To prove that $\overline{z} = \frac{1}{z}$ and $\overline{w} = \frac{1}{w}$, let's break the problem down step by step for $z$. The same reasoning will apply for $w$.

Step 1: Use the property $|z| = 1$

Given that $|z| = 1$, recall that the modulus of a complex number $z = a + bi$, where $a$ and $b$ are real, is defined as:

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

However, since $|z| = 1$, this implies:

$\sqrt{a^2 + b^2} = 1 \quad \Rightarrow \quad a^2 + b^2 = 1$

This tells us that the complex number $z$ lies on the unit circle in the complex plane.

Step 2: Conjugate of $z$

The complex conjugate of $z = a + bi$ is:

$\overline{z} = a - bi$

Step 3: Inverse of $z$

The inverse of $z = a + bi$ is given by:

$\frac{1}{z} = \frac{1}{a + bi}$

To simplify $\frac{1}{a + bi}$, we multiply both the numerator and the denominator by the complex conjugate of $a + bi$, which is $a - bi$. This yields:

$\frac{1}{z} = \frac{a - bi}{(a + bi)(a - bi)} = \frac{a - bi}{a^2 + b^2}$

Since we know $a^2 + b^2 = 1$ (from Step 1), the expression simplifies to:

$\frac{1}{z} = a - bi = \overline{z}$

Thus, we have shown that:

$\overline{z} = \frac{1}{z}$

Step 4: Apply the same reasoning to $w$

Since $|w| = 1$, we can follow exactly the same steps as above to show that:

$\overline{w} = \frac{1}{w}$

Conclusion

We have proven that:

$\overline{z} = \frac{1}{z} \quad \text{and} \quad \overline{w} = \frac{1}{w}$

Would you like further clarification or details on any part?

Here are some related questions:

1. How does the unit circle help in understanding complex numbers?
2. Can you show a general proof for other moduli $|z| \neq 1$?
3. How would this change if $z$ and $w$ were not on the unit circle?
4. What other properties of complex numbers can be derived from $|z| = 1$?
5. How can this proof be extended to other forms of $z$, such as polar form?

Tip: When dealing with complex numbers on the unit circle, it's often helpful to use their polar form, $z = e^{i\theta}$, where computations involving modulus and conjugates become more straightforward.