International J. of Math. Sci. & Engg. Appls. (IJMSEA), Vol. 2 No. III (2008), pp. 211-223
A NEW APPROACH TO ORDERING COMPLEX NUMBERS Dharmendra Kumar Yadav Department Of Applied Mathematics, HMR Institute Of Technology & Management G.T.Karnal Road, Hamidpur, Delhi-36, India E-mail:
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[email protected] Mobile No.: +91 9891643856
ABSTRACT In the present paper a new technique to ordering complex numbers have been discussed by applying the concepts of ordering real numbers on the real number line. The hidden property of ordering complex numbers in the extended complex plane and in its stereographic projection has been explained with its geometrical meaning in a little attempt. To order the complex numbers, a property named as D-law of trichotomy has been introduced with a very new concept of equi-radii complex numbers. This property has been derived from the law of trichotomy defined on the real numbers by making use of modulus of complex numbers. Key Words: Real numbers, Real number line, order properties of real numbers, Complex Numbers, Law of trichotomy, Argand plane, Ordering of complex numbers, Stereographic projection, etc. Mathematics Subject Classification2000: 30A10, 00A08.
1. INTRODUCTION Complex numbers were being used by mathematicians long before they were first properly defined, so it is difficult to trace the exact origin. The earliest fleeting reference to square roots of negative numbers perhaps occurred in the work of the Hellenized Egyptian Mathematician and inventor Heron of Alexandria in the 1 st century CE, when he considered the volume of an impossible frustum of a pyramid, though negative numbers were not conceived in the Hellenistic world. It became more prominent in the 16 th century, when closed formulas for the roots of cubic and quartic polynomials were discovered by Italian mathematicians Niccolo Fontana Tartaglia and Gerolamo Cardano. They realized that these formulas sometimes required the manipulation of square roots of negative numbers. Two Indian Mathematicians Mahavira(850) who first stated in his book ‘Ganitsara Sangraha’ that ‘a negative quantity is not a square quantity’ and Bhaskara(1150) wrote in his book ‘Bijaganita’ that ‘ there is no square root of a negative quantity, for it is not a square’ were acquinted about this problem. The fact that square root of a negative number does not exist in the real number system was recognized by the ‘Greeks’. The term ‘imaginary’ for these quantity was coined by Rene Descartes in 1637 and was meant to be derogatory. Bombelli also used the square root of negative number in finding the cubic roots of x3 = 15x+4. He was probably the first mathematician to have a clear idea of a complex number. He discussed imaginary and complex numbers in a treatise written in 1572 called ‘L’ Algebra’. It was Swiss Mathematician Leonhard Euler(1707-1783) who introduced the imaginary unit ‘iota’ with symbol ‘i’ for the square root of (-1) with the property i 2 = -1 or -1 = i possibly in 1748 and complex numbers came into existence. Though the existence of complex numbers was not completely accepted until the
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International J. of Math. Sci. & Engg. Appls. (IJMSEA), Vol. 2 No. III (2008), pp. 211-223 geometrical interpretation had been described by Casper Wessel in 1797 and C.F.Gauss in 1799 as points in a plane. Although the idea of representing a complex number by a point in a plane had been suggested by several mathematicians earlier, it was Argand’s proposal that was accepted. Gauss used it and proved “the Fundamental Theorem Of Algebra” in his Ph.D. thesis in 1799 which had been given by Albert Girard. Hamilton, an Irish mathematician, in 1833 introduced the complex number notation a+ib and made the connection with the point (a,b) in the plane although many mathematicians argued that they had found this earlier. The idea of the graphic representation of complex numbers had appeared, however, as early as 1685, in Wallis’s De Algebra tractatus. In 1804 the Abbe Buee independently came upon the same idea which Wallis had suggested that ±√-1 should represent a unit line, and its negative, perpendicular to the real axis. Buee’s paper was not published until 1806, in which year Jean-Robert Argand also issued a pamphlet on the same subject. It is to Argand’s essay that the scientific foundation for the graphic representation of complex numbers is now generally reffered. Nevertheless, in 1831 Gauss found the theory quite unknown, and in 1832 published his chief memoir on the subject, thus bringing it prominently before the mathematical world. The 18 th century saw the labors of French Mathematician Abraham de Moivre and Leonhard Euler. To De Moivre is due (1730) the well-known formula which bears his name, de Moivre’s formula: (cos+ i sin)n = (cosn + i sinn) and to Euler(1748) Euler’s formula of complex analysis: ei = cos + i sin. At last but not the least, the credit must also be shared with a number of contributors of high rank in the advancement of the Complex Analysis: Madhava of Sangamagrama, Mourey(1828), A.L.Cauchy, N.H.Abel, Hankel(1867), Kummer(1844), L.Kronecker(1845), Scheffler(1845,1851,1880), Bellavitis(1835,1852), Peacock(1845), DeMorgan(1849), F. Eisenstein, F. Klein(1893), E. Galois, Weierstrass, Schwarz, R. Dedekind, O. Holder, Berloty, H. Poincare, Eduard Study and A. MacFarlance. Mobius must also be mentioned for his numerous memoirs on the geometrical applications of complex numbers, and Dirichlet for the expansion of the theory to include primes, congruences, reciprocity, etc., as in the case of real numbers. Recently Yadav[15] introduced some theorems and conjecture which give some relations between real and imaginary numbers. He proved that i0, i∞. He, then, introduced imaginary number line to represent imaginary numbers on it. He also introduced a new mathematical system ‘Imaginary Analysis’. These new concepts are totally different from the conventional Complex Analysis and their applications are still in progress.
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. The point corresponding to zero is called the origin. 2.2. Ordering Real Numbers: Every real number can be represented by a point on the real number line and conversely to each point on the line there is one and only one real number. If a point A corresponding to a real number ‘a’ lies to the right of a point B corresponding to real number ‘b’, we say that a>b or b
