Solving Complex Numbers Problems Step-by-Step

Published on August 22, 2026

Algebra

Learn how to perform operations with complex numbers including multiplication, finding coordinates, raising to powers, finding conjugates, and converting to polar form.

Question

حله

Answer

Based on the image provided, this is a mathematics test in Arabic focusing on complex numbers. I will translate and solve the visible problems step-by-step.

Problem 1: Complex Number Multiplication

Question: Find the complex number

resulting from the product:

.
(Note: In the text,

is represented by the Arabic letter ‘ت’).

Solution:
Using the distributive property (FOIL method):

Since

Problem 2: Finding a Complex Number from its Coordinates

Question: If the coordinates of

are

and

, find

and

.
(The text says: "If coordinates

and

, then

Solution:
A complex number is written as

, where

is the real part and

is the imaginary part.
Given

The real part
The imaginary part

Problem 3: Power of a Complex Number

Question: Find the value of the expression

Solution:
First, let’s find

Now, we can rewrite

as:

Since

Problem 4: Finding the Conjugate

Question: If

, then find

(the conjugate).
(The text says: "If

, then

Solution:
The conjugate of a complex number

.
If

, then

Problem 5: Polar Form (Question 7)

Question: The complex number

. Write it in polar form

Solution:

Find the modulus :
Find the argument :
Since both real and imaginary parts are negative, is in the 3rd quadrant.In the 3rd quadrant: or .
Answer: or .