Show that the additive inverse of \(z=r e^{j \theta}\) may be written as \(r e^{j(\theta+\pi)}\).

The additive inverse is an important concept in mathematics. It refers to a number, which, when added to the original number, results in zero. For any given complex number, like our example in polar form, finding the additive inverse helps solve various equations and understand symmetry in the complex plane.

The problem gives us a complex number in the polar form: \( z = r e^{j \theta} \).
The additive inverse of this complex number is simply its negative: \( -z \).
We express the additive inverse as: \( -z = -r e^{j \theta} \).
If we convert this expression back into polar form, it looks a bit different but represents the same concept. By understanding the additive inverse, we make it easier to deal with complex numbers and their properties.

Complex numbers combine real and imaginary parts, and they have the form \( a + bi \). Here, 'a' and 'b' are real numbers, and 'i' is the imaginary unit with the property \( i^2 = -1 \).

In our exercise, we're dealing with complex numbers in polar form, which is another way to represent them: \( z = re^{j \theta} \).
In this form:

• 'r' is the magnitude (distance from the origin)
• 'e^{j \theta}' represents the direction, where 'θ' is the angle from the positive x-axis

This polar representation leverages Euler's formula to transform complex arithmetic into more manageable forms, particularly beneficial when dealing with multiplication and division.

Euler's formula is a key tool to connect exponential functions with trigonometry. It states that:

\( e^{j \theta} = \text{cos} (\theta) + i \text{sin} (\theta) \).
This is particularly useful for representing complex numbers in polar form. Using Euler's formula, we convert the angle 'θ' into its real (cosine) and imaginary (sine) components, enabling easier handling of angles in mathematics.

In our proof, we use Euler's formula to show that adding 'π' to the angle effectively changes its direction by 180 degrees, which flips the number to its opposite (additive inverse). It shows the power of Euler's formula in simplifying complex calculations and understanding the geometric representation of complex numbers.

Polar coordinates offer a way to determine the position of points in the plane using a distance 'r' (radius) and an angle 'θ' (theta).
This is in contrast to Cartesian coordinates (x, y), where points are defined by their horizontal and vertical distances from the origin.

For complex numbers, polar coordinates simplify multiplication and division.

• Multiplying two complex numbers multiplies their magnitudes and adds their angles.
• Dividing two complex numbers divides their magnitudes and subtracts their angles.

In the given exercise, using polar coordinates simplifies finding the additive inverse by merely adding a constant angle (π), thus showing the efficiency of polar representation in complex arithmetic.