# Equality of Two Complex Numbers

Two complex numbers

*z*_{1}=*a*_{1}+*ib*_{1}and*z*_{2}=*a*_{2}+*ib*_{2}are equal if*a*_{1}=*a*_{2}and*b*_{1}=*b*_{2}, i.e., Re(*z*_{1}) = Re(*z*_{2}) and Im(*z*_{1}) = Im(*z*_{2}).Complex numbers do not possess the property of order, i.e., (

*a*+*ib*) < (or) > (*c*+*id*) is not defined.# Square root of a complex number

Let = , for

*b*> 0= , for

*b*< 0To find the square root of

*a*â€“*ib*, replace*i*by â€“*i*in the above result.# Geometrical representation of a complex number on argand plane

Distance of any complex number from the origin is called the modulus of complex number and is denoted by |

*z*|, i.e., |*z*| = .Angle of any complex number with positive direction of the

*x*-axis is called amplitude or argument of*z*. i.e., amp (*z*) = arg(*z*) = tan^{â€“1}(here*z*lies in the first quadrant)**Argument or amplitude of**

*a**complex number**z**=**x*+*iy*for different signs of*x*and*y**z*=*x*+*i y*when*x*> 0 and*y*> 0*z*=*x*+*i y*when*x*< 0 and*y*> 0*z*=*x*+*i y*when*x*< 0 and*y*< 0*z*=*x*+*i y*when*x*> 0 and*y*< 0

# Polar form of a complex number

*z*=

*x*+

*iy*=

=

This is a polar form of the complex number.

Here, |

Here, |

*z*| = distance of*z*from origin and*Î¸*is argument.

**Properties of modulus**If

*z*,*z*_{1},*z*_{2}âˆˆ*C*, then- |
*z*| = 0 â‡”*z*= 0, i.e., Re(*z*) = Im(*z*) = 0 - |
*z*| = |*z*| = |â€“*z*| = |â€“*z*| - â€“|
*z*| â‰¤ Re(*z*) â‰¤ |*z*|; â€“|*z*| â‰¤ Im(*z*) â‰¤ |*z*| *z**z*= |*z*|^{2}- |
*z*_{1}*z*_{2}| = |*z*_{1}| |*z*_{2}| - ;
*z*_{2}â‰ 0 - |
*z*_{1}+*z*_{2}|^{2}= |*z*_{1}|^{2}+ |*z*_{2}|^{2}+ 2 Re(*z*_{1}*z*_{2}) - |
*z*_{1}â€“*z*_{2}|^{2}= |*z*_{1}|^{2}+ |*z*_{2}|^{2}â€“ 2 Re(*z*_{1}*z*_{2}) - |
*z*_{1}+*z*_{2}|^{2}+ |*z*_{1}â€“*z*_{2}|^{2}= 2(|*z*_{1}|^{2}+ |*z*_{2}|^{2}) - |
*az*_{1}â€“*bz*_{2}|^{2}+ |*bz*_{1}+*az*_{2}|^{2}= (*a*^{2}+*b*^{2}) (|*z*_{1}|^{2}+ |*z*_{2}|^{2}), where*a*,*b*âˆˆ*R* - |
*z*| = |^{n}*z*|, where^{n}*n*âˆˆ*Q* - |
*z*_{1}Â±*z*_{2}| â‰¤ |*z*_{1}| + |*z*_{2}| - |
*z*_{1}Â±*z*_{2}| â‰¥ ||*z*_{1}| â€“ |*z*_{2}||

# Conjugate of a complex number

If there exists

=

*a**complex number**z*=*a*+*ib*, (*a*,*b*) âˆˆ*R*, then its conjugate is defined as=

*a*â€“ i*b*.Hence, we have Re(

*z*) = .Geometrically, the conjugate of

*z*is the reflection or point image of*z*in the real axis.

*Properties of conjugate*

If

*z*,*z*_{1},*z*_{2}are complex numbers, then- (
) =__z__*z* *z*+*z*= 2 Re(*z*)*z*+*z*= 2*i*Im(*z*)*z*=*z*â‡”*z*is purely real*z*+*z*= 0 â‡” is purely imaginary-
*z*_{2}â‰ 0

**Properties of arguments**- arg(
*z*_{1}*z*_{2}) = arg(*z*_{1}) + arg(*z*_{2})*z*_{1}*z*_{2}*z*_{3}â€¦*z*) = arg(_{n}*z*_{1}) + arg(*z*_{2}) + arg(*z*_{3}) + â€¦ + arg(*z*)_{n} - arg(
*z*_{1}*z*_{2}) = arg(*z*_{1}) â€“ arg(*z*_{2}) - arg = arg
*z*_{1}â€“ arg*z*_{2} - arg = 2 arg
*z* - arg(
*z*) =^{n}*n*arg*z* - arg
*z*= â€“arg*z*= arg *z*_{1}*z*_{2}+*z*_{1}*z*_{2}= 2 |*z*_{1}| |*z*_{2}| cos (*Î¸*_{1}â€“*Î¸*_{2}), where*Î¸*_{1}= arg (*z*_{1}) and*Î¸*_{2}= arg(*z*_{2})- If
*z*is purely imaginary then arg(*z*) = - If
*z*is purely real then arg (*z*) = 0 or*Ï€* - Angle between two line segments joining
*z*_{1}and*z*_{2}and joining*z*_{3}and*z*_{4}is*Î¸*= Â±arg -
- If
*z*_{1},*z*_{2},*z*_{3}be the vertices of an equilateral triangle, then