Problem 26 - Precalculus Mathematics for Calculus

Question

Evaluate the expression and write the result in the form \(a+b i\) $$(5-3 i)(1+i)$$

Step-by-Step Solution

Answer

The result is \(8 + 2i\).

Step 1: Apply the Distributive Property

To solve (5-3i)(1+i), we'll need to multiply each term in the first complex number by each term in the second complex number. This is also known as the distributive property or FOIL method for complex numbers. So, we calculate:\[ 5(1+i) - 3i(1+i) \]

Step 2: Multiply the Terms Separately

Let's multiply each term separately:1. Multiply 5 by (1+i):\[ 5 \times 1 + 5 \times i = 5 + 5i \]2. Now, multiply -3i by (1+i):\[ -3i \times 1 - 3i \times i = -3i - 3i^2 \] Remembering that \(i^2 = -1\), then \(-3i^2 = 3\). Thus:\[ -3i - 3i^2 = -3i + 3 \]

Step 3: Combine Like Terms

Now, add together the real parts and the imaginary parts from Step 2:1. Real parts: \(5 + 3 = 8\)2. Imaginary parts: \(5i - 3i = 2i\)Thus, the expression becomes:\[ 8 + 2i \]

Step 4: Write Final Answer in Standard Form

The final expression \(8 + 2i\) is already written in the standard form \(a + bi\), where \(a = 8\) and \(b = 2\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that lets you multiply a single term by every term inside a parenthesis. When multiplying complex numbers, this means distributing each part of the complex number across the other. In our exercise, with the expression

• \((5-3i)(1+i)\),

you apply the distributive property by multiplying each part of the first complex number, \(5\) and \(-3i\), by each part of the second complex number, \(1\) and \(i\). This results in: - \(5(1+i) - 3i(1+i)\).This step breaks the expression into simpler parts so we can concentrate on solving smaller, more manageable multiplications.
Multiplying separately like this may seem complex at first glance, but this systematic breakdown helps simplify the process significantly.

FOIL Method

The FOIL method is a specific way to apply the distributive property to multiply two binomials. FOIL stands for First, Outer, Inner, Last, referring to the position of the terms in each binomial that need to be multiplied together. For example, with the expression:

• \((5-3i)(1+i)\),

we perform the following operations:- **First**: multiply the first terms of each binomial: \(5 \times 1 = 5\).- **Outer**: multiply the outer terms of the binomials: \(5 \times i = 5i\).- **Inner**: multiply the inner terms: \(-3i \times 1 = -3i\).- **Last**: multiply the last terms: \(-3i \times i = -3i^2\).
Remember, because \(i^2 = -1\), this last multiplication simplifies to \(-3i(-1) = 3\). Using FOIL efficiently breaks down the process and simplifies the multiplication, resulting in \(5 + 5i - 3i + 3\). By organizing terms using FOIL, complex number multiplication becomes clear and straightforward.

Imaginary Unit

The imaginary unit, denoted as \(i\), is the cornerstone of complex numbers. By definition, \(i\) is the square root of \(-1\). This creation was a breakthrough that solved equations which were otherwise unsolvable with just real numbers.

• The key property of \(i\) is that \(i^2 = -1\).
• This characteristic turns challenging multiplications into something manageable.

For instance, when multiplying \(-3i(i)\), it simplifies as: \(-3i^2 = -3(-1) = 3\).
By understanding this, you'll be equipped to deal with the squares of imaginary numbers, which otherwise could seem confusing initially.
As you continue to work with complex numbers, keep in mind this pivotal role of the imaginary unit \(i\). It not only allows us to extend our number system beyond the real numbers, but also introduces the fascinating world of both real and imaginary components in mathematics.

Standard Form

The standard form of a complex number is given as \(a + bi\), where \(a\) represents the real part and \(b\) represents the imaginary part (multiplied by the imaginary unit \(i\)).

• This form, often called the rectangular form, helps clearly distinguish between the real and imaginary components.
• It provides a clear structure for further mathematical operations or solutions.

In the final step of our solution, we combine the terms from our distributed and FOIL stages to express the result in standard form:

• Real parts are added to obtain \(8\).
• Imaginary parts are added to yield \(2i\).

This gives us \(8 + 2i\), which is our solution in standard form.
Recognizing and using standard form allows for easy interpretation, communication, and further calculations in complex number mathematics. It's a key step in ensuring complex numbers are neatly expressed and understood.