Page 1 - Chapter 2. Complex Numbers and Quadratic Equations

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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
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