Problem 58 - Precalculus

Question

Give an example of irrational numbers a and b such that the indicated expression is (a) rational and (b) irrational. $$a / b$$

Step-by-Step Solution

Answer

Use \(a = \sqrt{2}, b = \sqrt{2}\) for a rational result; and \(a = \sqrt{2}, b = 1\) for an irrational result.

Step 1: Understanding Rationality and Irrationality

Recall that a rational number is one that can be expressed as the quotient of two integers (i.e., a fraction \( \frac{m}{n} \) with \( n eq 0 \)), while an irrational number cannot be expressed in this form. Examples of irrational numbers include \( \sqrt{2} \), \( \pi \), and others.

Step 2: Create an Expression with a Rational Result

Choose \( a = \sqrt{2} \) and \( b = \sqrt{2} \). Then, \( \frac{a}{b} = \frac{\sqrt{2}}{\sqrt{2}} = 1 \), which is a rational number.

Step 3: Create an Expression with an Irrational Result

Choose \( a = \sqrt{2} \) and \( b = 1 \). Then, \( \frac{a}{b} = \frac{\sqrt{2}}{1} = \sqrt{2} \), which is an irrational number.

Key Concepts

Quotient of Integers

Understanding the concept of the quotient of integers is key when dealing with rational numbers. A quotient is the result of dividing one number by another. When discussing quotients of integers, we are referring to two whole numbers being divided to produce a result. This is essential in distinguishing rational numbers.

A rational number is defined as a number that can be expressed as a fraction \( \frac{m}{n} \), where \( m \) and \( n \) are integers and \( n eq 0 \). In simpler terms, a rational number is any number that can be written as a simple fraction. For instance:

• \( \frac{1}{2} \) is a rational number because both 1 and 2 are integers and 2 is not zero.
• \( \frac{-5}{3} \) is also a rational number.

This tells us that if you can find integers such that their division results in a specific number, then that number is rational.

Irrational Numbers

Irrational numbers are quite fascinating because they cannot be expressed as a simple fraction of integers. This means there are no two integers you can divide to get an irrational number.

Irrational numbers include well-known numbers like:

• \( \sqrt{2} \)
• \( \pi \)
• \( e \)

These numbers have decimal expansions that never end and never settle into a repeating pattern. Unlike rational numbers, irrational numbers cannot be completely written out as decimals, they go on forever without repeating.

To think of irrational numbers in another way, consider the diagonal length of a square with side length 1, which is \( \sqrt{2} \). No fraction with whole numbers will exactly equal this figure. That makes \( \sqrt{2} \) an irrational number.

Rational Numbers

Rational numbers, as previously mentioned, are numbers that can be written as a fraction \( \frac{m}{n} \) where both \( m \) and \( n \) are integers and \( n eq 0 \). This definition allows for a variety of numbers that fulfill these criteria:

• Whole numbers, like 7, as they can be expressed as \( \frac{7}{1} \).
• Negative numbers, such as \( \frac{-8}{3} \).
• Decimals that end, like 0.75, which can be written as \( \frac{3}{4} \).
• Decimals that repeat, such as 0.666..., which is equal to \( \frac{2}{3} \).

Rational numbers play a crucial role in everyday mathematics as they allow us to express quantities in a clear and precise manner. Understanding their structure helps in exploring other number concepts, such as converting between decimals and fractions or understanding the density of rational numbers on the real number line.