Complex number - Practice Questions & MCQ

A number of the form a + ib is called a complex number (where a and b are real numbers and i is iota). We usually denote a complex number by letter z, z₁, z₂, etc

For example, z = 5 + 2i is a complex number.

5 here is called the real part and is denoted by Re(z), and 2 is called imaginary part and is denoted by Im(z)

 

Note: 2i is not the imaginary part, only 2 is called the imaginary part.

For z = - 2 - i, Re(z) = -2, Im(z) = -1

We denote the set of all complex numbers by C.

 

 

Purely Real and Purely Imaginary Complex Number

• A complex number is said to be purely real if its imaginary part is zero, Im(z) = 0

            i.e. z = 4 + 0i, z = 4.

• A complex number is said to be purely imaginary if its real part is zero, Re(z) = 0

            i.e. z = 0 + 3i, z = 3i

All real numbers are also complex numbers (with b=0). Eg 4 can be written as 4 + 0i

So, R is a proper subset of C. 

 

Equality of Complex Numbers

Two complex numbers are said to be equal if and only if their real parts are equal and their imaginary parts are equal.

a + ib = c + id

⇒ a = c and b = d

a, b, c, d ∈ R and i = √-1

Note: In complex number, inequalities do not exist. z₁ > z₂ does not make any sense in complex numbers

Eg,

• 4 + 3i > 1 + i  is wrong (as two complex numbers cannot be compared)
• 1000 + 7000i > -39 - 800i is also wrong 
• 1000 + 40i > 2 is wrong

We can only compare two complex numbers if their imaginary parts are 0.  Eg, 5 + 0i > 4 + 0i is correct.

 

Argand Plane

A complex Number can be represented on a rectangular coordinate system called Argand Plane.

In this z = a + ib is represented by a point whose coordinates are (a,b) 

So, x-coordinate of the point is the Real part of z and y coordinate is imaginary part of z

Eg z = - 2 + 3i is represented by the point (-2,3) and it lies in second quadrant.