If P and Q are two complex numbers, then the modulus of the quotient of P and Q is

Options

A. Greater than the quotient of their moduli
B. Less than the quotient of their moduli
C. Less than or equal to the quotient of their moduli
D. Equal to the quotient of their moduli

Answer

To find the modulus of the quotient of two complex numbers \( P \) and \( Q \), we can follow these steps:

### Step-by-Step Solution:

1. **Define the Complex Numbers**:
Let \( P \) and \( Q \) be two complex numbers. We can express them in polar form:
\[ P = |P| e^{i \theta_1} \]
\[ Q = |Q| e^{i \theta_2} \]
where \( |P| \) and \( |Q| \) are the moduli (magnitudes) of \( P \) and \( Q \), and \( \theta_1 \) and \( \theta_2 \) are their respective arguments (angles).

2. **Calculate the Quotient**:
The quotient of the two complex numbers \( P \) and \( Q \) can be expressed as:
\[ \frac{P}{Q} = \frac{|P| e^{i \theta_1}}{|Q| e^{i \theta_2}} = \frac{|P|}{|Q|} e^{i (\theta_1 - \theta_2)} \]

3. **Find the Modulus of the Quotient**:
The modulus of the quotient \( \frac{P}{Q} \) is given by:
\[ \left| \frac{P}{Q} \right| = \left| \frac{|P|}{|Q|} e^{i (\theta_1 - \theta_2)} \right| \]
Since the modulus of a product is the product of the moduli, we have:
\[ \left| \frac{P}{Q} \right| = \frac{|P|}{|Q|} \cdot |e^{i (\theta_1 - \theta_2)}| \]

4. **Simplify the Expression**:
The modulus of \( e^{i (\theta_1 - \theta_2)} \) is 1 (since it lies on the unit circle), thus:
\[ \left| \frac{P}{Q} \right| = \frac{|P|}{|Q|} \]

5. **Conclusion**:
Therefore, the modulus of the quotient of the complex numbers \( P \) and \( Q \) is:
\[ \left| \frac{P}{Q} \right| = \frac{|P|}{|Q|} \]

### Final Answer:
The modulus of the quotient of two complex numbers \( P \) and \( Q \) is equal to the quotient of their moduli:
\[ \left| \frac{P}{Q} \right| = \frac{|P|}{|Q|} \]