To determine whether the given equation $|z_1 \cdot z_2| = |z_1||z_2|$ holds for two complex numbers $z_1, z_2 \in \mathbb{C}$, we will break it down step by step:

Step 1: Definition of Complex Numbers

Let $z_1$ and $z_2$ be two complex numbers. We can express them in their general form: $z_1 = a_1 + b_1i \quad \text{and} \quad z_2 = a_2 + b_2i$ where $a_1, a_2$ are real parts, and $b_1, b_2$ are imaginary parts of the complex numbers.

Step 2: Modulus of a Complex Number

The modulus (or absolute value) of a complex number $z = a + bi$ is defined as: $|z| = \sqrt{a^2 + b^2}$ This means the modulus of $z_1$ and $z_2$ are: $|z_1| = \sqrt{a_1^2 + b_1^2} \quad \text{and} \quad |z_2| = \sqrt{a_2^2 + b_2^2}$

Step 3: Modulus of the Product of Two Complex Numbers

Now consider the product of the two complex numbers: $z_1 \cdot z_2 = (a_1 + b_1i)(a_2 + b_2i)$ Expanding this product: $z_1 \cdot z_2 = a_1a_2 + a_1b_2i + b_1a_2i + b_1b_2i^2$ Since $i^2 = -1$, the expression simplifies to: $z_1 \cdot z_2 = (a_1a_2 - b_1b_2) + (a_1b_2 + b_1a_2)i$ Thus, the real part is $a_1a_2 - b_1b_2$ and the imaginary part is $a_1b_2 + b_1a_2$.

The modulus of the product $z_1 \cdot z_2$ is: $|z_1 \cdot z_2| = \sqrt{(a_1a_2 - b_1b_2)^2 + (a_1b_2 + b_1a_2)^2}$

Step 4: Simplifying the Modulus of the Product

To verify whether $|z_1 \cdot z_2| = |z_1||z_2|$, we need to check if the modulus of the product matches the product of the moduli.

Let's compute $(a_1a_2 - b_1b_2)^2 + (a_1b_2 + b_1a_2)^2$: $(a_1a_2 - b_1b_2)^2 + (a_1b_2 + b_1a_2)^2 = a_1^2a_2^2 - 2a_1a_2b_1b_2 + b_1^2b_2^2 + a_1^2b_2^2 + 2a_1b_2b_1a_2 + b_1^2a_2^2$ Simplifying further, we see that the cross terms cancel out: $a_1^2a_2^2 + b_1^2b_2^2 + a_1^2b_2^2 + b_1^2a_2^2 = (a_1^2 + b_1^2)(a_2^2 + b_2^2)$ This simplifies to: $|z_1 \cdot z_2| = \sqrt{(a_1^2 + b_1^2)(a_2^2 + b_2^2)} = |z_1||z_2|$

Step 5: Conclusion

Hence, we have shown that the modulus of the product of two complex numbers is equal to the product of their moduli: $|z_1 \cdot z_2| = |z_1||z_2|$ This equation holds true for any two complex numbers $z_1$ and $z_2$.

Would you like more details on any specific step or additional examples?

Follow-up Questions:

1. How is the modulus of a complex number related to its geometric interpretation in the complex plane?
2. Can the same property hold for sums of complex numbers, i.e., $|z_1 + z_2| = |z_1| + |z_2|$?
3. How does the formula for the modulus of the product of complex numbers change if one or both numbers are purely imaginary?
4. What is the geometric interpretation of the product of two complex numbers in terms of their moduli and arguments?
5. How can the property $|z_1 \cdot z_2| = |z_1||z_2|$ be used to simplify solving complex equations?

Tip:

When working with complex numbers, always remember that their modulus behaves similarly to absolute values for real numbers, which can often help in solving problems involving distances or magnitudes in the complex plane.