# Introduction

The modulus of a complex number denotes its magnitude and conjugate of a complex number is obtained by changing the sign of imaginary part of a complex number. If the imaginary part is positive we make it negative and if it is negative, we make it positive. The sign of real part doesn’t change.

Let z=a + ib be a complex number, then the modulus of ‘z’ is denoted by |z| and is defined as |z| = √a^{2}+b^{2}

For example: 1) |5-i| = √5^{2} + (-1)^{2}

= √25 + 1

= √26

2) |-3+4i| = √(-3)2 + 42

= √9+16

= √25

= 5

The conjugate of a complex number z=a + ib is denoted by z and is defined as z = a-ib

## Properties of Modulus and conjugate of complex numbers

For any two complex numbers z1 and z2, we have

## Square Root of a negative real number

We know i^{2}=-1 and (–i)^{2}=i^{2} =-1

Hence the square roots of -1 are i, -i but the symbol √-1 mean ‘i’ only.

Now we can see that i and –i both are the solutions of the equation x^{2}+1=0

Similarly, (√5i)^{2} = 5i^{2} = 5 x -1 = -5

(-√5i)^{2} = 5i^{2} = 5 x -1 = -5

Generally, if ‘a’ is a positive real number, √-a = √a √-1 = √a i

We have already learned that √a x √b = √ab for all positive real numbers ‘a’ and ‘b’

We can examine this result for i2,

i^{2} = i x i = √-1 x √-1 = √-1 x -1 = √1 =1, which is a contradiction to the fact that i^{2}=-1

Hence √a x √b ≠ √ab if both ‘a’ and ‘b’ are negative real numbers

If ‘a’ and ‘b’ are zero, then √a x √b = √ab =0

## Powers of ‘i’

We know that i^{2} = -1, different powers of ‘i’ are given below

i^{3} = i^{2} x i = -1 x i = -i

i^{4} = (i^{2})2 = (-1)^{2} = 1

i^{5} = i^{4} x i = 1 x i = i

i^{6} = i^{5} x i = i x i = i^{2} = -1 and so on.

Also we have,

In general, for any integer k, i4k=1, i4k+1=i, i4k+2 = -1, i4k+3 = -i

## Identities in Complex numbers

For any two complex numbers z1 and z2 we have the following identities

- (z
_{1}+ z_{2})^{2}= z_{1}^{2}+ z_{2}^{2}+ 2z_{1}z_{2}

(z_{1} + z_{2})^{2} = (z_{1} + z_{2}) (z_{1}+z_{2})

= z_{1}^{2} + z_{1}z_{2} + z_{1}z_{2} + z_{2}^{2}

= z_{1}^{2} + 2z_{1}z_{2} + z_{2}^{2}

- (z
_{1}-z_{2})2 = z_{1}^{2}-2z_{1}z_{2}+z_{2}^{2} - (z1+z2)3 = z13+3z12z2+3z1z22+z23
- (z1-z2)3 = z13-3z12z2+3z1z22+z23
- z12-z22= (z1+z2)(z1-z2)

Express in the form a + ib

1) i^{9} + i^{19}

1) i^{9} + i^{19} = i^{8} i + i^{18} i

= 1 x i + -1 x i

= i + -i

= 0

= 0 + i0

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