Complex numbers are numbers of the form m + in, where m and n are real numbers. The number 'i' is called as imaginary number. The symbol i for $\sqrt{-1}$ having the property i$^2$ = -1 and the solution of the equation x$^2$ + 1 = 0.

A complex number is also defined as ordered pair (m, n) of real numbers. Thus, if Z is a complex number, then Z = m + in = (m, n) where m and n are real numbers. Complex numbers are normally denote by $\mathbb{C}$.

### Basics Properties of Complex Numbers

Below are following properties of the complex numbers:

- Two complex numbers are equal iff their corresponding real and imaginary numbers are equal. Example: Z$_1$ = m + in and Z$_2$ = p + iq are equal iff m = p and n = q.
- A complex number is purely real if imaginary part is zero. Example: Z = m
- A complex number is purely imaginary if real part is zero. Example: Z = i n
- A complex number is zero complex number if both real and imaginary parts are zero. Example: Z = 0 + i 0 = 0.

- Addition of two complex numbers:

Let Z$_1$ = m + in and Z$_2$ = p + iq

Z$_1$ + Z$_2$ = (m + ni) + (p + qi) = (m + p) + (n + q)i

- Subtraction of two complex numbers:

Z$_1$ + Z$_2$ = (m - p) + (n - q)i

- Multiplication of two complex numbers:

(m + ni) * (p + qi) = mn + pi + mqi + nq i

^{2} = (mp – pq) + (np + mq) i

- Division of two complex numbers:

`(m+ni)/ (p+qi)` (This can be solved by rationalizing the denominator).

### How to Solve Complex Numbers

Below are the examples on how to solve complex numbers -

Problem 1:

Find the real part and imaginary part for the following numbers : 4 − 3i

Solution:

Let z = 4 − 3i

Re(z) = 4

Im(z) = − 3

Problem 2:

Express the standard form of a + ib : (3 + 2i) + (− 7 − i)

Solution :

= 3 + 2i − 7 − i

= − 4 + i

Problem 3 :

Find the complex conjugate of (i) 2 + 7i (ii) − 4 − 9i

Solution:

By definition, the complex conjugates is obtained by reversing the sign of the imaginary part of the complex number. Hence the required conjugates are explained

(i) 2 − 7i, (ii) − 4 + 9i