Polar and Cartesian Forms

Complex numbers are a fascinating extension of the real numbers. They provide a way to solve equations that otherwise do not have a solution in the set of real numbers. Complex numbers have many applications in engineering, physics, and mathematics. In this lesson, we will explore two beautiful forms for representing them: the Cartesian form and the polar form.

Understanding complex numbers

A complex number is made up of two parts: a real part and an imaginary part. It can be expressed as z = a + bi, where:

• a is the real part of the complex number.
• b is the imaginary part and is multiplied by the imaginary unit i, where i is defined by the property i² = -1.

For example, 3 + 4i and 5 - 2i are complex numbers.

Cartesian form

Cartesian form, also known as rectangular form, represents a complex number as z = a + bi. This form is quite useful for basic addition and subtraction of complex numbers.

Let's consider two complex numbers, z1 = 3 + 4i and z2 = 1 + 2i. To add them, we simply add the real parts and the imaginary parts separately:

z1 + z2 = (3 + 4i) + (1 + 2i) = (3 + 1) + (4i + 2i) = 4 + 6i

Subtraction follows a similar process:

z1 – z2 = (3 + 4i) – (1 + 2i) = (3 – 1) + (4i – 2i) = 2 + 2i

Geometric interpretation: Cartesian coordinates

Every complex number can be represented as a point in a two-dimensional space known as the complex plane. The real part is the horizontal component and the imaginary part is the vertical component. This is consistent with the Cartesian coordinate system.

The complex number 3 + 4i corresponds to the point (3, 4) in the complex plane.

Polar form

In addition to addition and subtraction, complex numbers can also be multiplied and divided. For these operations, the polar form is often more convenient. The polar form of a complex number expresses the number in terms of its magnitude (also called the modulus) and its angle (also called the argument).

The polar form is written as z = r(cos θ + i sin θ), where:

• r represents the modulus of the complex number, which is calculated as r = sqrt(a² + b²).
• θ represents the argument, calculated as θ = atan(b/a), which is the angle made with the positive x-axis.

Conversion between forms

Let us convert the complex number 3 + 4i to its polar form.

Step 1: Find the modulus

The modulus r is calculated as follows:

R = sqrt(a² + b²) = sqrt(3² + 4²) = sqrt(9 + 16) = sqrt(25) = 5

Step 2: Find the logic

The argument θ is obtained from:

θ = atan(b/a) = atan(4/3)

Typically, a calculator is used to measure angles in degrees or radians.

The resulting polar form

Thus, the polar form of 3 + 4i is:

5(cos θ + i sin θ)

Where this is approximately 53.13° when converted to degrees using θ calculator.

Geometric interpretation: polar coordinates

In the complex plane, the modulus r represents the distance from the origin to the point (a, b), and the argument θ represents the angle from the positive real axis to the line joining the origin to the point (a, b). The polar form emphasizes the use of angles, increasing its utility in rotation and other angular displacement calculations.

The following figure shows a complex number in polar form:

Multiplication and division polarly

The polar form of complex numbers simplifies multiplication and division. Two complex numbers z1 = r1(cos θ1 + i sin θ1) and z2 = r2(cos θ2 + i sin θ2) are given.

The product is calculated as follows:

z1 * z2 = r1 * r2 [cos(θ1 + θ2) + i sin(θ1 + θ2)]

The division is calculated as follows:

z1 / z2 = (r1 / r2) [cos(θ1 - θ2) + i sin(θ1 - θ2)]

Note that multiplying moduli and adding angles simplifies multiplication, and so does division.

Complex conjugates

The complex conjugate of a complex number z = a + bi is a - bi. The concept of a conjugate is essential in many areas of complex analysis, geometry, and is often used to simplify complex division.

A complex number and its conjugate have the same modulus, while their arguments differ by a sign.

Visual example

Here's a visual comparison between Cartesian and polar forms using the number 5 + 5i.

Cartesian form

Polar form

Conclusion

The study of complex numbers in both Cartesian and polar forms serves as the basis for higher mathematical concepts and practical applications. From simple arithmetic to complex equations, the choice of representation often depends on the problem at hand.

By understanding both geometric and algebraic forms, one can harness the power of complex numbers to solve challenging mathematical problems with beauty and ease.

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