The image shows a question in Hebrew, which translates to:

"Given complex numbers $z_1, z_2$.

Is the following equality correct: $|z_1| + |z_2| = |z_1 + z_2|$? Explain."

To address this, let's analyze it using the polar representation of complex numbers.

Polar Representation of Complex Numbers

A complex number $z$ can be represented in polar form as $z = r e^{i\theta}$, where:

• $r$ is the magnitude (or modulus) of $z$, denoted as $|z|$,
• $\theta$ is the argument (or angle) of $z$.

Triangle Inequality for Complex Numbers

The modulus of the sum of two complex numbers $z_1$ and $z_2$ is given by: $|z_1 + z_2| \leq |z_1| + |z_2|$

Equality holds if and only if $z_1$ and $z_2$ are in the same direction (i.e., $z_1$ and $z_2$ are linearly dependent, which means they have the same argument or one is a scalar multiple of the other with a positive real scalar).

Explanation

1. When $z_1$ and $z_2$ have the same direction:

   • In this case, $z_1$ and $z_2$ are collinear in the complex plane.
   • Therefore, $|z_1 + z_2| = |z_1| + |z_2|$.

2. When $z_1$ and $z_2$ do not have the same direction:

   • In general, $|z_1 + z_2| < |z_1| + |z_2|$.
   • This is because the vector sum of $z_1$ and $z_2$ will have a resultant magnitude that is less than the sum of their magnitudes, except when they point in the same direction.

Thus, the given equality $|z_1| + |z_2| = |z_1 + z_2|$ is not always correct. It is only true if $z_1$ and $z_2$ are collinear (i.e., they have the same or exactly opposite argument).

Would you like more details or have any other questions?

Related Questions

1. What is the geometric interpretation of the modulus of a complex number?
2. How does the argument of a complex number affect its position in the complex plane?
3. What is the triangle inequality in the context of complex numbers?
4. Can you provide an example where $|z_1| + |z_2| > |z_1 + z_2|$?
5. What happens to the equality $|z_1 + z_2| = |z_1| + |z_2|$ when $z_1$ and $z_2$ are orthogonal?
6. How does the polar form of a complex number differ from its rectangular form?
7. What are some practical applications of complex numbers in polar form?
8. Can two complex numbers with different magnitudes but the same argument satisfy $|z_1| + |z_2| = |z_1 + z_2|$?

Tip

Always remember that the equality $|z_1 + z_2| = |z_1| + |z_2|$ only holds under specific conditions, typically when the complex numbers point in the same direction in the complex plane.