Math Problem Statement

Represent 7zi into polar form and find the argument of z and the principal value of the argument of 𝑧.

Solution

To represent z=7iz = 7i in polar form and find the argument and the principal value of the argument, we’ll proceed step-by-step.

Step 1: Convert z=7iz = 7i to Polar Form

  1. Identify zz in Cartesian coordinates: Here, z=7iz = 7i, which means it has no real part, so: z=0+7iz = 0 + 7i Therefore, x=0x = 0 (real part) and y=7y = 7 (imaginary part).

  2. Find the modulus z|z|: The modulus z|z| is calculated as: z=x2+y2=02+72=49=7|z| = \sqrt{x^2 + y^2} = \sqrt{0^2 + 7^2} = \sqrt{49} = 7

  3. Calculate the argument arg(z)\arg(z): Since zz lies on the positive imaginary axis, the argument θ\theta is: θ=π2\theta = \frac{\pi}{2}

Thus, the polar form of zz is: z=7(cosπ2+isinπ2)z = 7 \left( \cos \frac{\pi}{2} + i \sin \frac{\pi}{2} \right) or, in exponential form: z=7eiπ2z = 7 e^{i \frac{\pi}{2}}

Step 2: Find the Argument of zz and the Principal Value

  1. Argument of zz: The argument of zz, arg(z)\arg(z), is π2\frac{\pi}{2}.

  2. Principal Value of the Argument: The principal value of the argument, Arg(z)\mathrm{Arg}(z), is the unique value of the argument within (π,π](-\pi, \pi]. Here, the principal argument is also: Arg(z)=π2\mathrm{Arg}(z) = \frac{\pi}{2}

Summary:

  • Polar Form: z=7(cosπ2+isinπ2)z = 7 \left( \cos \frac{\pi}{2} + i \sin \frac{\pi}{2} \right) or z=7eiπ2z = 7 e^{i \frac{\pi}{2}}
  • Argument arg(z)\arg(z): π2\frac{\pi}{2}
  • Principal Argument Arg(z)\mathrm{Arg}(z): π2\frac{\pi}{2}

Would you like further details on any part of this process?

Here are some related questions:

  1. How do you convert complex numbers into polar form?
  2. What is the difference between argument and principal argument?
  3. Why is the modulus of a complex number always positive?
  4. How does the argument of a complex number change with quadrants?
  5. How do you express complex numbers on the Argand plane?

Tip: For complex numbers lying on the imaginary axis, the argument is always ±π2\pm \frac{\pi}{2}, depending on the sign of the imaginary part.

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Math Problem Analysis

Mathematical Concepts

Complex Numbers
Polar Coordinates
Trigonometry

Formulas

Polar form of a complex number: \( z = r(\cos \theta + i \sin \theta) \) or \( z = r e^{i\theta} \)
Modulus of a complex number: \( |z| = \sqrt{x^2 + y^2} \)
Argument of a complex number: \( \theta = \tan^{-1}(\frac{y}{x}) \)

Theorems

Principal Argument of a Complex Number

Suitable Grade Level

Grades 11-12