Problem 63 - Intermediate Algebra

Question

Find each of the products and express the answers in the standard form of a complex number. $$ (7 i)(-6 i) $$

Step-by-Step Solution

Answer

42

Step 1: Understand the Problem

We need to multiply two imaginary numbers, \(7i\) and \(-6i\), and express the result as a complex number in standard form, which is \(a + bi\), where \(a\) and \(b\) are real numbers.

Step 2: Multiply the Coefficients

First multiply the numerical coefficients of the imaginary numbers: \( 7 \times (-6) = -42 \).

Step 3: Multiply the Imaginary Units

When you multiply \(i \times i\), it results in \(i^2\). We have: \(i^2 = -1\). Therefore, the multiplication becomes \(-42 \times i^2 = -42 \times (-1)\).

Step 4: Simplify the Expression

Simplify the product \(-42 \times (-1)\) to get 42. Thus, the result is a real number 42.

Step 5: Express in Standard Form

Since the result is purely real, we can write it as \(42 + 0i\). This is already in the standard form of a complex number \(a + bi\).

Key Concepts

Imaginary Numbers

Imagine a number with a twist. That's what an imaginary number is! In mathematics, imaginary numbers stem from the need to find a solution to negative square roots. To break it down, the foundation of imaginary numbers is the imaginary unit, denoted as \(i\). The imaginary unit \(i\) is defined as the square root of -1, or mathematically expressed as:

• \(i = \sqrt{-1}\)

The interesting part is that when you square \(i\) (\(i^2\)), you arrive at -1. That might seem unusual at first glance, but it gives mathematicians a powerful tool to solve equations that otherwise have no solutions in the realm of merely real numbers.
When we talk about imaginary numbers, they are generally expressed as \(bi\) where \(b\) is a real number. For example, in the exercise \(7i\) and \(-6i\) are imaginary numbers because the value in front of \(i\) (which is a real number) gives its magnitude.
Imaginary numbers are essential not only in theoretical mathematics but also in practical fields like engineering and physics, where they help to solve real-world problems by providing a fuller set of numbers for calculation.

Standard Form

Expressing complex numbers in a clear format helps maintain consistency and understanding. The standard form of a complex number is typically written as \(a + bi\). This form is easy to read and clearly indicates the real part \(a\) and the imaginary part \(bi\). For instance:

• \(a\) represents the real portion of the number.
• \(bi\) signifies the imaginary part, where \(b\) is a real number and represents the coefficient of the imaginary unit \(i\).

In the standard form, both \(a\) and \(b\) are real numbers. Sometimes either \(a\) or \(b\) can be zero. For example, in the exercise, the final result \(42 + 0i\) shows that the imaginary part is nonexistent (or \(b = 0\)), thus leaving the number completely real. This approach of using standard form helps to categorize numbers as real or complex efficiently. It's a useful way to ensure clarity when dealing with various mathematical operations, like addition, subtraction, or multiplication of complex numbers.

Multiplication of Complex Numbers

Multiplying complex numbers might seem daunting at first, but it's quite methodical. The process involves multiplying both the real and imaginary parts just like you would perform multiplication of binomials.

• Start by multiplying the coefficients of the imaginary parts.
• Then multiply the imaginary units.

In the given exercise, the numbers \(7i\) and \(-6i\) are multiplied:

• Multiply the coefficients: \(7 \times -6 = -42\)

Now, since both have an imaginary unit, it's crucial to multiply those units as well:

• \(i \times i = i^2\) and we know \(i^2 = -1\)

Thus, the product becomes \(-42 \times i^2 = -42 \times (-1) = 42\).
This result is a real number. Lastly, express the result in standard form \(42 + 0i\). Multiplication of complex numbers is a fundamental skill in mathematics, opening doors to understanding further operations and branches within the subject. It becomes easier with practice and aids in numerous applications across various fields.