Problem 17 - A Graphical Approach to College Algebra

Question

Determine whether each statement is true or false. If 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 with zero real part.

Step 1: Understand the Definitions

A pure imaginary number is of the form \( bi \), where \( b \) is a non-zero real number and \( i \) is the imaginary unit \( (i^2 = -1) \). A complex number is of the form \( a + bi \), where \( a \) and \( b \) are real numbers. The real part \( a \) can be zero.

Step 2: Relate Pure Imaginary to Complex Form

Notice that a pure imaginary number \( bi \) fits the complex number format \( a + bi \) with \( a = 0 \).

Step 3: Determine Truth Value

Since a pure imaginary number can be expressed as \( 0 + bi \), which is a special case of a complex number where the real part is zero, the statement "Every pure imaginary number is a complex number" is true.

Key Concepts

Pure Imaginary Numbers

Pure imaginary numbers play a unique role in the world of complex numbers. They are expressed in the form \( bi \), where \( b \) is a non-zero real number. The letter \( i \) is known as the imaginary unit, which is a crucial element in dealing with imaginary numbers. In contrast to real numbers, pure imaginary numbers have no real part; only their imaginary component contributes to their value.

• Form: \( bi \)
• Real part: 0
• Imaginary part: \( bi \)

To grasp pure imaginary numbers better, remember that they behave like multiples of \( i \), meaning they point in the vertical direction on the complex plane. They allow us to explore mathematical operations that extend beyond the limitations of real numbers alone. Pure imaginary numbers are indeed complex numbers because they fall within the broad category of complex numbers as they take the form \( 0 + bi \) which is a very special case in the set of complex numbers.

Real Numbers

Real numbers are the fundamental building blocks you encounter in everyday mathematics. These are numbers that can either be rational or irrational. They can be plotted on a number line, making them quite intuitive to understand and use. Unlike complex or pure imaginary numbers, real numbers do not include any multiple of the imaginary unit \( i \).

• Examples: 1, -3, 0, 0.5, \( \sqrt{2} \)
• Part of a complex number: \( a \)
• Imaginary part: 0

Real numbers are considered to be complex numbers with the form \( a + 0i \). In other words, every real number can be viewed as a complex number whose imaginary component is zero. They align horizontally on the complex number plane. Real numbers are essential in understanding how complex numbers are constructed and how they interact with each other.

Imaginary Unit

The imaginary unit, denoted as \( i \), is a vital concept when dealing with complex numbers. It is defined purely as the square root of -1, an expression that is not defined within the real number system.

• Mathematical definition: \( i^2 = -1 \)
• Origin: Result of trying to find solutions to equations like \( x^2 + 1 = 0 \)
• Utilization: Construction of complex numbers

The introduction of \( i \) broadens the spectrum of the number system, enabling calculations that include roots of negative numbers and solutions to polynomials that do not have real roots. With the imaginary unit, you can generate both pure imaginary numbers (like \( 3i \) or \(-4i\)) and complex numbers alike. The imaginary unit extends our mathematical toolkit, giving us more flexibility and power when handling different kinds of mathematical problems.