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If z = 1 + i, what is the argument of z?

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Question: If z = 1 + i, what is the argument of z?

Options:

  1. π/4
  2. π/2
  3. 0
  4. π

Correct Answer: π/4

Solution: 

The argument of a complex number z = a + bi is given by θ = tan⁻¹(b/a). Here, θ = tan⁻¹(1/1) = π/4.

If z = 1 + i, what is the argument of z?

Practice Questions

Q1
If z = 1 + i, what is the argument of z?
  1. π/4
  2. π/2
  3. 0
  4. π

Questions & Step-by-Step Solutions

If z = 1 + i, what is the argument of z?
  • Step 1: Identify the complex number z. Here, z = 1 + i.
  • Step 2: Recognize that in the complex number z = a + bi, 'a' is the real part and 'b' is the imaginary part. For z = 1 + i, a = 1 and b = 1.
  • Step 3: Use the formula for the argument of a complex number, which is θ = tan⁻¹(b/a).
  • Step 4: Substitute the values of a and b into the formula: θ = tan⁻¹(1/1).
  • Step 5: Calculate tan⁻¹(1/1), which equals tan⁻¹(1).
  • Step 6: Recognize that tan⁻¹(1) corresponds to an angle of π/4 radians.
  • 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 (or angle) of a complex number is calculated using the arctangent function, specifically θ = tan⁻¹(b/a).
  • Quadrants of the Complex Plane – Recognizing which quadrant the complex number lies in to determine the correct angle for the argument.
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