Problem 1 - College Algebra with Corequisite Support

Question

Is \(\sqrt{2}\) an example of a rational terminating, rational repeating, or irrational number? Tell why it fits that category.

Step-by-Step Solution

Answer

\( \sqrt{2} \) is an irrational number because it cannot be expressed as a fraction.

Step 1: Understand the Type of Number

We first need to recall the definitions of different types of numbers. A rational number is any number that can be expressed exactly as the quotient or fraction \( \frac{p}{q} \) of two integers, with \( q eq 0 \). Rational numbers can be either terminating or repeating in decimal form. An irrational number cannot be expressed as a simple fraction and its decimal form is non-terminating and non-repeating.

Step 2: Examine the Nature of \( \sqrt{2} \)

To determine if \( \sqrt{2} \) fits one of these categories, we must check if it can be written as a fraction (rational) or not (irrational). It is known that \( \sqrt{2} \) is not a ratio of two integers.

Step 3: Validate the Irrationality of \( \sqrt{2} \)

Historically, \( \sqrt{2} \) has been proven to be irrational through indirect proofs. A common proof involves assuming \( \sqrt{2} \) can be expressed as \( \frac{p}{q} \) in its lowest terms, leading to a contradiction that no such integers \(p\) and \(q\) exist.

Step 4: Conclude with Properties of Decimal Representation

Since \( \sqrt{2} \) cannot be expressed as \( \frac{p}{q} \), it is an irrational number. Additionally, its decimal representation (approximately 1.414213...) does not terminate or repeat, consistent with properties of irrational numbers.

Key Concepts

Rational Numbers

Rational numbers are a fundamental part of the number system. They are numbers that can be expressed in the form of a fraction, where the numerator is an integer and the denominator is a non-zero integer. In mathematical terms, a rational number is expressed as \( \frac{p}{q} \), where \( p \) and \( q \) are integers, and \( q eq 0 \).

• Examples of rational numbers include \( \frac{1}{2} \), \( -3 \), and \( 0.75 \).
• They encompass integers, finite decimals, and repeating decimals.
• All fractions are rational numbers because they represent a division between two integers.

Rational numbers can be either positive or negative, and they encompass both whole numbers and numbers with decimal parts, as long as their decimal expansion is either finite or eventually becomes periodic, i.e., repeating.

Square Root

The symbol \( \sqrt{} \) denotes the square root and refers to a value that, when multiplied by itself, yields the original number. The square root of a number \( x \), written as \( \sqrt{x} \), represents the quantity that, when squared, equals \( x \).

• For example, \( \sqrt{9} = 3 \) because \( 3 \times 3 = 9 \).
• The square root function is inherently linked to exponentiation, as it acts inversely by finding the base number in the equation \( x^2 = n \).
• Square roots of non-perfect squares often result in irrational numbers.

The challenge with square roots, like \( \sqrt{2} \), emerges when the original number isn't a perfect square, as it can't be represented as a fraction of two integers, thus rendering \( \sqrt{2} \) irrational.

Decimal Representation

Decimal representation is a way of expressing numbers. When considering rational and irrational numbers, their decimals reveal distinct patterns:

• Rational numbers have decimal representations that either terminate (end after a finite number of digits) or repeat (due to a periodic sequence of digits).
• Examples are \( 0.25 \) (terminating) and \( 0.333... \) (repeating).
• Irrational numbers, however, do not fit this pattern, displaying non-terminating and non-repeating decimals.

Take \( \sqrt{2} \) as an example of an irrational number. Its approximate decimal expansion is \( 1.414213... \), a sequence that neither settles into a finite sequence nor into a predictable repeating cycle. This continual non-repeating decimal pattern highlights the unique quality of irrational numbers, separating them distinctly from rational numbers.