1. 

What property is primarily used when reordering terms as in (1+i)+(3−i)=(3−i)+(1+i)?

a.

Commutative Property

b.

Associative Property

c.

Distributive Property

d.

Absorptive Property

2. 

Calculate the sum of 2+3i, 1−3i, and −3+0i.

a.

0 + i

b.

5i

c.

0 + 0i

d.

−1+0i

3. 

When adding the complex numbers 2+3i and 1−2i, what is the sum?

a.

3+5i

b.

3+i

c.

1+i

d.

3−i

4. 

What is the real part of the complex number 12i?

a.

12

b.

0

c.

-12

d.

12i

5. 

Subtract (7+8i) from (10+10i).

a.

17 + 18i

b.

-3 + 2i

c.

3 + 2i

d.

3 - 2i

6. 

What is the imaginary part of the complex number 10−6i?

a.

6

b.

-6i

c.

10

d.

-10

7. 

What is the standard form of the complex number represented by the point (-5, 8) in the Cartesian plane?

a.

-5 + 8i

b.

8 - 5i

c.

-8 - 5i

d.

-8 + 5i

8. 

What happens when (14+4i)−(7+2i) is calculated?

a.

-14

b.

14 + 4i

c.

7 + 2i

d.

7

9. 

Which of the following is a real number?

a.

3

b.

2i

c.

3+3i

d.

i

10. 

What is (5+5i)−(3+3i)−(2+i)?

a.

2

b.

0 + i

c.

10 + 10i

d.

5 + 5i

11. 

Which complex number has a real part of 15?

a.

15i

b.

4+10i

c.

15−7i

d.

−15

12. 

Subtract (8+8i)−(4+4i).

a.

4 + 4i

b.

8 + 8i

c.

12 + 12i

d.

4

13. 

Which of the following represents a correct application of the distributive property with complex numbers?

a.

(2+3i)(4+5i)=2(4+5i)+3i(4+5i)

b.

(2+3i)+(4+5i)=2+3+4+5i

c.

(2+3i)+(4+5i)=(2+4)+(3+5)i

d.

(2+3i)−(4+5i)=2−4+3i−5i

14. 

Subtract (2+3i) from (5+5i).

a.

-3 + 2i

b.

3 + 2i

c.

7 + 8i

d.

-3 - 2i

15. 

Which of the following represents the Cartesian form of the complex number −8−7i?

a.

(-8, 7)

b.

(-7, -8)

c.

(-7, 8)

d.

(-8, -7)

Here are the correct answers to the questions:

1. **Commutative Property**  
   - **Answer:** a. Commutative Property  
     The commutative property states that the order of addition does not affect the sum, which is used in the reordering of terms.

2. **Calculate the sum:**  
   - **Expression:** \( (2+3i) + (1-3i) + (-3+0i) \)  
   - **Calculation:** \( 2 + 1 - 3 + (3i - 3i + 0i) = 0 + 0i \)  
   - **Answer:** c. 0 + 0i

3. **Adding the complex numbers:**  
   - **Expression:** \( (2+3i) + (1-2i) \)  
   - **Calculation:** \( 2 + 1 + (3i - 2i) = 3 + i \)  
   - **Answer:** b. 3 + i

4. **Real part of the complex number 12i:**  
   - **Answer:** b. 0  
     The real part of \( 12i \) is 0, as it only has an imaginary component.

5. **Subtracting:**  
   - **Expression:** \( (10+10i) - (7+8i) \)  
   - **Calculation:** \( 10 - 7 + (10i - 8i) = 3 + 2i \)  
   - **Answer:** c. 3 + 2i

6. **Imaginary part of 10 - 6i:**  
   - **Answer:** b. -6i  
     The imaginary part is -6i.

7. **Standard form of the complex number (-5, 8):**  
   - **Answer:** a. -5 + 8i  
     The standard form is \( -5 + 8i \).

8. **Subtracting:**  
   - **Expression:** \( (14+4i) - (7+2i) \)  
   - **Calculation:** \( 14 - 7 + (4i - 2i) = 7 + 2i \)  
   - **Answer:** c. 7 + 2i

9. **Real number among the options:**  
   - **Answer:** a. 3  
     3 is the only real number; the others are either purely imaginary or complex.

10. **Subtracting:**  
   - **Expression:** \( (5+5i) - (3+3i) - (2+i) \)  
   - **Calculation:** \( 5 - 3 - 2 + (5i - 3i - i) = 0 + i \)  
   - **Answer:** b. 0 + i

11. **Complex number with a real part of 15:**  
   - **Answer:** c. 15−7i  
     The real part is 15.

12. **Subtracting:**  
   - **Expression:** \( (8+8i) - (4+4i) \)  
   - **Calculation:** \( 8 - 4 + (8i - 4i) = 4 + 4i \)  
   - **Answer:** a. 4 + 4i

13. **Distributive property:**  
   - **Answer:** a. \( (2+3i)(4+5i) = 2(4+5i) + 3i(4+5i) \)  
     This is the correct application of the distributive property with complex numbers.

14. **Subtracting:**  
   - **Expression:** \( (5+5i) - (2+3i) \)  
   - **Calculation:** \( 5 - 2 + (5i - 3i) = 3 + 2i \)  
   - **Answer:** b. 3 + 2i

15. **Cartesian form of the complex number -8-7i:**  
   - **Answer:** d. (-8, -7)  
     The Cartesian coordinates correspond to the real and imaginary parts.

Would you like more details on any of these questions?

**Relative Questions:**

1. How does the associative property differ from the commutative property in complex number operations?
2. What is the significance of the imaginary unit \(i\) in complex number arithmetic?
3. How can the geometric representation of complex numbers on the Cartesian plane aid in understanding their operations?
4. In what scenarios would the distributive property be crucial when dealing with complex numbers?
5. What is the process to convert polar form to Cartesian form for complex numbers?

**Tip:** Remember that the real part of a complex number is always the coefficient of the real unit, while the imaginary part is the coefficient of \(i\).

Explore complex number operations with detailed solutions to problems involving addition, subtraction, and properties such as commutative and distributive. Suitable for high school students studying algebra or pre-calculus.

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