What are the rectangular forms of the complex numbers listed in these exercises?

The rectangular form of a complex number is expressed as \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit with the property \(i^2 = -1\). 

In the exercises listed in the PDF, all the complex numbers are already given in rectangular form. Here’s a breakdown of the rectangular forms provided:

### Rectangular Forms of Complex Numbers from the Exercises:

1. **Addition**:
   - \( (7 + 3i) + (2 + 5i) \)
     - Both numbers are already in rectangular form: \(7 + 3i\) and \(2 + 5i\).

2. **Subtraction**:
   - \( (-2 + 3i) - (3 + 5i) \)
     - Both numbers are in rectangular form: \(-2 + 3i\) and \(3 + 5i\).
   - \( (4 + i) - (1 + 3i) \)
     - Both numbers are in rectangular form: \(4 + i\) and \(1 + 3i\).
   - \( (6 + 3i) - (2 - 4i) \)
     - Both numbers are in rectangular form: \(6 + 3i\) and \(2 - 4i\).
   - \( (7 - 4i) - (-6 + 4i) \)
     - Both numbers are in rectangular form: \(7 - 4i\) and \(-6 + 4i\).

3. **Combination of Addition and Subtraction**:
   - \( (3 + 7i) + (5 + 3i) - (-2 + 9i) \)
     - All numbers are in rectangular form: \(3 + 7i\), \(5 + 3i\), and \(-2 + 9i\).
   - \( (-4 - 5i) + (11 - 7i) - (8 + 6i) \)
     - All numbers are in rectangular form: \(-4 - 5i\), \(11 - 7i\), and \(8 + 6i\).
   - \( (9 + 7i) - (-9 + 7i) + (-18 + i) \)
     - All numbers are in rectangular form: \(9 + 7i\), \(-9 + 7i\), and \(-18 + i\).

4. **Multiplication**:
   - \( (2 + 5i)(3 + 7i) \)
     - Both numbers are in rectangular form: \(2 + 5i\) and \(3 + 7i\).
   - \( (2 - 3i)(3 + 5i) \)
     - Both numbers are in rectangular form: \(2 - 3i\) and \(3 + 5i\).
   - \( (4 - 3i)^2 \)
     - The number is in rectangular form: \(4 - 3i\).
   - \( (5 + 2i)^2 \)
     - The number is in rectangular form: \(5 + 2i\).
   - \( (6 + 5i)^3 \)
     - The number is in rectangular form: \(6 + 5i\).
   - \( (1 + 2i)(2 - i)(1 + i) \)
     - All numbers are in rectangular form: \(1 + 2i\), \(2 - i\), and \(1 + i\).

5. **Multiplication with Multiple Complex Numbers**:
   - \( (3 + 4i)(4 + 3i)(2 - 5i) \)
     - All numbers are in rectangular form: \(3 + 4i\), \(4 + 3i\), and \(2 - 5i\).
   - \( (3 + 5i)(5 + 3i)(2 - i) \)
     - All numbers are in rectangular form: \(3 + 5i\), \(5 + 3i\), and \(2 - i\).

6. **Division**:
   - \( \frac{-2}{4 + i} \)
     - \(-2\) and \(4 + i\) are in rectangular form.
   - \( \frac{-2 + 3i}{5 - 2i} \)
     - Both numbers are in rectangular form: \(-2 + 3i\) and \(5 - 2i\).
   - \( \frac{(2 + i)(1 - i)}{4 - 3i} \)
     - All numbers are in rectangular form: \(2 + i\), \(1 - i\), and \(4 - 3i\).
   - \( \frac{(2 + 3i)(3 + 2i)}{4 - 3i} \)
     - All numbers are in rectangular form: \(2 + 3i\), \(3 + 2i\), and \(4 - 3i\).

### Conclusion:
All the complex numbers provided in the exercises are already given in rectangular form, \(a + bi\). If you have any specific complex number that you want to convert to a different form or have additional questions, let me know!

### Would you like to solve any of these exercises or have any further questions?

### Here are 5 related questions you might explore:

1. How do you convert a complex number from rectangular form to polar form?
2. What are the rules for adding, subtracting, multiplying, and dividing complex numbers?
3. How do you calculate the modulus and argument of a complex number in rectangular form?
4. What are some applications of complex numbers in electrical engineering?
5. How can you visualize complex numbers on the complex plane?

### **Tip:** When performing operations with complex numbers, remember to always simplify the result and combine like terms, particularly when dealing with powers of \(i\).

Explore the rectangular forms of complex numbers in exercises, including addition, subtraction, multiplication, and division. Gain insights into operations with complex numbers expressed as a + bi. Suitable for advanced high school students studying complex numbers.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation