Problem 3 - College Algebra with Corequisite Support

Question

Give an example to show that the product of two imaginary numbers is not always imaginary.

Step-by-Step Solution

Answer

The product of \( 2i \) and \( 3i \) is \(-6\), a real number.

Step 1: Understanding Imaginary Numbers

Imaginary numbers are numbers of the form \( a + bi \), where \( a \) and \( b \) are real numbers and \( i \) is the imaginary unit, defined by \( i^2 = -1 \). Here, the term \( bi \) is the imaginary component.

Step 2: Choosing Two Purely Imaginary Numbers

Let's consider two purely imaginary numbers \( z_1 = 2i \) and \( z_2 = 3i \). A purely imaginary number is one where the real part is zero (i.e., of the form \( 0 + bi \)).

Step 3: Multiplying the Imaginary Numbers

To find the product \( z_1 \cdot z_2 \), use multiplication of complex numbers:\[(2i)(3i) = 2 \cdot 3 \cdot i^2 = 6i^2.\]

Step 4: Simplifying the Product Using \( i^2 = -1 \)

Substitute \( i^2 = -1 \) into the equation:\[6i^2 = 6(-1) = -6.\]

Step 5: Interpreting the Result

The result \( -6 \) is a real number, not an imaginary number. This shows that the product of two imaginary numbers can result in a real number.

Key Concepts

Complex Numbers

Complex numbers are an intriguing mathematical concept that extends the idea of the traditional number line into a complex plane. These numbers take the form of \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit. The imaginary unit \( i \) is defined by its unique property: \( i^2 = -1 \). This definition allows complex numbers to incorporate quantities that involve the square root of negative numbers.

- **Real Part**: The real part of a complex number is the \( a \) in \( a + bi \). It lies along the horizontal axis in the complex plane.- **Imaginary Part**: The imaginary part is \( bi \). This component defines the vertical position on the complex plane.

Complex numbers are essential in fields like engineering and physics as they effectively describe waveforms, oscillations, and alternating currents. They provide a comprehensive framework to solve equations that don't have real solutions, offering insights that are not possible in the realm of real numbers alone.

Purely Imaginary Numbers

A purely imaginary number is a special type of complex number where the real part is zero. In mathematical terms, it can be expressed as \( 0 + bi \) or simply \( bi \). Here, \( b \) is a real number, and \( i \) is again the imaginary unit. Observing this form, it's clear that purely imaginary numbers contain only an imaginary component and are "imaginary" in their entirety.

For example:

• \( 2i \) is purely imaginary because it contains no real part.
• \( -3.5i \) is another example, where the coefficient of \( i \) is a real number (in this case, \(-3.5\)).

Purely imaginary numbers are typically represented on the vertical axis of the complex plane. They can be used in mathematical computations that involve rotations and oscillations. Importantly, though they only contain the 'imaginary' component, they are very real in their applications and utility.

Multiplication of Complex Numbers

Multiplying complex numbers follows a specific set of rules similar to multiplying binomials, due to their extended form \( (a+bi) \). The crucial step is remembering the fundamental property \( i^2 = -1 \). This allows the conversion of imaginary parts into real ones when they are squared.

Here's a simple breakdown of how multiplication works:

• Write both complex numbers in their standard form.
• Use the distributive property: \((a+bi)(c+di) = ac + adi + bci + bdi^2\).
• Since \( i^2 = -1 \), simplify: \( adi + bci + bdi^2 = (ac - bd) + (ad + bc)i \).

When multiplying purely imaginary numbers, like \( 2i \cdot 3i \), the result initially yields \( 6i^2 \). Substituting \( i^2 \) with \(-1\), this becomes \(-6\). This specific computation shows that the product can be a real number, not just another imaginary number. Multiplication in the complex plane involves transformations and offers results that extend beyond those in the real number system, highlighting the depth and versatility of complex numbers.