To express $|z|$ in terms of $|w|$ for the equation $z + w = |z|w$, where $z$ and $w$ are complex numbers and $w > 1$, we use the Euler form of complex numbers.

Step 1: Express $z$ and $w$ in Euler form

Let: $z = |z|e^{i\theta_z}, \quad w = |w|e^{i\theta_w}.$

Step 2: Substitute into the equation

Substituting $z$ and $w$ into $z + w = |z|w$, we get: $|z|e^{i\theta_z} + |w|e^{i\theta_w} = |z||w|e^{i\theta_w}.$

Step 3: Separate real and imaginary parts

Let: $|z| = r, \quad |w| = R, \quad \text{and align the phases: } \theta_z = \theta_w.$

This simplifies the equation to: $r + Re^{i\theta_w} = rRe^{i\theta_w}.$

Now, the magnitudes of both sides must equal: $r + R = rR.$

Step 4: Solve for $|z|$ ($r$) in terms of $|w|$ ($R$)

Rearrange to isolate $r$: $rR - r = R,$ $r(R - 1) = R,$ $r = \frac{R}{R - 1}.$

Final Expression

The magnitude of $z$ is: $|z| = \frac{|w|}{|w| - 1}.$

Key Points:

1. The Euler form simplifies the representation of complex numbers but the phase cancels due to alignment.
2. The resulting magnitude relationship is purely scalar.

Would you like further details or a deeper explanation on any step?

Related Questions:

1. How does Euler's formula simplify the computation for complex numbers?
2. Can this relationship hold for any value of $|w|$, even if $w \leq 1$?
3. How would the solution change if $\theta_z \neq \theta_w$?
4. Can you graphically represent $|z|$ and $|w|$ on the complex plane?
5. What are other practical applications of this equation in physics or engineering?

Tip: Always verify the alignment of phases ($\theta_z = \theta_w$) when simplifying equations involving complex numbers in Euler form.