Question

Show that the absolute value of a product of two complex numbers is equal to the product of the absolute values. Also show that the absolute value of the quotient of two complex numbers is the quotient of the absolute values. Hint: Write the numbers in the \(r e^{i \theta}\) form.

Short answer

The magnitude of the product is the product of the magnitudes, and the magnitude of the quotient is the quotient of the magnitudes.

Step-by-step solution

- Represent the Complex Numbers

Represent the two complex numbers in polar form. Let the first complex number be written as \( z_1 = r_1 e^{i \theta_1} \) and the second as \( z_2 = r_2 e^{i \theta_2} \), where \( r_1 \) and \( r_2 \) are the magnitudes (or absolute values), and \( \theta_1 \) and \( \theta_2 \) are the arguments (or angles).

- Calculate the Product in Polar Form

Using the properties of exponents and polar form, the product of the two complex numbers is \[ z_1 z_2 = (r_1 e^{i \theta_1})(r_2 e^{i \theta_2}) = r_1 r_2 e^{i (\theta_1 + \theta_2)} \].

- Find the Magnitude of the Product

The magnitude (or absolute value) of the product is \[ |z_1 z_2| = |r_1 r_2 e^{i (\theta_1 + \theta_2)}| = r_1 r_2 \]. Notice that this is the product of the magnitudes of \( z_1 \) and \( z_2 \), i.e., \[ |z_1 z_2| = |z_1||z_2| \].

- Calculate the Quotient in Polar Form

Similarly, the quotient of the two complex numbers is \[ \frac{z_1}{z_2} = \frac{r_1 e^{i \theta_1}}{r_2 e^{i \theta_2}} = \frac{r_1}{r_2} e^{i (\theta_1 - \theta_2)} \].

- Find the Magnitude of the Quotient

The magnitude of the quotient is \[ \left| \frac{z_1}{z_2} \right| = \left| \frac{r_1}{r_2} e^{i (\theta_1 - \theta_2)} \right| = \frac{r_1}{r_2} \]. This corresponds to the quotient of the magnitudes of \( z_1 \) and \( z_2 \), i.e., \[ \left| \frac{z_1}{z_2} \right| = \frac{|z_1|}{|z_2|} \].

Key concepts

polar form of complex numbers

To dive into the world of complex numbers, it's crucial to understand their representation in polar form. A complex number like 3 + 4i can be expressed in this form. In general, any complex number z can be written as \( z = r e^{i \theta} \). Here,
- \( r \) is the magnitude or absolute value of the complex number.
- \( \theta \) is the argument or angle.
This polar form helps in simplifying many operations, like multiplication and division, as we'll see further. By converting complex numbers into polar form, we can leverage the properties of exponents and angles to make our calculations easier.

absolute value of complex numbers

The absolute value, or magnitude, of a complex number tells you its distance from the origin in a complex plane. For a complex number \( z = x + yi \), its magnitude is given by \( |z| = \sqrt{x^2 + y^2} \). In polar form, if \( z = r e^{i \theta} \), the magnitude is simply \( r \).
- If you have two complex numbers \( z_1 = r_1 e^{i \theta_1} \) and \( z_2 = r_2 e^{i \theta_2} \), their magnitudes are \( r_1 \) and \( r_2 \) respectively.
- Magnitude simplifies multiplication and division to their simplest forms.
This is essential for ensuring that you understand how the length or distance of a complex number affects operations.

complex number multiplication

Multiplying two complex numbers is more straightforward in polar form. Consider \( z_1 = r_1 e^{i \theta_1} \) and \( z_2 = r_2 e^{i \theta_2} \). Their product is:
\[ z_1 z_2 = (r_1 e^{i \theta_1})(r_2 e^{i \theta_2}) = r_1 r_2 e^{i (\theta_1 + \theta_2)} \].
The magnitude of the product is \( |z_1 z_2| = r_1 r_2 \). This shows that the absolute value of a product is the product of the absolute values. This relationship is useful as it makes checking work or solving equations easier. Remember, when multiplying complex numbers, you:

• Multiply their magnitudes.
• Add their angles.

complex number division

Dividing one complex number by another is also simplified using polar form. Take \( z_1 = r_1 e^{i \theta_1} \) and \( z_2 = r_2 e^{i \theta_2} \). The quotient is:
\[ \frac{z_1}{z_2} = \frac{r_1 e^{i \theta_1}}{r_2 e^{i \theta_2}} = \frac{r_1}{r_2} e^{i (\theta_1 - \theta_2)} \].
The magnitude of the quotient is \( \left| \frac{z_1}{z_2} \right| = \frac{r_1}{r_2} \). This confirms that the absolute value of a quotient is the quotient of the absolute values. During division, you:

• Divide their magnitudes.
• Subtract their angles.

Understanding these steps makes handling complex arithmetic much more manageable.