# Complex Number

**Complex numbers**

Complex numbers are numbers that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, defined as i^{2} = -1.

The real part of a complex number a + bi is the real number a, and the imaginary part is the real number b multiplied by i.

For example, 2+5i is a complex number, where 2 is a real number (Re) and 5i is an imaginary number (Im).

The set of all complex numbers is denoted by the symbol C

Note:-1)Euler was the first mathematician to introduce the symbol i (iota) for the square root of -1 with property. i^{2}=-1 He also called this symbol as the imaginary unit.

2) Iota (i) is neither 0, nor greater than 0, nor less than 0.

3) The square root of a negative real number is called an imaginary unit.

4) For any positive real number a, we have √-a=i√a

5)The property √a√b = √ab is valid only if at least one of a and b is non-negative. If

a and b are both negative then √a√b = -√ab

**Integral powers of iota (i) :**

Since i= $\sqrt{-1}$√−1 hence we have i^{2}=-1,i^{3}=I and i^{4}=1 .

In general we have the following results

i^{4k+1} = i.i^{4k+2 }= -1 i^{4k+3} = -i.i^{4k} = 1

Where k can have an integral value (positive or negative).

Similarly, we can find for the negative power of *i*, which are as follows;

i^{-1} =-i, i^{-2}=-1,i^{-3}=i and i^{-4}=1

**Important Tips**

→The sum of four consecutive powers of i is always zero i.e i^{k}+i^{k+1}+i^{k+2}+i^{k=3}=0

→ (i+1)^{2}=2i , (1-i)^{2}=-2i

→ $\frac{1+i}{1-i}=i,\frac{1-i}{i+i}=-i$1+`i`1−`i` =`i`,1−`i``i`+`i` =−`i`

**Real Part of a Complex Number**

If x and y are two real numbers, then a number of the form z=x+iy is called a complex

number. Here ‘x’ is called the real part of z. The real

part of z is denoted by Re(z)

**Imaginary Part of a Complex Number**

If x and y are two real numbers, then a number of the form z=x+iy. Here ‘y’ is called the Imaginary part of z. The Imaginary

part of z is denoted by Im(z)

If z = 5 – 4i, then Re(z) = 5 and Im(z) = – 4

** Note **:

- A complex number z is purely real if its imaginary part is zero i.e., Im(z) = 0 and

purely imaginary if its real part is zero i.e., Re(z) = 0. - I can be denoted by the ordered pair (0,1).
- The complex number (a, b) can also be split as (a, 0) + (0, 1) (b, 0)

**Equality of Two Complex Numbers****.**

Two complex numbers *z*_{1} = *x*_{1} + *iy*_{1} and *z*_{2} = *x*_{2} + iy_{2} are said to be equal if and only if their

real parts and imaginary parts are separately equal.

*i.e., **z*_{1}=* z*_{2} Þ* x*_{1} + *iy*_{1}* =* *x*_{2} + iy_{2} Û *x*_{1} = *x*_{2} and *y*_{1}=* y*_{2}

Thus , one complex equation is equivalent to two real equations.

Note :

- A complex number z=x+iy.=0
*iff*. x=0, y=0 - The complex number do not possess the property of order
*i.e.,*(*a*+*ib*) < (*or*) > (*c*+*id*) is

not defined. For example, the statement 9 + 6*i *> 3 + 2*i *makes no sense.

**Representation of Complex Number****.**

A complex number can be represented in the following from:

(1) **Geometrical representation (Cartesian representation): **The complex number *z *= *a *+ *ib *=(*a*, *b*)

is represented by a point *P *whose coordinates are referred to rectangular axes *XOX *¢ and *YOY *¢ which are called real and imaginary axis respectively. Thus a complex number *z *is represented by a point in a plane, and corresponding to every point in this plane there exists a complex number such a plane is called argand plane or argand diagram or complex plane or gaussian plane.

*Note* :

- Distance of any complex number from the origin is called the modules of complex number and is denoted by |
*z*|,*i.e |z|= $\sqrt{a^2+b^2}$√*`a`^{2}+`b`^{2} - Angle of any complex number with positive direction of
*x–*axis is called amplitude or argument of*z. i.e.,**amp*(*z*)=*arg*(*z*)= tan^{-1}(b/a)

(2)**Trigonometrical (Polar) representation : **In D *OPM*, let * OP *=* r * , then *a*=* r*cosq * * and* b*=.* r* sinq

Hence *z *can be expressed as *z *= *r*(cosq + *i *sinq )

where *r *= |*z*| and q = principal value of argument of *z*.

For general values of the argument *z *= *r*[cos(2*n*p +q )+ *i *sin(2*n*p +q )]

(3) **Vector representation : **

If *P *is the point (*a*, *b*) on the argand plane corresponding to the complex number *z *= *a *+ *ib *.

Then OP̅=aî+bĵ, \| OP̅|=√a²+b² and *arg z *= direction of the vector OP̅= tan^{-1}(b/a)

Therefore, complex number z can also be represented by OP̅

(4) **Eulerian representation (Exponential form) : **

Since we have e^{i}^{q} =cosq + *i *sinq and thus *z *can be expressed as *z *= *r *e^{i}^{q} , where *r *= |*z*| and q = arg(*z)*.

**Logarithm of complex number**

**Geometry of complex number**

**Square Root of a Complex Number**