Problem 64 - Intermediate Algebra

Question

Find each of the products and express the answers in the standard form of a complex number. $$ (-5 i)(-12 i) $$

Step-by-Step Solution

Answer

The product is \(-60\), expressed as \(-60 + 0i\).

Step 1: Understand the Problem

We need to find the product of two complex numbers, \((-5i)\) and \((-12i)\), and express the result in the standard form of a complex number \(a + bi\).

Step 2: Multiply the Imaginary Parts

Multiply the two imaginary numbers: \((-5i) \times (-12i) = 60i^2\).

Step 3: Simplify Using \(i^2 = -1\)

Substitute \(i^2\) with \(-1\) to simplify: \(60i^2 = 60(-1) = -60\).

Step 4: Express in Standard Form

The result is \(-60\), which is a real number. In the standard form \(a + bi\), it can be expressed as \(-60 + 0i\).

Key Concepts

Imaginary Numbers

Imaginary numbers are mathematically essential yet can seem a bit abstract at first. They are used to represent numbers that are not real in the usual sense. An imaginary number consists of a real number multiplied by the imaginary unit. The symbol for the imaginary unit is "i", which is defined by the property \(i^2 = -1\).

Imaginary numbers are crucial in various fields of mathematics, engineering, and physics. They extend our number system beyond the real numbers. Complex numbers, which include both real and imaginary numbers, provide deeper solutions in algebra and calculus.

• Imaginary numbers are written as \( bi \), where \( b \) is a real number.
• The number 0i is just 0; it's neither strictly imaginary nor real.
• i is often referred to as the basic imaginary unit.

Understanding imaginary numbers is the first step in mastering complex numbers, as they provide symmetry in algebra and allow us to solve equations that would otherwise have no real solutions.

Standard Form

The standard form of a complex number is expressed as \(a + bi\). This form is crucial for effectively handling and interpreting operations involving complex numbers. Here, \(a\) represents the real part, and \(bi\) represents the imaginary part of the complex number.

Writing numbers in this form allows for a clear distinction between real and imaginary components, making addition, subtraction, and multiplication straightforward. It is important to keep these parts separated when dealing with complex numbers.

• The real number \(a\) can be any real value, including zero.
• The imaginary part \(bi\) involves an imaginary unit \(i\).
• Complex numbers include all real numbers too, as any real number \(a\) can be seen as \(a + 0i\).

Expressing results in this standard form is essential for ensuring that any number can be clearly understood and utilized within equations and real-world applications.

Complex Multiplication

Multiplying complex numbers involves a few straightforward steps, especially when dealing with imaginary coefficients. These steps are essential for solving problems involving products of complex numbers in algebra.

The process usually involves distributing and simplifying, similar to multiplying binomials. Here’s a simplified breakdown when multiplying two imaginary numbers such as \(-5i\) and \(-12i\):

• Multiply the coefficients: \(-5 \times -12 = 60\).
• Multiply the imaginary units: \(i \times i = i^2\).
• Combine the results: \(60i^2\).

Using the property that \(i^2 = -1\), you simplify the result to \(60(-1) = -60\).

The product, initially expressed in terms of \(i^2\), can then be represented in standard form as \(-60 + 0i\), a pure real number. This demonstrates how imaginary number multiplication involves converting imaginary products into real components.