Question

Answers are given at the end of these exercises Find the sum and the product of the complex numbers \(3-2 i\) and \(-3+5 i\)

Short answer

The sum is \(3i\) and the product is \(1 + 21i\).

Step-by-step solution

Identify the Complex Numbers

The given complex numbers are: \(3 - 2i\) and \(-3 + 5i\).

Sum the Complex Numbers

To find the sum, add the real parts and the imaginary parts separately.\((3 - 2i) + (-3 + 5i) = (3 + (-3)) + (-2i + 5i)\)Summing the real parts: \(3 + (-3) = 0\).Summing the imaginary parts: \(-2i + 5i = 3i\).So, the sum is \(0 + 3i\) or simply \(3i\).

Product of Complex Numbers

To find the product, use the distributive property (FOIL method): \((3 - 2i)(-3 + 5i) = 3(-3) + 3(5i) + (-2i)(-3) + (-2i)(5i)\)Calculate each term separately:\(3(-3) = -9,\)\(3(5i) = 15i,\)\((-2i)(-3) = 6i,\)\((-2i)(5i) = -10i^2\)Since \(i^2 = -1\), \(-10i^2 = 10\).Now add all the terms together:\(-9 + 15i + 6i + 10 = 1 + 21i\)So, the product is \(1 + 21i\).

Key concepts

Sum of Complex Numbers

Complex numbers have both real and imaginary parts. To add complex numbers, you handle the real and imaginary parts separately. This makes the operation straightforward.
For example, if we have two complex numbers, such as\(3 - 2i\) and \(-3 + 5i\), the sum can be found by:

• Adding the real parts: \(3 + (-3) = 0\)
• Adding the imaginary parts: \(-2i + 5i = 3i\)

Combining these results, the sum is\(0 + 3i\) or simply \(3i\).

Product of Complex Numbers

Multiplying complex numbers is a bit more involved but can be easily managed using properties of these numbers. To find the product, we multiply the complex numbers like binomials.
Given the complex numbers \(3 - 2i\) and \(-3 + 5i\):

1. Multiply the real and imaginary parts: \

• \(3 \times (-3) = -9\)
• \(3 \times 5i = 15i\)
• \(-2i \times (-3) = 6i\)
• \(-2i \times 5i = -10i^2\)

2. Simplify the \(-10i^2\): since \(i^2 = -1\), we have \(-10(-1) = 10\).

3. Add up all the terms: \(-9 + 15i + 6i + 10 = 1 + 21i\).

Thus, the product of these two complex numbers is \(1 + 21i\).

FOIL Method

The FOIL Method is a specific case of the distributive property used to multiply two binomials. FOIL stands for First, Outer, Inner, Last, describing which terms to multiply together.
For the complex numbers \(3 - 2i\) and \(-3 + 5i\), applying FOIL means:

• First: Multiply the first terms: \(3 \times (-3) = -9\)
• Outer: Multiply the outer terms: \(3 \times 5i = 15i\)
• Inner: Multiply the inner terms: \(-2i \times (-3) = 6i\)
• Last: Multiply the last terms: \(-2i \times 5i = -10i^2\)

By following these steps, you ensure that all necessary pairs of terms get multiplied, leading to the correct final product of\(1 + 21i\).

Distributive Property

The distributive property simplifies the multiplication of expressions by distributing one term to each term inside a parenthesis. When it comes to complex numbers, this property lets us handle each part separately.
Given \((3 - 2i)(-3 + 5i)\), we distribute \(3\) and \(-2i\) through to\((-3 + 5i)\):

• \(3 \times (-3) = -9\)
• \(3 \times 5i = 15i\)
• \(-2i \times (-3) = 6i\)
• \(-2i \times 5i = -10i^2\)

Next, simplify the\(-10i^2\), as \(i^2 = -1\), making it \(10\). Finally, add all the terms together: \(-9 + 15i + 6i + 10 = 1 + 21i\).
The distributive property ensures that you multiply each component correctly and simplify step by step to reach the final answer.