If z = re^(iθ), what is the value of |z|?

1 question

1 item

Q: If z = re^(iθ), what is the value of |z|?

Solution: The modulus |z| = r, as |re^(iθ)| = r.

Steps: 7

Step 1: Understand that z is given in the form z = re^(iθ), where r is a real number and θ is an angle in radians.
Step 2: Recognize that |z| represents the modulus (or absolute value) of the complex number z.
Step 3: Recall the property of the modulus of a complex number in exponential form: |re^(iθ)| = |r| * |e^(iθ)|.
Step 4: Since r is a real number, |r| = r (assuming r is non-negative).
Step 5: The modulus of e^(iθ) is 1, because it represents a point on the unit circle in the complex plane.
Step 6: Combine the results: |z| = |r| * |e^(iθ)| = r * 1 = r.
Step 7: Conclude that the value of |z| is simply r.