Question: If z = re^(iθ), what is the value of |z|?
Options:
r
θ
re
1
Correct Answer: r
Solution:
The modulus |z| = r, as |re^(iθ)| = r.
If z = re^(iθ), what is the value of |z|?
Practice Questions
Q1
If z = re^(iθ), what is the value of |z|?
r
θ
re
1
Questions & Step-by-Step Solutions
If z = re^(iθ), what is the value of |z|?
Correct Answer: r
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.
Complex Numbers – Understanding the representation of complex numbers in polar form, where z = re^(iθ) consists of a modulus r and an argument θ.
Modulus of a Complex Number – The modulus |z| of a complex number is the distance from the origin in the complex plane, which is given by the absolute value of the coefficient r in the polar form.
Soulshift Feedback×
On a scale of 0–10, how likely are you to recommend
The Soulshift Academy?
