Problem 9 - Precalculus

Question

Compute each of the following. (a) \((3-4 i)(5+i)\) (b) \((5+i)(3-4 i)\) (c) \(\frac{3-4 i}{5+i}\) (d) \(\frac{5+i}{3-4 i}\)

Step-by-Step Solution

Answer

(a) 19 - 17i, (b) 19 - 17i, (c) \(\frac{11}{26} - \frac{23}{26}i\), (d) \(\frac{11}{25} + \frac{23}{25}i\)."

Step 1: Multiply Two Complex Numbers (a)

To compute \((3-4i)(5+i)\), apply the distributive property: \((3)(5) + (3)(i) + (-4i)(5) + (-4i)(i)\). Simplify each term: 1. \(3 \times 5 = 15\)2. \(3 \times i = 3i\)3. \(-4i \times 5 = -20i\)4. \(-4i \times i = -4i^2 = 4\) (since \(i^2 = -1\))Combine the terms together: \(15 + 3i - 20i + 4 = 19 - 17i\).

Step 2: Multiply Two Complex Numbers (b)

To compute \((5+i)(3-4i)\), use the distributive property again: \((5)(3) + (5)(-4i) + (i)(3) + (i)(-4i)\). Calculate each component:1. \(5 \times 3 = 15\)2. \(5 \times -4i = -20i\)3. \(i \times 3 = 3i\)4. \(i \times -4i = -4i^2 = 4\) (since \(i^2 = -1\))Combine the terms:\(15 - 20i + 3i + 4 = 19 - 17i\).

Step 3: Divide Complex Numbers (c)

To divide \(\frac{3-4i}{5+i}\), multiply the numerator and the denominator by the conjugate of the denominator, which is \(5-i\).Numerator:\((3-4i)(5-i)\) using the distributive property:1. \(3 \times 5 = 15\)2. \(3 \times -i = -3i\)3. \(-4i \times 5 = -20i\)4. \(-4i \times -i = 4i^2 = -4\) (since \(i^2 = -1\))Arranging terms gives: \(15 - 3i - 20i - 4 = 11 - 23i\).Denominator:\((5+i)(5-i) = 5^2 - (i)^2 = 25 + 1 = 26\).Result:\(\frac{11-23i}{26} = \frac{11}{26} - \frac{23}{26}i\).

Step 4: Divide Complex Numbers (d)

For \(\frac{5+i}{3-4i}\), multiply numerator and denominator by the conjugate of the denominator, \(3+4i\).Numerator:\((5+i)(3+4i)\) leads to:1. \(5 \times 3 = 15\)2. \(5 \times 4i = 20i\)3. \(i \times 3 = 3i\)4. \(i \times 4i = 4i^2 = -4\) (since \(i^2 = -1\))Combine the terms: \(15 + 20i + 3i - 4 = 11 + 23i\).Denominator:\((3-4i)(3+4i) = 3^2 + 4^2 = 9 + 16 = 25\).Result:\(\frac{11+23i}{25} = \frac{11}{25} + \frac{23}{25}i\).

Key Concepts

Multiplication of Complex Numbers

To multiply two complex numbers, we use the distributive property, which involves spreading each term of the first complex number over each term of the second. Let's break it down with an example: consider the multiplication of (3-4i) and (5+i). First, distribute the terms:

• Multiply 3 by 5 to get 15.
• Multiply 3 by i to get 3i.
• Multiply -4i by 5 to get -20i.
• Multiply -4i by i to get -4i².

Notice that i² equals -1, which simplifies -4i² to 4. Combine the like terms to get the result: 15 + 4 + 3i - 20i, which simplifies to 19 - 17i. This method applies to any multiplication of complex numbers. By understanding the process of distributing and combining terms, multiplying complex numbers becomes straightforward.

Division of Complex Numbers

Dividing complex numbers involves using the conjugate of the denominator. Take (3-4i)/(5+i). The conjugate of 5+i is 5-i. Multiply both the numerator and the denominator by the conjugate:

• Numerator: (3-4i)*(5-i).
• Distribute using the same method as multiplication: 15 - 3i - 20i + 4.
• Combine terms to get 11 - 23i.
• Denominator: (5+i)*(5-i) simplifies to 25 + 1 = 26 because you use the identity (a+b)(a-b) = a² - b².

Finally, divide each part by the denominator to get: (11/26) - (23/26)i. Remember, the conjugate method ensures that the denominator becomes a real number, making the division process simpler and resulting in a complex number in standard form.

Conjugate of Complex Numbers

The conjugate of a complex number reverses the sign of the imaginary part. If our complex number is a + bi, then its conjugate is a - bi. For example, the conjugate of 5 + i is 5 - i. The importance of conjugates lies in their use in division. When you multiply a complex number by its conjugate, you get a real number. This property is crucial for dividing complex numbers because it helps eliminate complex terms from the denominator, making the arithmetic manageable. By applying this principle, you can transform a division problem like (3-4i)/(5+i) into (3-4i)(5-i)/(5+i)(5-i), resulting in a real number in the denominator. Understanding how to use conjugates is essential in not only simplifying complex number division but also in various advanced mathematical computations where clarity and precision are needed.

Distributive Property with Complex Numbers

The distributive property is central to performing algebraic operations on complex numbers. It states that for three numbers a, b, and c, a(b + c) equals ab + ac. This property allows us to multiply complex numbers by breaking them into simpler terms. When multiplying two complex numbers, such as (3-4i) and (5+i), you distribute each part of the first complex number through each part of the second:

• (3)(5) = 15, which is the real part
• (3)(i) = 3i, imaginary contribution from the first number
• (-4i)(5) = -20i, imaginary contribution from the second number
• (-4i)(i) = -4i², which becomes 4 due to i² = -1

Adding these results gives 15 + 4 + 3i - 20i = 19 - 17i, demonstrating how the distributive property is used to simplify the multiplication of complex numbers. The process translates easily to any complex multiplication, ensuring that even the most intricate calculations remain accessible and straightforward.