We start with the equation:

$z + w = |z|w$

Here, $z$ and $w$ are complex numbers, and $|z|$ represents the modulus of $z$, while $|w|$ represents the modulus of $w$.

Step 1: Expressing $z$

Let $z = a + bi$, where $a, b \in \mathbb{R}$. The modulus of $z$ is given by:

$|z| = \sqrt{a^2 + b^2}.$

Similarly, let $w = c + di$, where $c, d \in \mathbb{R}$, and $|w| = \sqrt{c^2 + d^2}$.

Step 2: Rewrite the equation

Substitute $z = a + bi$ and $|z| = \sqrt{a^2 + b^2}$ into the original equation:

$(a + bi) + (c + di) = \sqrt{a^2 + b^2}(c + di).$

Separate real and imaginary parts:

$(a + c) + (b + d)i = \sqrt{a^2 + b^2}(c + di).$

Equating real and imaginary parts, we have:

1. $a + c = \sqrt{a^2 + b^2} \cdot c$,
2. $b + d = \sqrt{a^2 + b^2} \cdot d$.

Step 3: Solve for $|z|$

From these equations, we observe that both equations are proportional. Divide the imaginary part equation by the real part equation:

$\frac{b + d}{a + c} = \frac{\sqrt{a^2 + b^2} \cdot d}{\sqrt{a^2 + b^2} \cdot c}.$

This simplifies to:

$\frac{b + d}{a + c} = \frac{d}{c}.$

Returning to the modulus equation, recall $|z|$ depends only on the relationship between $|z|$ and $|w|$. We derive directly by rewriting the given equation:

$|z| = \frac{|z + w|}{|w|}.$

Here, the modulus form simplifies to:

$|z| = |w| - 1.$

Final Answer:

$|z| = |w| - 1.$

Would you like further details or examples?

---

1. How do