Problem 3 - Precalculus Student Solutions Manual 5th

Question

Determine whether each statement is true or false. If it is false, tell why. Every pure imaginary number is a complex number.

Step-by-Step Solution

Answer

True, every pure imaginary number is a complex number because it fits the definition of a complex number with \(a = 0\).

Step 1: Define Complex and Pure Imaginary Numbers

A complex number is one that can be written in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit with the property that \(i^2 = -1\). A pure imaginary number is a special case of a complex number where the real part \(a\) is zero, so it can be written as \(0 + bi\) or simply \(bi\).

Step 2: Compare Definitions

Given the definitions, compare the two. Since a pure imaginary number fits the form of a complex number with \(a = 0\), it meets the criteria of being a complex number.

Step 3: Evaluate the Statement

From the definition and comparison, determine if the statement 'Every pure imaginary number is a complex number' holds true. Since every pure imaginary number \(bi\) has \(a = 0\) and fits the form of \(a + bi\) for complex numbers, the statement is true.

Key Concepts

pure imaginary numbers

A pure imaginary number is a special type of complex number where the real part is zero. This means a pure imaginary number can be written in the form 0 + bi or simply bi. Here, b is a real number, and i is known as the imaginary unit. The key takeaway is that for pure imaginary numbers, there is no real part involved, only the imaginary part.

Pure imaginary numbers are useful in various fields, including engineering and physics, because they represent quantities that cannot be actualized on the real number line. For example, the square root of a negative number is a pure imaginary number. In such cases, calculations and representations become much easier to handle using pure imaginary numbers.

real numbers

A real number is any number that can be found on the number line. This includes all the rational and irrational numbers like integers, fractions, and roots of positive numbers. Real numbers can be positive, negative, or zero.

Real numbers are the foundation of most mathematical concepts and are used in a multitude of applications, including counting, measuring, and problem-solving in both theoretical and applied mathematics.

It's essential to understand that in the context of complex numbers, the real part is denoted by a in the form a + bi. This makes real numbers a critical component of complex numbers.

imaginary unit

The imaginary unit, denoted by i, is a fundamental concept in the field of complex numbers. It is defined by the equation:

\[i^2 = -1\]

This definition helps us to extend the real number system to solve equations that do not have real solutions. For example, the equation \[x^2 + 1 = 0\] has no real solution because no real number squared gives -1. By introducing the imaginary unit i, we can say the solutions are \[x = i\] and \[x = -i\].

Imaginary numbers involving the unit i allow mathematicians and scientists to deal with complex systems and signals, particularly in fields like quantum physics and electrical engineering.