To find the modulus of a complex number, follow these simple steps:

1. What is a Complex Number?

A complex number looks like this: ( z = a + bi )

Here:

• ( a ) is the real part.
• ( b ) is the imaginary part.
• ( i ) is a special number that helps with imaginary parts. It means that ( i^2 = -1 ).

2. How to Find the Modulus

The modulus, or size, of a complex number can be found using this formula: ( |z| = \sqrt{a^2 + b^2} )

3. Steps to Calculate the Modulus

Let's break it down:

• Find the Real and Imaginary Parts: From the number ( z = a + bi ), identify ( a ) (the real part) and ( b ) (the imaginary part).

• Square the Real Part: Take ( a ) and multiply it by itself to get ( a^2 ).

• Square the Imaginary Part: Do the same for ( b ) to get ( b^2 ).

• Add the Squares: Now, add together the squares you calculated: ( a^2 + b^2 ).

• Take the Square Root: Finally, find the square root of that sum. This gives you the modulus: ( |z| = \sqrt{a^2 + b^2} ).

4. Example

Let’s look at an example with the complex number ( z = 3 + 4i ):

• Here, ( a = 3 ) and ( b = 4 ).

• Calculate ( |z| ):

  • First, square the real part: ( a^2 = 3^2 = 9 ).
  • Next, square the imaginary part: ( b^2 = 4^2 = 16 ).
  • Now, add those two results: ( a^2 + b^2 = 9 + 16 = 25 ).
  • Finally, take the square root: ( |z| = \sqrt{25} = 5 ).

So, the modulus of the complex number ( 3 + 4i ) is 5.

5. What Does This Mean Geometrically?

The modulus shows us how far the point ( (a, b) ) is from the starting point ( (0, 0) ) on a graph. It connects the math we do with complex numbers to the shapes we see in geometry.