Complex Numbers

COMPLEX NUMBERS

Complex number is a combination of real and imaginary numbers. It can be generally expressed as (a + ib).

Where 'a' is the real number and 'i' is the imaginary number.

Complex Number System :

A complex number is of the form a + ib where ‘a’ and ‘b’ are real numbers and i is called the imaginary unit, having the property that i² = − 1.

If z = a + ib then a is called the real part of z, denoted by Re(z) and b is called the imaginary part of z and is denoted by Im(z)

Some examples of complex numbers are 3 − 2i, 2 + 3i

Let z₁ = 3 - 2i and z₂ = 2 + 3i

z₁ = 3 - 2i

Re (z₁) = 3

Imz (z₁) = -2

z₂ = 2 + 3i

Re (z₂) = 2

Imz (z₂) = 3

Negative of a Complex Number :

If z = a + ib is a complex number then the negative of z is denoted by − z and it is defined as − z = − a + i(− b)

Basic Algebraic Operations :

Addition :

(a + ib) + (c + id) = (a + c) + i (b + d)

To add two or more complex numbers, we have to combine the real parts and imaginary parts respectively.

Subtraction :

(a + ib) − (c + id) = (a − c)+ i(b − d)

To subtract two or more complex numbers, we have to combine the real parts and imaginary parts respectively.

To perform the operations with complex numbers we can proceed as in the algebra of real numbers replacing i² by − 1 whenever it occurs.

Multiplication :

(a + ib) (c + id) = ac + iad + ibc + i2bd

= (ac − bd) + i (ad + bc)

Division :

(a + ib) / (c + id) = ((a + ib) / (c + id)) ((c - id) / (c - id))

To divide two complex numbers, we have to multiply the given fraction by the conjugate of denominator.

Conjugate of a complex number :

If z = a + ib, then the conjugate of z is denoted by