Problem 13 - Algebra 2 and Trigonometry

Question

In \(3-14,\) determine whether each of the numbers is rational or irrational. $$ \frac{\pi}{\pi} $$

Step-by-Step Solution

Answer

\( \frac{\pi}{\pi} = 1 \), which is a rational number.

Step 1: Understand the Problem

We are asked to determine if the number \( \frac{\pi}{\pi} \) is rational or irrational. A rational number can be expressed as a fraction \( \frac{a}{b} \), where \(a\) and \(b\) are integers and \(b eq 0\). An irrational number cannot be expressed as such a fraction.

Step 2: Simplify the Expression

The expression \( \frac{\pi}{\pi} \) involves division. Any non-zero real number divided by itself equals 1, hence \( \frac{\pi}{\pi} = 1 \).

Step 3: Determine the Nature of the Result

Since the result of \( \frac{\pi}{\pi} \) is 1, we recognize that 1 is a rational number. This is because 1 can be expressed as a fraction \( \frac{1}{1} \), where both the numerator and the denominator are integers.

Key Concepts

Simplifying Expressions

When faced with an expression like \( \frac{\pi}{\pi} \), it's important to simplify it first. Simplifying expressions makes them easier to understand and work with. In mathematics, simplification involves performing operations or factoring out common terms. For instance:

• Any non-zero number divided by itself equals one. Thus, \( \frac{\pi}{\pi} = 1 \).
• This rule applies to all numbers, including famous constants like \( \pi \).

By simplifying expressions, we reduce them to their simplest form, which can be vital in determining the properties of the resulting number, such as whether it's rational or irrational. This process not only makes calculations simpler but also helps reveal deeper mathematical insights.

Properties of Rational Numbers

Rational numbers are an integral part of mathematics. To determine whether a number is rational, it must meet specific criteria. A rational number can be defined as any number that can be expressed as the fraction \( \frac{a}{b} \) where:

• Both \(a\) and \(b\) are integers.
• \(b eq 0\), as division by zero is undefined.

Here are some key properties of rational numbers:

• They include integers, whole numbers, and fractions (e.g., \( \frac{1}{2}, -3, \text{and} \ 4\)).
• The decimal representation of a rational number can either terminate (e.g., 0.5) or repeat (e.g., 0.333...).
• They are dense, meaning between any two rational numbers, there exists another rational number.

In our original problem, since \( \frac{\pi}{\pi} \) simplifies to 1, and 1 can be expressed as \( \frac{1}{1} \), it's a rational number exhibiting these properties.

Definition of Irrational Numbers

While rational numbers can be expressed as fractions, irrational numbers cannot. This difference is crucial when distinguishing between the two. An irrational number:

• Cannot be represented as \( \frac{a}{b} \), where \(a\) and \(b\) are integers, and \(b eq 0\).
• Has a non-repeating, non-terminating decimal expansion.
• Famous examples include \( \pi \) and \( \sqrt{2} \).

Understanding the nature of irrational numbers is essential in many areas of math, including geometry and number theory. They often arise in situations where square roots of non-perfect squares, heights, or other measurements are involved. In the problem given, since the expression \( \frac{\pi}{\pi} \) simplifies to a rational result, it does not exemplify irrational properties. However, individually, \( \pi \) is a quintessential irrational number.