Problem 8 - Algebra for College Students

Question

Label each statement true or false. The sum of two pure imaginary numbers is always a pure imaginary number.

Step-by-Step Solution

Answer

True.

Step 1: Understanding Pure Imaginary Numbers

A pure imaginary number is of the form \( bi \), where \( b \) is a non-zero real number, and \( i \) is the imaginary unit, such that \( i^2 = -1 \). In this context, examples include \( 3i, -7i, \) and so on.

Step 2: Setting Up the Problem

Suppose we have two pure imaginary numbers \( a = bi \) and \( c = di \), where \( b \) and \( d \) are real numbers. Their sum is \( a + c = bi + di \).

Step 3: Adding Pure Imaginary Numbers

The sum \( a + c = bi + di = (b+d)i \) can still be expressed in the pure imaginary form \( ei \), where \( e = b + d \). Since \( b \) and \( d \) are real numbers, \( e \) is also real.

Step 4: Conclusion Based on Calculation

Because \( (b+d)i \) is a pure imaginary number as long as \( e = b + d eq 0 \) (though it includes zero, which is often treated as not being pure imaginary for strict definitions), the sum is typically still considered a pure imaginary number. Therefore, the statement is true.

Key Concepts

Imaginary Unit

The imaginary unit, represented by the symbol \( i \), is a fundamental concept in mathematics, especially useful when dealing with complex numbers. In essence, the imaginary unit \( i \) is defined such that \( i^2 = -1 \). This might seem peculiar at first, since we are accustomed to the idea that squaring a real number always yields a non-negative value. However, the brilliance of \( i \) is exactly in this defiance of conventional algebraic rules, making it possible to solve equations that otherwise would have no real solutions, such as \( x^2 + 1 = 0 \).

Using \( i \), mathematicians unlocked a whole new number system that extends our comprehension of numbers beyond the real line. This allows us to perform various calculations that rely on this imaginary aspect, serving as a building block for the broader concept of complex numbers.

Complex Numbers

Complex numbers are numbers that have both a real part and an imaginary part, typically expressed in the form \( a + bi \), where \( a \) and \( b \) are real numbers and \( i \) is the imaginary unit. This form elegantly captures numbers on a different plane known as the complex plane, where the horizontal axis represents the real part and the vertical axis represents the imaginary part.

• The number \( a \) is called the real part.
• The number \( b \) is called the imaginary part, primarily because it is paired with \( i \).

The addition of complex numbers is straightforward, involving the separate addition of real parts and imaginary parts. This makes it not only easy to work with but also to visualize and comprehend. Complex numbers are fundamental in various fields, from engineering to physics, offering solutions to equations and observations that do not limit themselves within the real number line.

Addition of Imaginary Numbers

The addition of imaginary numbers, specifically pure imaginary numbers, is both interesting and conceptually straightforward. Pure imaginary numbers are in the form \( bi \), where \( b \) is a non-zero real number. When you add two pure imaginary numbers, say \( bi \) and \( di \), the result is \((b+d)i\).

Here's how the addition works:

• Consider two pure imaginary numbers: \( a = bi \) and \( c = di \).
• When these are added: \( a + c = bi + di \).
• This simplifies to \( (b + d)i \), which is also in the form of a pure imaginary number.

The essence of this addition is that the resultant is determined by adding the coefficients \( b \) and \( d \) together, as \( i \) remains a common factor. Provided the sum \( b + d eq 0 \), the result is a pure imaginary number and maintains its form. This aligns perfectly with the exercise statement that the sum of two pure imaginary numbers typically remains imaginary.