Problem 4 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

False.

Step 1: Understand the definitions

A real number is any number that can be found on the number line. This includes rational numbers (numbers that can be expressed as fractions of integers) and irrational numbers (numbers that cannot be expressed as fractions, like the square root of 2 or pi). A rational number, on the other hand, is a number that can be expressed as the quotient of two integers, such that the denominator is not zero.

Step 2: Consider the set relationships

The set of real numbers includes both rational and irrational numbers. Therefore, while all rational numbers are included in the set of real numbers, not all real numbers are rational because irrational numbers also lie within the set of real numbers.

Step 3: Identify counterexamples

To determine if a statement is true, a single counterexample is enough to prove it false. Consider the number \(\pi\), which is a well-known irrational number. It is a real number, but not a rational number. Therefore, there exists at least one real number that is not rational.

Step 4: Draw Conclusion

Since \(\pi\) is a real number but not a rational number, this shows that not every real number is a rational number. This contradicts the statement, meaning the statement is false.

Key Concepts

Rational Numbers

Rational numbers are numbers that can be represented as a fraction where both the numerator and the denominator are integers, and the denominator is not zero. This definition implies that any whole number, fractional number, and decimal that ends or repeats is a rational number. For example, the number 5 can be expressed as \(\frac{5}{1}\), and 0.75 can be represented as \(\frac{3}{4}\).

• They include numbers like \(\frac{2}{3}\), \(0.5\,\text{or}\,\frac{1}{2}\), and \(\-4\,\text{or}\,\frac{\-8}{2}\).
• Rational numbers are dense on the number line; between any two rational numbers, there exists another rational number.

Because they can be precisely expressed as fractions, rational numbers are a subset of real numbers.

Irrational Numbers

Irrational numbers are numbers that cannot be written as a simple fraction; their decimal forms are non-repeating and non-terminating. Examples of irrational numbers include \(\pi\) and \(\sqrt{2}\).

• These numbers cannot be expressed exactly as the ratio of two integers.
• They often arise in geometry, such as the length of the diagonal of a square with side lengths of 1, given by \(\sqrt{2}\).

Irrational numbers, like rational numbers, are real numbers, and together they construct the set of real numbers. Despite their complex nature, irrational numbers can still be located on the number line uncompromisingly in terms of position.

Number Line

A number line is a straight horizontal line used to represent real numbers as points. Each point on the line corresponds to a real number, allowing a visual representation of the number's order and magnitude.

• The center point often represents zero, with positive numbers stretching to the right and negative numbers to the left.
• This simple diagram helps conceptualize the size and order of numbers in mathematical operations.

Both rational and irrational numbers can be placed on the number line, representing the full set of real numbers. The line is continuous, capturing the essence of real numbers, including gaps filled by irrational numbers.