Complex Numbers

What is a Complex Number?

A complex number is one that may be expressed as x+ iy,where x and y are real numbers and i is the imaginary unit defined by i = √-1. While dealing with equations like x2+ 1= 0, the real number system is incapable of finding out the roots, whereas in complex number systems the equation x2 + 1= 0 has a solution and can be used to find out its roots.

Let us denote -1 by the symbol i. Then, we have i2 = −1. This means that i is a solution of the equation x2 = −1

Now, suppose there are two complex numbers z1 = a + ib and z2 = c + id. Now if a = c and b = d then z1 = z2.

Complex Number Algebra

We can perform algebra operations like addition, subtraction, multiplication and division on complex numbers. 

Addition of Complex Numbers

Let us take z1 and z2 as two complex numbers. Then,z1 = x + iy and  z2 = p + iq complex numbers.

We can perform addition in few simple steps

Step 1:  Couple the real part and imaginary part together

z1+z2 = (x+p) + (iy+iq)

Step 2: Combine the like terms and simplify

z1+z2 = (x+p)+(iy+iq)

Subtraction of Complex Numbers

Let us take z1 and z2 as two complex numbers. Then, z1= x + iy and z2 = p + iq, then the difference of these two complex numbers that is. z1 – z2  is calculated as:

Step 1: Distribute the negative parts

z1 – z2 = z1 + (– z2)

z1 – z2= x + iy-p– iq

Step 2: Couple the real part and imaginary part together

z1 – z2= (x – p) + i (y – q)

Step 3: Combine the like terms and simplify

Multiplication of Complex Numbers

Let us take z1 and z2 as two complex numbers. Then, z1= x + iy and z2 = p + iq, then the multiplication of these    two complex numbers is z1 × z2 is calculated as:

z1 × z2 = (x + iy) × (p + iq)

z1 × z2 = (xp – yq) + i(xq + yp)

Division of Complex Numbers

Given any two complex numbers z1 and z2, where z2 ≠ 0 , the quotient of z1/z2 defined by z1/z2 = z1 x (1/z2)

What is Modulus of Complex Numbers?

By definition, the modulus of a complex number is equal to the square root of the sum of the squares of its real and imaginary parts. 

Let’s consider z as a complex number. Now, the modulus of z will be equal to, √{[Real (z)]2 + [Imaginary (z)]2} = |z|. Here modulus of z is denoted by the symbol of |z|.

Properties of Modulus of Complex Number

Following are some of the general properties of modulus of a complex number:

• If z is a complex number, the n modulus of z is equal to modulus of -z i.e. |z| = |-z|
• When the modulus of a complex number is found to be 0 then it implies that the complex number itself is zero, i.e., |z| = 0 only if z = 0
• If z and w are two complex numbers, then the modulus of the product of z and w must be equal to the product of modulus of z and modulus of w, i.e., |z x w| = |z|x|w|
• If z and w are two complex numbers, then the modulus of the quotient of z and w must be equal to the quotient of modulus of z and modulus of w, i.e., |z/w| = |z|/|w|
• If z is a complex number then the modulus of z will always be equal to the modulus of the conjugate of z.
• If z is a complex number then the modulus of z to power a will be equal to the modulus of z to power a, i.e., |za| = |z|a