## Complex Numbers and Quadratic Equations

# Complex Numbers

- A number of the form a + ib, where a and b are real numbers, is called a
*complex number,*a is called the real part and b is called the imaginary part of the complex number.

**Tips:**

- If a < 0 then \sqrt{a} = \sqrt{|a|} i
- The system of complex number includes the system of real numbers.
- Every real number is a complex number.
- ‘O’ is both purely real and purely imaginary number.
- A complex number is an imaginary number if and only if its imaginary part is non-zero. Here real part may or may not be zero.
- All purely imaginary numbers except zero are imaginary numbers but an imaginary number may or may not be purely imaginary.
- Inequality relation does not hold good in case of complex numbers having non-zero imaginary parts. Ex: “8 + 5i > 3 + 7i” makes no sense.
- Two complex numbers are said to be equal if and only if their real parts and imaginary parts are separately equal.

### View the Topic in this video From 00:40 To 19:56

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**Tricks:**

- For any n∈N i
^{2n}= (i^{2})^{n}= (−1)^{n}= 1 (if n is even)

= −1 (if n is odd) - For any n∈N i
^{2n+1}= (i^{2n}).i = (−1)^{n}.i = i (if n is even)

= −i (if n is odd) - The sum of 4 consecutive powers of ‘i’ is ‘o’.
- Also for any n∈N, the value of i
^{−n}is found out by writing this as \frac{1}{i^{n}} and solving i^{n}.