If z = 2 + 2i, find the value of z/z*.

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Question: If z = 2 + 2i, find the value of z/z*.

Options:

Correct Answer: i

Solution:

z/z* = (2 + 2i)/(2 - 2i) = (2 + 2i)(2 + 2i)/(4 + 4) = (4 + 8i - 4)/(8) = i.

If z = 2 + 2i, find the value of z/z*.

If z = 2 + 2i, find the value of z/z*.

Correct Answer: i

Step 1: Identify the complex number z. Here, z = 2 + 2i.
Step 2: Find the conjugate of z, denoted as z*. The conjugate of z = 2 + 2i is z* = 2 - 2i.
Step 3: Set up the expression z/z*. This means we will divide z by its conjugate: (2 + 2i) / (2 - 2i).
Step 4: Multiply the numerator and the denominator by the conjugate of the denominator to simplify. This gives us: (2 + 2i)(2 + 2i) / (2 - 2i)(2 + 2i).
Step 5: Calculate the denominator: (2 - 2i)(2 + 2i) = 2^2 - (2i)^2 = 4 - (-4) = 4 + 4 = 8.
Step 6: Calculate the numerator: (2 + 2i)(2 + 2i) = 2^2 + 2*2*2i + (2i)^2 = 4 + 8i - 4 = 8i.
Step 7: Now we have (8i) / (8).
Step 8: Simplify the fraction: 8i / 8 = i.

Complex Numbers – Understanding the operations involving complex numbers, including division and conjugates.
Conjugate of a Complex Number – Using the conjugate to simplify the division of complex numbers.
Polar Form of Complex Numbers – Recognizing that the division of complex numbers can also relate to their polar forms.