Problem 15 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

True, every real number is a complex number because it can be expressed as \(a + 0i\).

Step 1: Define Real Numbers

Real numbers include all the rational and irrational numbers. They can be positive, negative, or zero, represented on a number line. Examples of real numbers are 3, -1.5, 0, and π.

Step 2: Define Complex Numbers

Complex numbers extend real numbers by adding an imaginary part. A complex number is of the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit with the property \(i^2 = -1\).

Step 3: Relationship between Real and Complex Numbers

Every real number can be considered a complex number with its imaginary part equal to zero. For example, the real number 5 can be expressed as a complex number 5 + 0i.

Step 4: Evaluate the Statement

Since every real number can be written as a complex number (having the form \(a + 0i\)), it can be concluded that every real number is indeed a complex number.

Key Concepts

Real Numbers

Real numbers are a fundamental concept in mathematics and encompass both rational and irrational numbers. Rational numbers can be expressed as a fraction or ratio, like 1/2 or 5. They also include integers, whole numbers, and even numbers like -44 and 0.
Irrational numbers cannot be written as a simple fraction. Examples include numbers like \( \pi \) and \( \sqrt{2} \), which have non-repeating and non-terminating decimal places.

• Positive numbers: Any real number greater than zero (e.g., 3, 5, \( \pi \)).
• Negative numbers: Real numbers less than zero (e.g., -2, -7.8).
• Zero: A neutral element in the set of real numbers.

Real numbers are typically represented on a number line where each point corresponds to a unique real number.

Imaginary Unit

The imaginary unit, denoted as \( i \), is a mathematical concept that allows us to work with numbers that have a negative square. The defining property of \( i \) is that \( i^2 = -1 \). This concept is crucial for extending the set of real numbers to complex numbers.

• Complex form: Combining real and imaginary numbers forms complex numbers. These numbers are expressed as \( a + bi \), where \( a \) and \( b \) are real numbers.
• Applications: Imaginary numbers are used in advanced engineering fields, such as signal processing and fluid dynamics, to solve equations that traditional real numbers cannot.

The imaginary unit forms the foundation for complex numbers, allowing us to describe phenomena that go beyond the limitations of the real number system.

Number Line

A number line provides a visual representation of real numbers, allowing us to comprehend their order and magnitude. Each point on a number line corresponds to a real number, arranged sequentially.

• Zero point: Located in the center of the number line, representing the integer zero.
• Positive direction: Numbers increasing from zero to the right.
• Negative direction: Numbers decreasing from zero to the left.

On the number line, distances between numbers represent differences in their value. Real numbers such as 3, -1.5, and \( \pi \) have their own distinct positions, portraying their relationship to one another.