Problem 23 - Elementary Algebra

Question

Perform the operations. $$ 2 i(7-4 i) $$

Step-by-Step Solution

Answer

The result of the operation is \(8 + 14i\).

Step 1: Distribute the term

We will distribute the term \(2i\) to both elements inside the parentheses \((7 - 4i)\). This gives us: \[2i \cdot 7 - 2i \cdot 4i\].

Step 2: Multiply the terms

First, multiply \(2i\) by \(7\): \(2i \cdot 7 = 14i\). Then, multiply \(2i\) by \(-4i\): \(2i \cdot -4i = -8i^2\).

Step 3: Substitute \(i^2\) with \(-1\)

Recall that \(i^2 = -1\). Substitute this into our expression: \(-8i^2 = -8(-1) = 8\).

Step 4: Combine the real part and imaginary part

Combine the results from Step 2 and Step 3: \(8 + 14i\).

Key Concepts

Imaginary Unit

The imaginary unit is a fundamental concept in the realm of complex numbers. It is denoted by the symbol \(i\) and is defined by the property that \(i^2 = -1\). This is quite peculiar because ordinarily, we do not have real numbers whose square is negative.

• This special unit allows us to extend our number system to include solutions to equations like \(x^2 + 1 = 0\), which have no real solutions.
• Essentially, the imaginary unit creates a whole new dimension or direction on the number line, which is very useful in various fields of mathematics, physics, and engineering.

To work with the imaginary unit, we often encounter it multiplied by real numbers, creating terms such as \(3i\) or \(-4i\). In such expressions, \(i\) behaves much like a variable in algebraic operations, but with the crucial importance of using the definition \(i^2 = -1\) whenever squaring it is involved.

Multiplication of Complex Numbers

Multiplying complex numbers might initially seem daunting, but it's pretty much a systematic process. Let's break it down.

Complex numbers usually come in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. When you multiply two complex numbers, you apply the distributive property (which we'll talk about more later) to expand the expression, ensuring you adhere to the rule \(i^2 = -1\).

For instance, in the exercise task to evaluate \(2i(7-4i)\), we are essentially doing the following:

• First, multiply \(2i\) by each term inside the parentheses.
• Calculate \(2i \times 7\) to get \(14i\).
• Compute \(2i \times -4i\). Remember that \(i \times i = i^2\), which simplifies from \(-8i^2\) to just \(8\) since \(i^2 = -1\).

Ultimately, the result leads to a new complex number that consists of a real part and an imaginary part. In this example, it results in \(8 + 14i\).

Distributive Property

The distributive property is a cornerstone of mathematics that comes in handy especially when dealing with expressions involving sums or differences inside parentheses.

It states that the multiplication of a single term distributed over several terms in parentheses can be expressed as the sum of individual products. If you have \(a(b + c)\), applying the distributive property gives you \(ab + ac\).

In complex numbers, this property is applied in much the same way. Consider the example in our exercise: \(2i(7 - 4i)\).

• Here, you distribute \(2i\) across both terms within the parenthesis, resulting in two separate products: \(2i \cdot 7\) and \(2i \cdot -4i\).
• This approach simplifies the process of multiplication by breaking it down into manageable parts.

Utilizing the distributive property not only keeps things clear and organized but also significantly reduces the potential for mistakes, especially when dealing with complex numbers and their imaginary components.