Page 1 - Chapter 2. Complex Numbers and Quadratic Equations
P. 1

CHAPTER 2




                         Complex Numbers and Quadratic Equations


            Topics

                •   Complex numbers as ordered pairs of reals
                •   Representation of complex numbers in the form	   +      and their representation in a plane
                •   Argand diagram
                •   Algebra of complex numbers
                •   Modulus and argument (or amplitude) of a complex number
                •   Square root of a complex number
                •   Triangle inequality
                •   Quadratic equations in real and complex number system and their solutions
                •   Relation between roots and co-efficient, nature of roots
                •   Formation of quadratic equations with given roots
            Key Notes

            Complex  Numbers  as  Ordered  Pairs  of              The complex number is represented by the real numbers (x,
            Reals                                                 y) in a x-y plane and it is referred to as the complex plane or
                                                                    -plane.
            A  complex  number  z  is  defined  as  an  ordered  pair.  It  is
            given by,                                             |z| is modulus of the complex number that gives the vector
                                                                  representation.
                                  = (  ,   )
                                                                  arg z is the angle made by the complex number along the
            where    and    are a pair of real numbers.           plane.
            Sometimes  the  order  pairs  of  real  numbers  are  given  by   Argand Diagram
            (  ,   ) which are called the polar coordinates of the point.
                                                                  The plane having a complex number assigned to each of its
            The polar form is represented by,                     point is called the complex plane or the Argand plane.

                             =   (cos    +    sin   )             The representation of a complex number z = x + iy and its
            Representation  of  Complex  Numbers  in              conjugate z = x – iy in the Argand plane is,
            the form a+ib and their Representation in
            a Plane

            Complex number is represented in the form of	   +     ,    is
            the  real  part  and      is  the  imaginary  part  of  the  complex
            numbers. They are denoted by Re    and Im    respectively.










                                                                  Geometrically, the point (x, – y) is the mirror image of the
                                                                  point (x, y) on the real axis.
                                                                  P (x, y) is the real axis and Q (x, -y) is the imaginary axis.

                                                                  Algebra of Complex Numbers




                                            Year Book Maths 2021
   1   2   3   4   5   6