Polar coordinates form another set of parameters that characterize the vector from the origin to the point z = x + iy , with magnitude and direction.

**Polar and Euler form of a Complex Number**

When performing addition and subtraction of complex numbers, we
use rectangular form. This is because we
just add real parts and add imaginary parts; or subtract real parts, and
subtract imaginary parts. When performing multiplication or finding powers or
roots of complex numbers, use an alternate form namely, polar form, because it is easier
to compute in polar form than in rectangular form.

**Polar form of a complex number**

Polar coordinates form another set of parameters that characterize
the vector from the origin to the point *z *= *x *+
*iy *, with magnitude and direction. The polar coordinate system consists
of a fixed point O called the pole and the horizontal half line emerging from
the pole called the initial line (polar axis). If *r *is the distance from
the pole to a point *P *and *Î¸ *is an angle of
inclination measured from the initial line in the counter clockwise direction
to the line *OP*, then *r *and *Î¸ *of the ordered pair (*r*,*Î¸** *) are called the polar coordinates of *P*.
Superimposing this polar coordinate system on the rectangular coordinate
system, as shown in diagram, leads to

*x *= *r *cos*Î¸* ...(1)

*y *= *r *sin*Î¸ ** â€¦.*(2)

Any non-zero complex
number *z *= *x *+ *iy *can be expressed as *z *= *r *cos*Î¸
*+ *i r *sin*Î¸*.

Let *r *and *Î¸ *be polar coordinates of the point *P*(*x*, *y*) that corresponds to a
non-zero complex number *z *= *x *+ *iy *. The polar form or trigonometric form of a
complex number *P *is

*z *= *r*(cos*Î¸ *+ *i *sin*Î¸ *) *.*

For convenience, we can write polar form as

* z *= *x *+ *iy *= *r *(cos*Î¸** *+ *i *sin*Î¸** *) = *r cis**Î¸** *.

The value *r *represents the absolute value or modulus of the complex number *z *. The angle *Î¸** *is called the argument or amplitude of the complex number *z
*denoted by *Î¸** *= arg ( *z *).

(i) If *z *= 0 , the argument *Î¸** *is undefined; and so it is understood that *z* â‰ 0 whenever polar coordinates are used.

(ii) If the complex number *z *= *x *+
*iy *has polar coordinates (*r*, *Î¸** *) , its conjugate * *= *x *âˆ’ *iy has *polar
coordinates (*r*, âˆ’*Î¸** *) .

Squaring and adding (1) and (2), and taking square root, the value
of *r* is given by r = |z| =

Dividing (2) by (1),

Case (i) The real number *Î¸ *represents the angle, measured in
radians, that *z *makes with the positive real axis when *z *is
interpreted as a radius vector. The angle *Î¸ *has an infinitely many
possible values, including negative ones that differ by integral multiples of 2*Ï€*.
Those values can be determined from the equation tan Î¸ = *y/x* where the
quadrant containing the point corresponding to *z *must be specified.
Each value of Î¸* *is called an *argument
*of *z*, and the set of all such values is obtained by adding multiple
of 2Ï€* *to Î¸, and it is
denoted by arg *z*.

Case (ii) There is a unique value of *Î¸ *which satisfies the condition âˆ’*Ï€** *<*Î¸** *â‰¤ *Ï€** *.

This value is called a principal value of *Î¸ *or principal
argument of *z** *and is denoted by Arg *z*.

Note that, âˆ’ Ï€ < Arg(z) â‰¤ Ï€ or âˆ’Ï€ < Î¸ â‰¤ Ï€

Principal Argument of a complex number

The capital A is important here to distinguish the principal
value from the general value.

Evidently, in practice to find the principal angle *Î¸*, we
usually compute Î± = tan^{-1}|*y*/*x*| and adjust for the quadrant problem by
adding or subtracting *Î± *with *Ï€ *appropriately.

**arg z = Arg z + 2 n
Ï€, n ****âˆˆ Z ****.**

Some of the properties
of arguments are

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

(2) arg (z_{1}/z_{2}) = arg z_{1} - arg z_{2}

(3) arg (z^{n}) = n arg z

(4) The alternate
forms of cos Î¸ + *i* sin Î¸ are cos (2kÏ€ + Î¸) + i sin (2kÏ€ + Î¸), k âˆˆ **Z**.

For instance the
principal argument and argument of 1, *i*, -1, and -*i* are shown
below:-

Plot the following
complex numbers in complex plane

Tags : Definition, Illustration, Formulas , 12th Mathematics : UNIT 2 : Complex Numbers

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12th Mathematics : UNIT 2 : Complex Numbers : Polar form of a complex number | Definition, Illustration, Formulas

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