Complex numbers for IIT-JEE MAINS /ADVANCED

Complex numbers basic concepts-Defining

A number in the form of a + ib, where a, b are real numbers and is called a complex
Number. A Complex Number can also be defined as an ordered pair of real numbers a and b any
may be written as (a, b), where the first number denotes the real part and the second number
denotes the imaginary part. If z = a + ib, then the real part of z is denoted by Re (z) and the
imaginary part by Im(z). A complex number is said to be purely real if Im(z)= 0, and is said to
be purely imaginary if Re(z) = 0. The complex number 0 = 0 + i0 is both purely real and imaginary.
Two complex numbers are said to be equal if and only if their real parts and imaginary parts are
separately equal i.e. a + ib = c + id implies a = c and b = d. However, there is no order relation
between complex numbers and the expressions of the type a + ib < (or > ) c + id are meaningless

Geometrical representation of complex numbers

A complex number z = x + iy, written as an ordered pair (x, y), can be represented by a point P whose Cartesian coordinates are (x, y) referred to axes OX and OY, usually called the real and the imaginary axes.

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The plane of OX and OY is called the Argand diagram or the complex plane. Since the origin O lies on both OX and OY, the corresponding complex number z = 0 is both purely real and purely imaginary.

 

Modulus and Argument of a Complex Number

We define modulus of the complex number z = x + iy as

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We can define the argument of a complex Number also as any value of the which satisfies the
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is the angle which OP makes with the positive x-axis.

Trigonometric ( or polar ) from a Complex Number

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Remark : Method of finding the principal value of the argument of a complex number z = x + iy.

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Unimodular Complex Number

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Algebraic Operations with Complex Number

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Geometrical Meaning of Product and Division

Let z1 = x1 + i y1 and z2 = x2 + i y2 be two complex numbers represented by the points P1(x1, y1) and P2(x2, y2) respectively. By definition z1 + z2 should be represented by the point (x1 + x2 , y1 + y2 ). This point is the vertex which completes the parallelogram with the line segments joining the origin with OP1 and OP2 as the adjacent sides.
| z1 + z2 | = OP.

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lso by definition z1 – z2 should be represents by the point (x1 – x2, y1 – y2). This point is the vertex
which completes the parallelogram with the line segments joining the origin with OP1 and
(where the point represents ; the point can be obtained by producing the directed
line P2O by length | z2 | ) as the adjacent sides.
| z1 – z2 | = OQ = P2 P1.
Remarks
(a) In any triangle, sum of any two sides is greater than the third side and difference of any two

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Construction for the point representing the product z1 z2

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Construction for the point representing the quotient z1/z2

Draw the triangle OQ1P directly similar to the triangle OQ2L
The P represents the quotient z1/z2 .

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Square Root of a Complex Number

How we get the square root of complex number

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Conjugate of a Complex Number

The conjugate of the complex number z = a + ib is defined to be a – ib and i denoted by . In other words is the mirror image of z in the real axis.

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Properties of Conjugate, Modulus, Argument

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Modulus of a Complex Number

Distance of a complex number z from origin is called the modulus of the complex number z and it is denoted by |z| .

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Properties of Arguments

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Complex Numbers Represented By Vectors
It can be easily seen that multiplication by real numbers of a complex number is subjected to the same rule as the vectors . The addition or the subtraction of two complex numbers is also the same as the addition or the substraction of two vectors. This fact is fundamental in theory and very useful in practice.
It should be noticed that if a number z is represented by points P and OP by a vector then
| z | is the length OP and arg(z) is the angle which the directed line OP makes with directed OX.
Please note that if z = x + iy and P is the point (x, y), a one–to–one correspondence exists between the number z and any of the following : (i) the point P; (ii) the displacement ; (iii) the vector
(or directed length ) .
Any one of these three things may therefore be said to represent z, or to be represented by z.

 

De-Moiver’s Theorem

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Application of De Moiver’s Theorem

This is a fundamental theorem and has various applications. Here we will discuss few of these
which are important from the examination point of view

The nth Roots of Unity

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Cube Roots of Unity

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Logarithm of Complex Number

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CONCEPT OF ROTATION

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Geometrical Application

Section Formula

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Equation of a straight line

Equation of straight line with the help of coordinate geometry

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Equation of straight line with the help of rotation formula
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General equation of the line

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Equation of Perpendicular Bisector

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arg (z – z0) =p  represents a line passing through z0 with slope (making angle p with the positive direction of x-axis).

Equation of a circle

Consider a fixed complex number z0 and let z be any complex number which moves in such a way
that it’s distance from z0 is always equals to ‘r’. This implies z would lie on a circle whose centre is
z0 and radius r. And its equation would | z – z0 | = r

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It represents the general equation of a circle in the complex plane.

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Note :
(i) If we take ‘C’ to be mid–point of A2A1, it can be easily proved that CA.CB = (CA1)2 i.e. | z1 – z0 | | z2 – z0 | = r2, where the point C is denoted z0 and r is the radius of the circle.

(ii) If | z1 – z0 | = | z2 – z0 | hence P(z) would lie on the right bisector of the line A(z1)
and B(z2). Note that in this case z1 and z2 are the mirror images of each other with respect
to the right bisector.

Equation of tangent a given circle at a given point

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Some important results

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