Last Updated on: 8th January 2024, 02:57 pm
Complex Number
We know Real Numbers (like 5, -7,
,
) and we know imaginary Numbers (like 
,). Complex Numbers are combination of Real Numbers and Imaginary Numbers (like 
). Complex Number has 2 parts (Real Number Part) and Imaginary Number part. The Imaginary Part is represented by i (i=).
Complex Number representation
So, a Complex Number may be represented as a+ib. Where a and b are Real Number. ‘a’ represents the Real part, and i‘b’ represents the imaginary part. ‘i’ represents ).
Complex Numbers Addition
Let z1 = a + ib and z2 = c + id be any two complex numbers. Then, the sum z1 + z2 is defined as:
z1 + z2 = (a + c) + i (b + d) [So, the sum is also a complex number].
For example, (5 + i4) + (– 3 +i3) = (5-3) + i (4+3) = 2 + i7
Complex Numbers Subtraction (Difference)
For any two complex numbers z1 and z2, the difference z1 – z2 is defined as z1 – z2 = z1 + (– z2).
Complex numbers Multiplication (Product)
For two Complex Numbers z1 = a + ib and z2 = c + id, the product z1 ×z2 = (ac – bd) + i(ad + bc)
Complex numbers Division of two Division
Given any two complex numbers z1 and z2 (where z2≠ 0), the quotient 
is defined by z1 × 
.
Negative Real Number Square roots
i2 = –1 and (– i)2 = i2 = – 1. So, the square roots of – 1 are i, – i. We represent , by i.
Modulus and Conjugate of a Complex Number
Modulus of complex number z = a + ib (represented as |z|),is defined as non-negative real number 
(a2+b2). So, |z|= |(a2+b2)|.
Conjugate of complex number z =a + ib (represented as 
) is defined as = a-ib.
Modulus and Conjugate of a Complex Number – Properties
(i) z× =|z|2. (ii) |z1×z2|=|z1|×| z2|(iii) |z1/z2|=|z1|/| z2|. (iv) 
(v) 
Argand Plane and Polar Representation
We have seen that corresponding to each ordered pair of real numbers (x, y), we get a unique point in the XY plane and vice-versa with reference to a set of mutually perpendicular lines x-axis and the y-axis. The complex number x + iy corresponding to the ordered pair (x, y) can be geometrically represented as the unique point P(x, y) in the XY-plane and vice-versa.
The plane having a complex number assigned to each of its point is called the complex plane or Argand plane. So, in the Argand plane, the modulus of the complex number
x + iy = √(x2 + y2 ) is the distance between the point P(x, y) and the origin O (0, 0). The points on the x-axis corresponds to the complex numbers of the form a + i 0 and the points on the y-axis corresponds to the complex numbers of the form 0 + i b (F-1). The x-axis and y-axis in the Argand plane are called, respectively, the real axis and the imaginary axis. So, the representation of a complex number z = x + iy and its conjugate z = x – iy in the Argand plane are, the points P (x, y) and Q (x, – y) respectively. Geometrically, the point (x, – y) is the mirror image of the point (x, y) on the real axis (F-2).
Polar representation of Complex Number
Complex Number Polar representation
A point in the Argand plane, represented the nonzero complex number z = x + iy, whose distance from origin is r, making angle q with the positive direction of x-axis (F-3). So, point is uniquely determined by the ordered pair of real numbers (r, q), called the polar coordinates of the point. Here, the origin is considered as the pole and q the angle with positive direction of the x axis as initial line.
So, we can write x = r cos q, y = r sin q. So, z = r (cos q + i sin q), referred as polar form of the complex number. Here r = √(x2 + y2) = |z|. It is referred as modulus of z and q is called the argument (or amplitude) of z, denoted by arg z.
Modulus and Amplitude of Complex Numbers
Modulus of Complex Number : The modulus of a complex number in the form z=a+bi is denoted by |z|, is the real number given by √(a2+b2)
Modulus of Complex Number : The angle with the positive direction of x-axis in anticlockwise sense is called the argument or amplitude of complex number z. It is denoted by arg(z) or amp(z).
Click here to play a video on Problem of Modulus and Amplitude explained by a teacher, in Bengali
