Determine whether the statement is true or false. If true,...

Determine whether the statement is true or false. If true, explain why. If false, give a counter example.
Prove
$$
\frac{z_{1}}{z_{2}}=\frac{r_{1} e^{i \theta_{1}}}{r_{2} e^{i \theta_{2}}}=\frac{r_{1}}{r_{2}} e^{i\left(\theta_{1}-\theta_{2}\right)}
$$

Key Concepts

Complex Numbers in Polar Form

This concept involves expressing complex numbers in the form r e^(i?), where r represents the magnitude and ? represents the angle (or argument). This representation simplifies many operations, such as multiplication and division, by decoupling the magnitude and directional (angle) components.

Euler's Formula

Euler's formula, which states that e^(i?) = cos(?) + i sin(?), is fundamental for translating between the exponential and trigonometric representations of complex numbers. It underpins the polar form representation and aids in understanding how operations on complex numbers affect their magnitudes and angles.

Division of Complex Numbers in Polar Form

When dividing complex numbers expressed in polar form, the division of the magnitudes (moduli) is performed directly and the subtraction of the angles (arguments) is applied. This property directly leads to the result that the quotient of two complex numbers is given by (r1/r2) e^(i(?1 - ?2)).

Properties of Moduli and Arguments

The modulus of a complex number represents its distance from the origin in the complex plane, and the argument represents its direction. These properties allow for simple algebraic manipulations during multiplication or division, as the modulus of the quotient is the quotient of the moduli, and the argument of the quotient is the difference of the arguments.

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