If z = 2 + 2i, find the argument of z.

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Question: If z = 2 + 2i, find the argument of z.

Options:

Correct Answer: π/4

Solution:

The argument is given by tan^(-1)(2/2) = tan^(-1)(1) = π/4.

If z = 2 + 2i, find the argument of z.

If z = 2 + 2i, find the argument of z.

Step 1: Identify the complex number z, which is given as z = 2 + 2i.
Step 2: Recognize that the argument of a complex number is the angle it makes with the positive x-axis in the complex plane.
Step 3: The real part of z is 2 and the imaginary part is also 2.
Step 4: Use the formula for the argument, which is given by the tangent of the angle: tan(θ) = imaginary part / real part.
Step 5: Substitute the values: tan(θ) = 2 / 2.
Step 6: Simplify the fraction: tan(θ) = 1.
Step 7: Find the angle whose tangent is 1. This angle is θ = tan^(-1)(1).
Step 8: The angle θ = π/4 radians (or 45 degrees).
Step 9: Therefore, the argument of z is π/4.

Complex Numbers – Understanding the representation of complex numbers in the form z = a + bi, where a is the real part and b is the imaginary part.
Argument of a Complex Number – The argument of a complex number is the angle formed with the positive real axis in the complex plane, typically calculated using the arctangent function.
Polar Coordinates – Converting complex numbers from rectangular form (a + bi) to polar form (r(cos θ + i sin θ)), where r is the modulus and θ is the argument.