Problem 2 - Algebra for College Students

Question

Identify each statement as true or false. Every rational number is a real number.

Step-by-Step Solution

Answer

True

Step 1: Understand the Sets in Question

A rational number is any number that can be expressed as a fraction \( \frac{a}{b} \) where \( a \) and \( b \) are integers and \( b eq 0 \). Examples include 1/2, -3, and 7. A real number includes all rational numbers and also all irrational numbers, such as \( \sqrt{2} \) and \( \pi \).

Step 2: Examine the Relationship

The real number set encompasses both rational and irrational numbers. Therefore, all rational numbers are naturally part of the real numbers since the real number set is a broader category that includes them, along with irrational numbers.

Step 3: Make a Conclusion

Since rational numbers, by definition, fit within the broader category of real numbers, the statement "Every rational number is a real number" is true.

Key Concepts

Real Numbers

Real numbers are a fundamental set in mathematics. They include both rational and irrational numbers, making them a vast and inclusive category.

• Rational numbers are numbers that can be expressed as a fraction, such as \( \frac{1}{2} \) or \( \frac{-3}{1} \).
• Irrational numbers, like \( \sqrt{2} \) and \( \pi \), cannot be expressed as simple fractions.

By understanding this, we see that real numbers cover everything on the number line without any gaps.
Examples range from simple whole numbers to complex numbers with endless decimal places like \( \pi \). This set is essential because it represents every possible magnitude or quantity.

Set Theory

Set theory forms the basis of modern mathematics and allows us to understand collections of objects, or sets. Each set groups elements that share common properties. For instance, the set of rational numbers contains all numbers that can be expressed as fractions. Meanwhile, the set of real numbers includes both rational and irrational numbers.
This means:

• The set of rational numbers is a subset of the set of real numbers.
• Every element within the set of rational numbers is also found within the set of real numbers.

This relationship reflects the hierarchy of number systems, where smaller, more specific sets nest within larger, more general ones.

Number Classification

Number classification helps us categorize numbers based on their properties. Real numbers are broadly divided into two main categories:

• Rational Numbers: These are numbers like 3, \( \frac{5}{7} \), and 0.75, which can all be expressed as a ratio of two integers.
• Irrational Numbers: These include numbers like \( \sqrt{5} \) and \( \pi \) that cannot be written as fractions.

When classifying numbers, it’s crucial to recognize whether they fit into more than one category. Sometimes, a number like -1 is both an integer and a rational number. This classification system enables us to analyze mathematical properties and solve problems more efficiently.

Irrational Numbers

Irrational numbers are an intriguing part of the real number system. They can't be expressed as simple fractions, and their decimal forms are non-repeating and endless. Examples include \( \pi \) and \( e \).Characteristics of irrational numbers include:

• Non-terminating decimals, meaning the decimal goes on forever without repeating.
• They fill the gaps between the rational numbers on the number line, contributing to the completeness of real numbers.

Understanding irrational numbers expands our number system, allowing for a comprehensive mathematical framework that covers all possible quantities and magnitudes.