shown below:
The point corresponding to zero is called the origin. 2.2. Imaginary Unit : It is denoted by ‘i’ with the property defined in the introduction. 2.3. Imaginary Numbers : A number of the form ‘ix’ where x is a real number, is called an imaginary number. The complex number a + ib is also an imginary number as proved by Yadav [1]. 2.4. Imaginary Number Line : To represent the imaginary numbers on a line like the real numbers, Yadav [1] introduced imaginary number line by extending the interval ] − ∞, +∞[ as shown below:
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2.5. Imaginary Circle : An imaginary circle of very large indeterminate radius (−r2 ) with its centre at the point (h, k) or at the centre of the earth (0, 0) is given by (x − h)2 + (y − k)2 = (−r2 ) = (ir)2 or x2 + y 2 = (−r2 ) = (ir)2 where r is non-zero real number, respectively. 2.6. Imaginary Sphere : An imaginary sphere of very large indeterminate radius {−r2 = (ir)2 } with its centre at (j, h, k) or at the centre of the earth (0, 0, 0) is given by (x − j)2 + (y − h)2 + (z − k)2 = (−r2 ) = (ir)2 or x2 + y 2 + z 2 = (−r2 ) = (ir)2 respectively, used in locating points in the sky. To locate stars and other objects in the sky, we imagine all the objects to lie on the surface of a sphere with ourselves at the centre of it. The imaginary sphere is also called the Cellestial sphere.
3. Discussion, Result and Application In this section, we shall discuss the geometrical meaning of imaginary number line given by Yadav [1] by considering the values of in , n being natural numbers and taking in as the sum of the distances i + i + i + i + i · · · i times in the negative direction (left hand side of the origin O) on the imaginary number line. According to Yadav [1], imaginary numbers can be located on the imaginary number line as:
Figure 3.1 By considering this imaginary number line, let us discuss the geometrical meaning of
in
for different values of n. We have
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D. K. YADAV, SARITA RANI & SIPRA MOHANTY
(i) i1 = i × 1 = i (ii) i2 = i × i = i + i + i + i + i + i + i + i + · · · , i times = OG + 0G + OG + OG + · · · , i times = OG + GH + HI + IJ + JK + KL + · · · i times = −1
which means that if we start moving in the left direction on the imaginary number line from the point O and cover a distance i × i, we shall reach at the point −1(M ) as shown below:
Figure 3.2 (iii) i3 = −1 × i = −i, which means that if we start to move in the left direction on the imaginary number line from the point O and cover a distance i × i × i, we shall reach at the point −i(A) as shown below:
Figure 3.3
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(iv) i4 = 1, which means that if we start to move in the left direction on the imaginary number line from the point O and cover a distance i × i × i × i, we shall reach at the point +1(N ) as shown below:
Figure 3.4 (v) i5 = i, which means that if we start to move in the left direction on the imaginary number line from the point O and cover a distance i × i × i × i × i, we shall reach at the point +i(G) as shown below:
Figure 3.5 Similarly in = i, −1, −i, 1 for n = (4r + 1), (4r + 2), (4r + 3), (4r), where r = whole numbers 0, 1, 2, 3, · · · . From the above discussion it is clear that in takes only four different values i, −1, −i, 1, i.e. in varies from +i to −i via −1 and +1 on the imaginary number line.
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The above discussion suggests that the values of in lie on a circle C. Since it has four values i, −1, −i, 1 and i, −i lie beyond ] − ∞, +∞[, whereas −1, 1 lie in ] − ∞, +∞[. This implies that the imaginary number line lies on a circle and the real number line being the subset of imaginary number line also lies on this circle. This circle C can be named as the Circular Number Line and it can be shown geometrically as follows:
Figure 3.6 Thus the circular number line is the union of the real number line and the imaginary number line. Applications : To apply these concepts in studying the axioms of Elliptical geometry, Hyperbolic geometry and Euclidean geometry, let us discuss and introduce some new concepts as below: 3.1 Imaginary Number Line : Yadav [1] has already given the imaginary number line as shown below:
Figure 3.1.1 3.2 Imaginary Circles Of Imaginary Radii : Rotate the imaginary number line Figure 3.1.1 through an angle 1800 clockwise or anti-clockwise, we get the circles as shown below:
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Figure 3.2.1 where C0 , C1 , C∞ , C−i , C−2i , C−3i are circles of radii 0, 1, ∞, −i, −2i, −3i respectively, in which C0 , C1 are of finite radius, C∞ is of infinite radius and C−i , C−2i , C−3i are of imaginary radius. Let us call the plane on which imaginary circles have been shown, Imaginary Circular Plane, the superset of Circular Complex Plane earlier proposed by Yadav [2]. This imaginary circular plane is itself an imaginary circle and not a rectangular plane. This circular number line is the superset of all the circles in the sense that all the circles that can be drawn or imagined lie within the circle of circular numer line. 3.3 Imaginary Spheres of Imaginary Radii : Rotate the imaginary circular plane Figure 3.2.1 about X-axis or Y -axis or any straight line passing through the origin O, we shall get the spheres of radii 0, 1, ∞, −i, −2i, −3i and so on. The spheres of radii 0, 1 are finite sphere, the sphere of radius ∞ is infinte sphere, and the spherers of radii −i, −2i, −3i can be named as Imaginary Spheres of Imaginary Radii, as shown below:
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D. K. YADAV, SARITA RANI & SIPRA MOHANTY
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Figure 3.3.1 The imaginary sphere formed by rotating the imaginary circular plane about a line passing through the origin of the circular number line or imaginary circular plane is the superset of all the spheres (real or imaginary) in the sense that all the spheres that can be drawn or imagined lie within the imaginary sphere. Axioms : From the above discussion and results, we can have the following axioms: Axiom 3.1 : Every line segment/straight line is a part of a circle of finite or infinite or imaginary radius. This property follows from the concept of circular number line. The real number line is a straight line in the interval ] − ∞, +∞[ but when it crosses the boundary of ] − ∞, +∞[, it bends and becomes a curved line in the form of imaginary number line and finally becomes a circular number line i.e. a circle. Therefore, we can say that −∞ and +∞ are the mid-points of the real number line and the imaginary number line on the circular number line and it divides the circular number line into two parts: the real number line and the imaginary number line. Axiom 3.2 : −∞ and +∞ are neither purely real nor purely imaginary/complex numbers, but it can be regarded as an imaginary number. As discussed above, −∞ and +∞ are the two dividing points of the real number
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line and the imaginary number line on the circular number line. −∞ can be regarded as the least real number and +∞ as the largest real number, whereas +∞ can be regarded as the least imaginary number and −∞ as the greatest imaginary number; i.e. in one side |∞ is the greatest real number and in another side, it is the lowest imaginary number. Therefore we can say that −∞ and +∞ are neither purely real nor purely imaginary/complex numbers. But since every real number is a finite number upto some extent, so −∞ and +∞ can not be a real number. It is just an assumption that there exist two numbers −∞ and +∞ on the real number line. So it can be regarded as imaginary numbers as imaginary numbers are only assumptions on the imaginary number line. Axiom 3.3 : A real number line contains infinitely many points, and an imaginary number line contains imaginary number of points; whereas the circular number line contains infinitely imaginary number of points. Axiom 3.4 : Every plane is a surface of a sphere (finite, infinite or imaginary). Axiom 3.5 : Every line is a great circle on the sphere (finite, infinite or imaginary). Axiom 3.6 : A line can not be extended indefinitely in both directions. Axiom 3.7 : The circular number line contains all the finite, infinite or imaginary circles. Axiom 3.8 : The imaginary sphere contains all the finite, infinite or imaginary spheres. Axioms 3.3-3.8 can be explained by the concepts of real number line, imaginary number line, circular numer line, imaginary circles and imaginary spheres.
4. Conclusion From the above discussion and results, we observe that the circular number line plays an important role to explain many axioms of Plane and Spherical geometry. Infact the spherical geometry can be studied in detail with many new concepts by the help of Imaginary spheres. The concept of imaginary sphere can also be applied to study the universe mathematically, which is infact in progress by the same author.
References [1] Yadav D.K., An analysis of the imaginary unit ‘i’ and its position on the imaginary number line, Int. J. of Mathematical Sciences and Engg. Applications, 2(1) (2008), 203-209, Pune, India.
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[2] Yadav D.K., A New Approach To Ordering Complex Numbers, Int. J. of Mathematical Sciences and Engg. Applications, 2(3) (2008), 211-223, Pune, India. [3] Gakkhad, S. C., Teaching of Mathematics, N.M.Prakashan, Chandigarh, India, 1991. [4] www.en.wikipedia.org/wiki/spherical− geometry. [5] www.en.wikipedia.org/wiki/elliptical− geometry. [6] www.en.wikipedia.org/wiki/euclidean− geometry. [7] www.en.wikipedia.org/wiki/hyperbolic− geometry. [8] Loney, S.L., Co-ordinate Geometry(I), Book Palace, New Delhi, India, 1994. [9] Bell, R.J.T., Co-ordinate Geometry(3-D), Macmillan India Ltd, India, 1994. [10] Sen, D.K. and Dass I. K. L., Analytical Geometry(3-D), Syndicate Pub., Patna, India, 1994. [11] Nelson D., Dictionary of Mathematics, Penguin Reference, Penguin Book, England, 2003.
Dharmendra Kumar Yadav, Department Of Applied Mathematics, HMR Institute Of Technology & Management, G.T.Karnal Road, Hamidpur, Delhi-36, India E-mail:
[email protected],
[email protected] URL: www.dkyadav.110mb.com Sarita Rani, Lecturer in Applied Mathematics Guru Premsukh Memorial College of Engineering, Buddhpur, G.T.Karnal Road, Delhi-36, India E-mail: sarita−
[email protected] Sipra Mohanty, Lecturer in Applied Mathematics, HMR Institute Of Technology & Management, G.T.Karnal Road, Hamidpur, Delhi-36, India E-mail:
[email protected] 
