Notes On Polar Representation of Complex Numbers - CBSE Class 11 Maths

Consider a complex number x + iy.

The real part of the complex number is 'x' and the imaginary part is 'y'. Both x and y are real numbers.

An ordered pair of real numbers can represent a unique point in a XY-plane.

Let's consider an ordered pair (x, y), which is represented by point P in the XY plane.

If this complex number corresponds to the ordered pair (x, y),, then it can be represented by the unique point P(x, y), in the XY-plane. Conversely, each point in the XY-plane represents a unique complex number.

Each point in the XY-plane represents an ordered pair corresponding to a complex number.

A plane having complex numbers corresponding to each of its points is called the complex plane or the Argand plane. In the Argand plane, the x-axis is called the real axis and the y-axis is called the imaginary axis. The x-coordinate of a point corresponds to the real part and the y-coordinate corresponds to the imaginary part of a complex number.

All the points on the real axis are represented in the form (x, 0), corresponding to the complex number x + 0i.

Similarly, all the points on the imaginary axis are represented in the form (0, y), corresponding to the complex number 0+ iy.

Let's consider the complex number 8 + 6i.

Point P (8, 6) on the Argand plane represents the complex number 8 + 6i.

The perpendicular distance of point P from the x-axis is 6 units, while the perpendicular distance of point P from the y-axis is 8 units.

Representation of the conjugate of a complex number in an Argand Plane:

Consider the complex number 8 + 6i.

Conjugate of 8 + 6i = 8 - 6i.

The ordered pair corresponding to the complex number is (8, -6)

Point Q(8, -6) in the Argand plane represents the conjugate of the given complex number.

The distance of point P from the origin O = √(82 + 62) = 10.

The distance of point P from the origin is the modulus of the complex number 8+ 6i.

The modulus of z = x + iy can be geometrically interpreted as the distance of point P(x,y) from the origin in the Argand plane.

In the Argand plane, z = x + iy is represented by the point (x,y), and its conjugate z^ = x - iy is represented by the point (x,-y), which is geometrically interpreted as the mirror image of the point (x,y) on the real axis.

Polar coordinates

Complex number z (x + iy) can be uniquely located in an XY-plane by locating the ordered pair (x, y).

Let the length of the directed line joining point P and the origin (OP), be r, which is equal to the modulus of the complex number z.

Let q be the angle that OP makes with the positive direction of the x-axis.

The complex number z can also be uniquely located in the XY-plane by the ordered pair of real number (r, q).

The coordinates (r, q) are called the polar coordinates of point P. The origin is considered as the pole and the positive direction of x-axis is called the initial line.

Relationship between the Cartesian coordinates and the polar coordinates:

The length of PM a perpendicular from point P to the x-axis is y units and the length of OM is x units.

cos θ = OM/OP = x/r

x = rcos θ

sin θ = PM/OP = y/r

y = rsin θ

z = x + iy = r cos θ + i r sin θ

z = r (cos θ + i sin θ)

This is called the polar form of the complex number. Value of r = Modulus of z = |z| = √(x2 + y2).

'q' is called the Argument or amplitude of 'z', denoted by arg z.

tan θ = sin θ / cos θ = (y/r)/(x/r) = y/x

⇒ θ = tan-1(y/x) , where x ≠ 0

So, argument q of a complex number can only be defined for non-zero complex numbers.

For any non-zero complex number, we commonly consider the value of q, such that -π < θ < π, called the principal argument of the complex number, unless specified otherwise.

Properties of arguments:
   •  arg(z1 x z2) = arg(z1) + arg(z2)
   •  arg(z1 / z2) = arg(z1) - arg(z2)

Summary

Consider a complex number x + iy.

The real part of the complex number is 'x' and the imaginary part is 'y'. Both x and y are real numbers.

An ordered pair of real numbers can represent a unique point in a XY-plane.

Let's consider an ordered pair (x, y), which is represented by point P in the XY plane.

If this complex number corresponds to the ordered pair (x, y),, then it can be represented by the unique point P(x, y), in the XY-plane. Conversely, each point in the XY-plane represents a unique complex number.

Each point in the XY-plane represents an ordered pair corresponding to a complex number.

A plane having complex numbers corresponding to each of its points is called the complex plane or the Argand plane. In the Argand plane, the x-axis is called the real axis and the y-axis is called the imaginary axis. The x-coordinate of a point corresponds to the real part and the y-coordinate corresponds to the imaginary part of a complex number.

All the points on the real axis are represented in the form (x, 0), corresponding to the complex number x + 0i.

Similarly, all the points on the imaginary axis are represented in the form (0, y), corresponding to the complex number 0+ iy.

Let's consider the complex number 8 + 6i.

Point P (8, 6) on the Argand plane represents the complex number 8 + 6i.

The perpendicular distance of point P from the x-axis is 6 units, while the perpendicular distance of point P from the y-axis is 8 units.

Representation of the conjugate of a complex number in an Argand Plane:

Consider the complex number 8 + 6i.

Conjugate of 8 + 6i = 8 - 6i.

The ordered pair corresponding to the complex number is (8, -6)

Point Q(8, -6) in the Argand plane represents the conjugate of the given complex number.

The distance of point P from the origin O = √(82 + 62) = 10.

The distance of point P from the origin is the modulus of the complex number 8+ 6i.

The modulus of z = x + iy can be geometrically interpreted as the distance of point P(x,y) from the origin in the Argand plane.

In the Argand plane, z = x + iy is represented by the point (x,y), and its conjugate z^ = x - iy is represented by the point (x,-y), which is geometrically interpreted as the mirror image of the point (x,y) on the real axis.

Polar coordinates

Complex number z (x + iy) can be uniquely located in an XY-plane by locating the ordered pair (x, y).

Let the length of the directed line joining point P and the origin (OP), be r, which is equal to the modulus of the complex number z.

Let q be the angle that OP makes with the positive direction of the x-axis.

The complex number z can also be uniquely located in the XY-plane by the ordered pair of real number (r, q).

The coordinates (r, q) are called the polar coordinates of point P. The origin is considered as the pole and the positive direction of x-axis is called the initial line.

Relationship between the Cartesian coordinates and the polar coordinates:

The length of PM a perpendicular from point P to the x-axis is y units and the length of OM is x units.

cos θ = OM/OP = x/r

x = rcos θ

sin θ = PM/OP = y/r

y = rsin θ

z = x + iy = r cos θ + i r sin θ

z = r (cos θ + i sin θ)

This is called the polar form of the complex number. Value of r = Modulus of z = |z| = √(x2 + y2).

'q' is called the Argument or amplitude of 'z', denoted by arg z.

tan θ = sin θ / cos θ = (y/r)/(x/r) = y/x

⇒ θ = tan-1(y/x) , where x ≠ 0

So, argument q of a complex number can only be defined for non-zero complex numbers.

For any non-zero complex number, we commonly consider the value of q, such that -π < θ < π, called the principal argument of the complex number, unless specified otherwise.

Properties of arguments:
   •  arg(z1 x z2) = arg(z1) + arg(z2)
   •  arg(z1 / z2) = arg(z1) - arg(z2)

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