Problem 50 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \text {. Find two irrational numbers whose sum is rational. } $$

Step-by-Step Solution

Answer

Two irrational numbers are \(\sqrt{2}\) and \(-\sqrt{2}\). Their sum is 0, a rational number.

Step 1: Understanding the Problem

We need to find two irrational numbers, say \(a\) and \(b\), such that their sum \(a + b\) equals a rational number.

Step 2: Choosing an Irrational Number

Let's choose \(a = \sqrt{2}\), which is a known irrational number. We need to find another irrational number \(b\) such that their sum is rational.

Step 3: Calculating the Required Irrational Number

For the sum to be rational, we need \(b = c - a\), where \(c\) is a rational number. If we choose \(c = 0\), then \(b = 0 - \sqrt{2} = -\sqrt{2}\), which is also irrational.

Step 4: Confirming the Solution

Now, we check the sum: \(a + b = \sqrt{2} + (-\sqrt{2}) = 0\), which is a rational number. Therefore, \(a = \sqrt{2}\) and \(b = -\sqrt{2}\) are two irrational numbers whose sum is rational.

Key Concepts

Rational Numbers

Rational numbers are numbers that can be expressed in the form of a fraction \( \frac{p}{q} \), where both \( p \) and \( q \) are integers, and \( q eq 0 \).
Examples of rational numbers include integers, fractions, and finite or repeating decimals.
These numbers represent a broad category where most of the numbers we frequently encounter in daily life reside.

• An integer like 5 can be considered as \( \frac{5}{1} \), making it rational.
• A fraction like \( \frac{3}{4} \), where both numerator and denominator are integers, is rational.
• Decimal numbers that terminate or repeat, like 0.75 or 0.333..., are rational because they can be converted to fractions.

The rational numbers are dense on the number line, meaning between any two rational numbers, there exists another rational number.
Understanding rational numbers is critical in basic algebra as they form the foundation upon which more complex concepts are built.

Properties of Irrational Numbers

Irrational numbers are numbers that cannot be expressed as a simple fraction. They are often the result of taking roots of certain numbers, such as \( \sqrt{2} \), \( \pi \), or \( e \).
Unlike rational numbers, irrational numbers have non-repeating, non-terminating decimal expansions.

• \( \sqrt{2} \) is irrational because there is no fraction \( \frac{p}{q} \) that equals its value.
• \( \pi \), the ratio of the circumference to the diameter of a circle, has a decimal expansion that doesn’t repeat.

One interesting property of irrational numbers is that the sum or product of two irrational numbers isn't always irrational.
For example, when you sum \( \sqrt{2} \) and \( -\sqrt{2} \), you get 0, which is rational.
Recognizing when operations with irrational numbers can yield rational results is a useful skill in algebra and number theory.

Basic Algebra

Basic algebra involves working with unknowns or variables and applying basic arithmetic operations such as addition, subtraction, multiplication, and division.
It uses algebraic expressions and equations to represent real-world situations and solve problems.
In this context, considering the sum of irrational numbers, we can apply basic algebra principles by setting up equations to find how they relate to rational numbers.

• Linear equations, such as \( x + y = 0 \), where \( x = \sqrt{2} \) and \( y = -\sqrt{2} \), can illustrate the sum of irrational numbers totaling a rational number.
• Understanding algebraic manipulation, such as rearranging terms to isolate variables, helps in solving these types of problems.

Basic algebra forms the groundwork for more advanced mathematical topics, enabling students to explore complex scenarios through simple rules and logical reasoning.
It acts as a bridge, connecting the properties of different types of numbers and allowing for deeper analysis of their interactions and results.