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### Exercise 1
Let \( z_1 \) and \( z_2 \) be two complex numbers. Prove that:
\[ |z_1 + z_2| \leq |z_1| + |z_2| \]

This is known as the triangle inequality for complex numbers. 

### Exercise 2
Let \( z_1 \) and \( z_2 \) be two non-zero complex numbers. Prove that:
\[ \left| \frac{z_1}{z_2} \right| = \frac{|z_1|}{|z_2|} \]

### Exercise 3
Prove that for a complex number \( z \):
\[ |z|^2 = z \overline{z} \]
where \( \overline{z} \) is the complex conjugate of \( z \).

### Exercise 4
Let \( z = re^{i\theta} \) where \( r = |z| \) and \( \theta = \arg(z) \). Prove that:
\[ |e^{i\theta}| = 1 \]

### Exercise 5
Let \( z_1 \) and \( z_2 \) be complex numbers such that \( |z_1| = |z_2| \). Prove that the points representing \( z_1 \) and \( z_2 \) on the Argand diagram lie on a circle of radius \( |z_1| \) centered at the origin.

### Exercise 6
If \( z \) is a complex number such that \( \text{Re}(z) = 0 \), show that \( z \) is purely imaginary.

### Exercise 7
If \( z \) is a complex number and \( |z| = 1 \), show that:
\[ \overline{z} = \frac{1}{z} \]

### Exercise 8
Let \( z \) be a variable complex number such that \( |z - 4i| \leq 3 \). Show that the locus of \( z \) in the Argand diagram lies inside or on a circle centered at \( 4i \) with a radius of 3.

### Exercise 9
Show that \( \left| |z_1| - |z_2| \right| \leq |z_1 - z_2| \).

### Exercise 10
If \( z \) is a non-zero complex number such that \( z + \frac{1}{z} \) is real, prove that \( z \) lies on the unit circle.

### Exercise 11
Find the greatest and least values of \( |z| + \left| \frac{

Explore exercises on complex numbers covering topics such as the triangle inequality, complex conjugates, Argand diagram representations, and more. Learn how to prove various properties of complex numbers and their geometrical representations.

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