Master Complex Number Operations: A Comprehensive Guide for Beginners

Question 1

Find the square of the complex number (3 + 2i).

Accepted Answer

13 + 4i

Answer

9 - 4i

Answer

9 + 4i

Answer

13 - 4i

Question 2

Find the cube root of the complex number 8i.

Accepted Answer

2i

Answer

-2i

Answer

2i³

Answer

-2i³

Question 3

Which of the following is NOT a property of complex numbers?

Accepted Answer

Closure property under division

Answer

Commutative property of addition

Answer

Associative property of multiplication

Answer

Distributive property

Question 4

How do complex numbers relate to real numbers in terms of their properties and the imaginary unit i?

Accepted Answer

Complex numbers extend real numbers by adding the imaginary unit i, while sharing many properties.

Answer

Complex numbers are identical to real numbers and have the same properties.

Answer

Complex numbers and real numbers have completely different properties.

Question 5

Find the sum of the complex numbers (3 + 4i) and (5 - 2i).

Accepted Answer

8 + 2i

Answer

8 - 2i

Answer

2 + 6i

Answer

-2 + 6i

Question 6

Which of the following is the real part of the complex number (7 + 3i)?

Accepted Answer

7

Answer

3

Answer

-3

Answer

-7

Question 7

Prove that the product of two complex numbers, $z_1$ and $z_2$, is equal to the product of their moduli, $|z_1||z_2|$, and the product of their arguments, $arg(z_1) + arg(z_2)$.

Accepted Answer

Let $z_1 = r_1(cos(	heta_1) + isin(	heta_1))$ and $z_2 = r_2(cos(	heta_2) + isin(	heta_2))$. Then, $z_1z_2 = r_1r_2(cos(	heta_1 + 	heta_2) + isin(	heta_1 + 	heta_2))$. Therefore, $|z_1z_2| = |r_1r_2|$ and $arg(z_1z_2) = 	heta_1 + 	heta_2$.

Answer

The product of two complex numbers is always a complex number.

Answer

The product of two complex numbers is equal to the sum of their moduli.

Answer

The product of two complex numbers is equal to the product of their real parts.

Question 8

How can the complex roots of unity of order n be used to construct a regular polygon with n sides?

Accepted Answer

Plot the roots on the complex plane and connect them with line segments.

Answer

Find the sum of the roots and divide it by n.

Answer

Multiply the roots together and take the nth root.

Answer

Subtract the roots from each other and find the absolute value.

Question 9

Divide (6 + 4i) by (2 - i).

Accepted Answer

4 + 2i

Answer

4 - 2i

Answer

2 - 4i

Answer

2 + 4i

Question 10

Calculate the square of (3 + 2i).

Accepted Answer

13

Answer

11

Answer

15

Answer

17

Question 11

Find the complex conjugate of (1 + 2i).

Accepted Answer

1 - 2i

Answer

-1 - 2i

Answer

1 + 2i

Answer

-1 + 2i

Question 12

Calculate the square of the complex number (2 - 3i).

Accepted Answer

13 - 12i

Answer

13 + 12i

Answer

-13 - 12i

Answer

-13 + 12i

Question 13

Simplify the following expression: (2 + 3i)² - (2 - 3i)². Provide the result in the form ai + bj, where a and b are real numbers.

Accepted Answer

24i

Answer

12i

Answer

-24i

Answer

-12i

Question 14

Obtain the conjugate of the complex number (3 - 4i).

Accepted Answer

3 + 4i

Answer

-3 + 4i

Answer

3 - 4i

Answer

-3 - 4i

Question 15

Determine the absolute value of the complex number (-5 + 12i).

Accepted Answer

13

Answer

5

Answer

-5

Answer

-13

Question 16

Convert the complex number (3 + 4i) into its polar form and express it as r(cos θ + i sin θ), where r is the modulus and θ is the argument in degrees.

Accepted Answer

5(cos 53.1° + i sin 53.1°)

Answer

-5(cos 53.1° + i sin 53.1°)

Answer

5(cos 36.9° + i sin 36.9°)

Answer

-5(cos 36.9° + i sin 36.9°)

Question 17

Simplify the given expression: (5i)² - (2 - 3i)². Express the result in the form ai + bj, where a and b are real numbers.

Accepted Answer

-29 - 34i

Answer

29 + 34i

Answer

29 - 34i

Answer

-29 + 34i

Question 18

Find the sum of the complex numbers z₁ = 3 + 2i and z₂ = 4 - 3i.

Accepted Answer

7 - i

Answer

-7 - i

Answer

-7 + i

Answer

7 + i

Question 19

Subtract the complex number z₂ = 4 - 3i from z₁ = 3 + 2i.

Accepted Answer

-1 + 5i

Answer

1 + 5i

Answer

-1 - 5i

Answer

1 - 5i