If z = re^(iθ), what is the value of r if z = 3 + 4i?

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Question: If z = re^(iθ), what is the value of r if z = 3 + 4i?

Options:

Correct Answer: 5

Solution:

r = |z| = √(3^2 + 4^2) = √25 = 5.

If z = re^(iθ), what is the value of r if z = 3 + 4i?

If z = re^(iθ), what is the value of r if z = 3 + 4i?

Step 1: Identify the complex number z, which is given as z = 3 + 4i.
Step 2: Recall that in the expression z = re^(iθ), r represents the magnitude (or absolute value) of the complex number z.
Step 3: To find the magnitude r, use the formula r = |z| = √(a^2 + b^2), where a is the real part and b is the imaginary part of z.
Step 4: In our case, a = 3 and b = 4.
Step 5: Substitute the values into the formula: r = √(3^2 + 4^2).
Step 6: Calculate 3^2, which is 9, and 4^2, which is 16.
Step 7: Add the results: 9 + 16 = 25.
Step 8: Take the square root of 25: √25 = 5.
Step 9: Therefore, the value of r is 5.

Complex Numbers – Understanding the representation of complex numbers in polar form and the relationship between rectangular and polar coordinates.
Magnitude of Complex Numbers – Calculating the magnitude (or modulus) of a complex number using the formula |z| = √(a^2 + b^2) for z = a + bi.