Complex Numbers

We are asked to find the argument of the complex number:

Step 1: Multiply the numerator First, multiply the numerator: Thus, the numerator becomes .

Step 2: Multiply by the conjugate of the denominator Next, multiply both the numerator and the denominator by the conjugate of , which is : Step 3: Multiply the numerator Now, multiply the numerator: Thus, the expression simplifies to: Step 4: Find the argument The complex number is , which lies on the imaginary axis with an argument of . Thus, the correct answer is .

Given the complex number , we want to compute and find the ordered pair .

Firstly, identify in polar form. The modulus of is calculated as:

The argument is determined by:

which gives . Therefore, the polar form is:

Using De Moivre’s Theorem, is:

Now, compute the powers:

Next, simplify the angle:

Find the equivalent angle within by subtracting (since ):

The angle simplifies to:

Thus,

Substitute and :

Thus, and , giving .

The ordered pair is .

• could be any of .
• follows in this order.
• follows , and so on.

• If , then the sum is .
• If , the sum is .
• If , the sum is .
• If , the sum is .