Complex Modulus: Step-by-Step Guide
The modulus (or absolute value) of a complex number z = a + bi is its distance from the origin on the complex plane. It is written |z|. You find it with the Pythagorean theorem: |z| = the square root of (a^2 + b^2).
Math Racer skill guide · Algebra 2 · Grade 11
How to solve it, step by step
- Find the real and imaginary parts. Write z as a + bi. The number a is the real part, and b is the imaginary part.\[z = a + bi\]
- Square each part. Work out a^2 and b^2. Both are positive or zero, even if a or b is negative.\[a^{2},\; b^{2}\]
- Add and take the square root. Add a^2 + b^2 and take the positive square root.\[|z| = \sqrt{a^{2}+b^{2}}\]
- Simplify the root. Pull out any perfect squares to write the answer in simplest form.\[|z| = \sqrt{a^{2}+b^{2}}\;\text{(simplified)}\]
Worked example
Find |z| for z = -3 + 4i.
Step 1: a = -3 and b = 4.
Step 2: a^2 = (-3)^2 = 9, and b^2 = 4^2 = 16.
Step 3: Add: 9 + 16 = 25. The square root of 25 is 5.
Step 4: |z| = 5.
Check: The point (-3, 4) is 5 units from the origin. It is the corner of a 3-4-5 right triangle.
Tips and common mistakes
- The modulus is never negative, even if a or b is negative.
- |a + bi| and |a - bi| are the same. A complex number and its conjugate are the same distance from the origin.
- |z|^2 = z times its conjugate = a^2 + b^2, which has no i in it.
- For a number with no imaginary part, |a| is the usual absolute value. For a number with no real part, |bi| = |b|.
- |zw| = |z| times |w|, and |z/w| = |z| / |w|.
- Plot z on the complex plane. The modulus is the hypotenuse of a right triangle with legs a and b.
- Simplify the final root. For example, the square root of 50 is 5 times the square root of 2.