Problem 50 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find two irrational numbers whose sum is rational.

Step-by-Step Solution

Answer

\( \sqrt{2} + (-\sqrt{2}) = 0 \) is a rational sum.

Step 1: Choose an Irrational Number

Select an irrational number such as \( \sqrt{2} \). Irrational numbers have decimal representations that neither terminate nor become periodic.

Step 2: Find a Sum for Which You Want the Sum to be Zero

For simplicity, we aim for a sum of 0. Thus, choose \(-\sqrt{2}\) as the number to pair with \(\sqrt{2}\).

Step 3: Calculate the Sum

Compute the sum of these two numbers as follows: \( \sqrt{2} + (-\sqrt{2}) = 0 \). The sum is rational since 0 is a rational number.

Key Concepts

Rational Numbers

Rational numbers are numbers that can be expressed as a fraction of two integers. This means a rational number is any number that can be written in the form \( \frac{a}{b} \), where \( a \) and \( b \) are integers, and \( b eq 0 \). Here are some examples to clarify:

• \( \frac{1}{2} \) is a rational number because it is the quotient of 1 and 2.
• 7 is a rational number as it can be written as \( \frac{7}{1} \).
• 0.75 is a rational number because it is equal to \( \frac{3}{4} \).

Although most integers, fractions, and finite decimals are rational, not all numbers fall under this category. Some numbers cannot be expressed as fractions of two integers; these are known as irrational numbers.

Sum of Irrational Numbers

The sum of two irrational numbers can sometimes be rational, depending on the choice of the numbers. For instance, if you take an irrational number like \( \sqrt{2} \) and another irrational number, \(-\sqrt{2}\), their sum will be 0, which is a rational number.In more general terms, when:

• The irrational parts of the numbers perfectly cancel each other out, resulting in a rational sum.
• This concept can be counterintuitive because it goes against the common assumption that combining irrational numbers will always yield something irrational.

This means that while irrational numbers are never expressed as a fraction of two integers, their combination may lead to a rational outcome under certain conditions.

Properties of Irrational Numbers

Irrational numbers have intriguing properties that make them different from rational numbers. Some key characteristics include:

• Non-terminating and Non-repeating Decimal Representation: One of the most notable properties is their decimal form—it neither terminates nor repeats.
• Cannot Be Expressed as a Fraction: Unlike rational numbers, no fraction of two integers can completely describe an irrational number.
• Infinite Variety: There are an infinite number of irrational numbers, yet they cannot be neatly listed like integers or fractions.
• Examples of Irrational Numbers: Some of the common examples include \( \pi \), the mathematical constant, and \( \sqrt{2} \), which arises from geometric calculations.

These properties highlight why irrational numbers are essential to understanding the full number system, even if their unpredictable nature makes them complex to work with in equations.