Question: If z = re^(iθ), what is the value of r if z = 1 + i?
Options:
√2
1
2
0
Correct Answer: √2
Solution:
r = |z| = √(1^2 + 1^2) = √2.
If z = re^(iθ), what is the value of r if z = 1 + i?
Practice Questions
Q1
If z = re^(iθ), what is the value of r if z = 1 + i?
√2
1
2
0
Questions & Step-by-Step Solutions
If z = re^(iθ), what is the value of r if z = 1 + i?
Correct Answer: √2
Step 1: Identify the complex number z. Here, z = 1 + i.
Step 2: Recall that in the form z = re^(iθ), r represents the magnitude (or absolute value) of z.
Step 3: To find r, we need to calculate the magnitude of z, which is given by the formula |z| = √(a^2 + b^2), where a is the real part and b is the imaginary part.
Step 4: In our case, the real part a = 1 and the imaginary part b = 1.
Step 5: Substitute the values into the formula: |z| = √(1^2 + 1^2).
Step 6: Calculate 1^2, which is 1, and then add it to the other 1^2, which is also 1. So, we have 1 + 1 = 2.
Step 7: Now, take the square root of 2: r = √2.
Complex Numbers – Understanding the representation of complex numbers in polar form and calculating their magnitude.
Magnitude of Complex Numbers – Calculating the modulus (magnitude) of a complex number using the formula |z| = √(a^2 + b^2) for z = a + bi.
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