Polar Form

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locating numbers

Representing Complex Numbers

We already have several ways of representing complex numbers.

$\blacktriangleright (4, 8)$: We represent complex numbers visually with points or dots on the Cartesian plane.

$\blacktriangleright \langle 4, 8 \rangle $: We represent complex numbers with vectors. These are illustrated with arrows on the Cartesian plane. Algebraically, we write vectors using triangular brackets.

Vectors have a visual arithmetic, where we arrange the vectors tail-to-head or tail-to-tail and then create a resultant vector from the arrangement. These resultant vectors represent the sum or difference of two Complex numbers.

Vectors have a symbolic arithmetic as well, which agrees with the geometric operations:

\[ \langle a, b \rangle + \langle c, d \rangle = \langle a+c, b+d \rangle \]

$\blacktriangleright 4 + 8 \, i$: We represent complex numbers with a 2-dimensional sum.

The left, first, or horizontal part is called the real part and the right, second, or vertical part is called the imaginary part.

When viewing the plane in this context, we call it the Complex Plane.

All of these representations share a common structure. They all describe complex numbers with rectangular measurements. All of the descriptions above give horizontal (left/right) and vertical (up/down) information.

There is a different way to describe complex numbers.

Context

Mathematics is famous for reusing notation. We use the exact same symbols to represent different mathematical objects and we let the context of the situation dictate our interpretation.

Same story here.

Just like with Real Numbers, we have geometric and algebraic representations of Complex Numbers. The geometric representations and symbols are borrowed from our Geometry.

$(a, b)$ might represent an interval in the real numbers. It might represent the coordinates of a point in the Cartesian plane. It might represent a Complex number.

$\langle a, b \rangle $ might represent a vector. It might represent an arrow in the Cartesian plane. It might represent a Complex number.

This is normal. It happens every day with the English langugae. We need to take into account the people and context of the discussion.

Polar Coordinates

Instead of rectangular information, we could give circular information. We could give the angle the vector makes with the positive horizontal axis together with the length or distance from the origin.

In keeping with the circular idea, the vector resembles a radius.

Our two pieces of information will be $r$ and $\theta $. These are known as polar coordinates

Polar Coordinates

Each Complex number can be described by $(r, \theta )$, where $r$ is a real number and $\theta $ is an angle measurement.

$r$ can be positive or negative.
$\theta $ is measured counterclockwise from the positive real axis.

Of course, that is how we write rectangular coordinates as well. The context of the situation will tell us how to interpret the coordinates.

Write $4 + 8i$ in polar form.

$r = \sqrt {4^2 + 8^2} = \sqrt {80} = 4\sqrt {5}$ (the modulus of $4 + 8 \, i$)
We need $\theta $ to give us $\tan (\theta ) = 2$, which is approximately $\theta \approx 63.43^{\circ }$

We, again, wrap the coordinates up with parentheses: $(r, \theta ) = (16\sqrt {5}, 63.43^{\circ })$.

And, we need to work equally well with radians and degrees: $(r, \theta ) = (16\sqrt {5}, 1.107)$

Multiple Polar Coordinates

Unlike with rectangular coordinates, different polar coordinates can describe the same complex number.

\[ \cdots = (r, \theta - 2\pi ) = (r, \theta ) = (r, \theta + 2\pi ) = (r, \theta + 4\pi ) = \cdots \]

Negative “Distance”

The $r$ in $(r, \theta )$ is the distance to move in the $\theta $-direction. However, $r$ can be negative. In this case, the interpretation is to move in the opposite direction from $\theta $.

\[ (r, \theta ) = (-r, \theta + \pi ) = (-r, \theta - \pi ) \]

Which of the following polar coordinates represent the same complex number?

$\left ( 5, \frac {\pi }{4} \right )$ $\left ( 5, \frac {9\pi }{4} \right )$ $\left ( 5, \frac {3\pi }{4} \right )$ $\left ( -5, \frac {5\pi }{4} \right )$ $\left ( -5, \frac {7\pi }{4} \right )$

Radians

There is a very tight relationship between angles and arc length in the unit circle.

A full unit circle is $2\pi $ radians and the circumference is $2\pi $. When measuring in radians, the angle is the same value as the arc length. As you rotate around the unit circle that angles changes at exactly the same rate as the arc length.

This relationship does not hold with degrees. A full unit circle is $360$ degrees, which is not matching the circumference at $2\pi $.

For this reason, Calculus uses radians almost exclusively. In Precalculus, we need to become radian thinkers.

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more examples can be found by following this link
More Examples of Polar Form of Complex Numbers

2025-05-18 00:25:21