To find the modulus (or absolute value) of a complex number, follow these steps:

Steps to Find the Modulus of a Complex Number

1. Write the Complex Number:

   • A complex number is usually written in the form $z = a + bi$, where $a$ is the real part and $b$ is the imaginary part.

2. Apply the Modulus Formula:

   • The modulus of the complex number $z = a + bi$ is given by: $|z| = \sqrt{a^2 + b^2}$
   • This formula is derived from the distance formula in the complex plane, where the modulus represents the distance from the origin (0, 0) to the point $(a, b)$ in the plane.

3. Calculate the Modulus:

   • Square the real part $a$ and the imaginary part $b$.
   • Add the squares.
   • Take the square root of the sum.

Example

Let's find the modulus of the complex number $z = 3 + 4i$.

1. Identify $a$ and $b$:

   • Here, $a = 3$ and $b = 4$.

2. Apply the Modulus Formula: $|z| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

So, the modulus of $z = 3 + 4i$ is 5.

Would you like more details or have any questions? Here are five related questions:

1. How does the modulus of a complex number relate to its graphical representation?
2. What is the modulus of a purely imaginary number, like $0 + bi$?
3. How would you find the modulus of a complex number given in polar form?
4. What is the effect of conjugating a complex number on its modulus?
5. How does the modulus help in dividing complex numbers?

Tip: The modulus is always a non-negative real number and represents the distance from the origin to the point representing the complex number in the complex plane.