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Complex Number Learning Outlines At the course of this chapter, you will be able to: 7-1, Understand the concepts of Conju gate Complex 7-2, Leam Properties of conjugate, Modulus al arguments Definiti aSumpyesh tai 7-3. Representation of complex numbers, 7-4, Understand Demoivre's Theorem, Roots of unity,and, Ptolemy's Theorem Complex numbers are defined as expressions of the form a+ ibwherea, be Randi = V-1. tris denoted by z, i.e, z=a + tb. ‘a’ is called as real part of z (Re z) and *h* is called as imaginary part of z (Im z). Purely real ifb=0 Purely imaginary ifa=0 Imaginary ifb40 1 1, The sct & of real numbers is a proper subset of the complex numbers. Hence, the complete num- bersystemisNCWCeICQCREC. 2. Zero is both purely real as well as purely imagi- nary but not imaginary. V-1 is called the imaginary unit. The algebraic operatioris on complex numbers are similar to those on real numbers treating ‘i’ as a polynomial. Inequalities in complex numbers are not defined. There is no validity if we say that complex number is positive or negative. eg, 2>0,4+2i<2+4/are meaningless. However, in real numbers, ifa? + =O thena=0=6 but in complex numbers, Also? =-1; P=-i; P= 1, etc. 4. Ja Vb = Jab only if atleast one of either a or 4 is non-negative. 5. ¥-Ixa=iJa, where a be any positive real number, -- 2) +2," = 0 does not imply z, = Equality In Complex Number Two complex numbers z, =a, + ib, andz,=a,+ ib, are equal ifand only if their real and imaginary parts coincide. Ifz =a + ib then its conjugate complex is obtained by changing the sign of its imaginary part and is denoted by 7. ie, z =a—ib. 
