Problem 119 - Introductory and Intermediate Algebra for College Students 4th

Question

Give an example of a real number that is not an irrational number. (Section \(1.3,\) Example 5 ).

Step-by-Step Solution

Answer

An example of a real number that is not an irrational number would be 2, as it can be expressed as the fraction 2/1.

Step 1: Understanding Real, Rational and Irrational Numbers

Real numbers are all numbers that can be plotted on a number line. They include rational numbers (which can be expressed as a fraction a/b where a and b are integers and b is not zero) and irrational numbers (which cannot be expressed as a simple fraction). By asking for a real number that is not an irrational number, the exercise essentially asks for a rational number.

Step 2: Chose an Example of a Rational Number

Literally any rational number would satisfy the requirement. Let's choose the number 2, which is an integer, and hence by definition also a rational number.

Key Concepts

Rational Numbers

Rational numbers are numbers that can be expressed as the quotient or fraction of two integers. The general form is \( \frac{a}{b} \), where \( a \) and \( b \) are integers, and \( b \) is not zero. This includes whole numbers like 5 (which can be seen as \( \frac{5}{1} \)), fractions like \( \frac{3}{4} \), and even repeating decimals such as 0.333... (which is equivalent to \( \frac{1}{3} \)).

There are several interesting characteristics of rational numbers:

• They include both positive and negative numbers.
• They can be finite decimals or repeating decimals.
• A simple way to test if a number is rational is to see if it can be written as a simple fraction.

Choosing a rational number can be straightforward, like using the number 2 in our exercise, as it can simply be expressed as \( \frac{2}{1} \). This fact helps illustrate that all integers are automatically rational numbers.

Irrational Numbers

Irrational numbers are, in a way, the opposite of rational numbers. Unlike rational numbers, they cannot be expressed as the fraction of two integers. Instead, they have an infinite, non-repeating decimal expansion.

Some classic examples of irrational numbers include:

• \( \pi \) (pi), which is approximately 3.14159, and represents the ratio of the circumference of a circle to its diameter.
• \( \sqrt{2} \) (the square root of 2), which is approximately 1.414 and arises in geometry when dealing with the diagonal of a square.
• The mathematical constant \( e \), which is important in the field of natural logarithms and calculus.

These numbers cannot be written down completely as decimals, and that's what makes them unique. The distinction between rational and irrational numbers is crucial for understanding the full scope of real numbers.

Number Line

The number line is a visual representation of all real numbers, stretching infinitely in both positive and negative directions. Plotting real numbers on the number line helps illustrate their relationships and magnitudes.

Here's how to visualize different categories of real numbers on the number line:

• Whole Numbers and Integers: These are plotted as equally spaced points along the number line, like 0, 1, 2, -1, -2, etc.
• Rational Numbers: These appear as points between the integers, and include fractions such as \( \frac{1}{2} \) or \( \frac{3}{4} \).
• Irrational Numbers: These also lie between the integers but cannot be pinpointed exactly. Instead, they are approximated, such as the point near where \( \sqrt{2} \) would be positioned.

By visualizing numbers on a number line, it’s easier to see the density and continuity of real numbers, meaning that between any two rational numbers, you can find an irrational number and vice versa, filling the number line without any "gaps."