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Chapter 3: Problem 17

If \(r\) and \(s\) are any two rational numbers with \(r

Short Answer

Expert verified
To prove that \(\frac{r+s}{2}\) is rational, we first represent the rational numbers, \(r\) and \(s\), as fractions: \(r = \frac{a}{b}\) and \(s = \frac{c}{d}\). Adding the fractions, we get \(r+s = \frac{ad + bc}{bd}\). Then divide by \(2\), resulting in \(\frac{r+s}{2} = \frac{ad + bc}{2bd}\). Since the numerator and denominator are both integers and the denominator is non-zero, this expression represents a rational number.

Step by step solution

01

Recall the definition of a rational number

A rational number is any number that can be expressed as the quotient (or fraction) of two integers, where the numerator is an integer and the denominator is a non-zero integer. In other words, if a number can be written in the form \(a/b\), where a and b are integers and b 鈮� 0, then it is a rational number.
02

Use the given information

We are given that r and s are rational numbers, which means they can be written as fractions with integer numerators and non-zero integer denominators. Let \(r = \frac{a}{b}\) and \(s = \frac{c}{d}\), where a, b, c, and d are integers and b 鈮� 0 and d 鈮� 0.
03

Calculate the expression of interest

We want to show that \(\frac{r+s}{2}\) is rational. Let's start by calculating r + s: \(r + s = \frac{a}{b} + \frac{c}{d}\) Now we need to find a common denominator for these two fractions. Since the product of two non-zero numbers is non-zero, we can use bd as the common denominator: \(r + s = \frac{ad}{bd} + \frac{bc}{bd}\) Now, we can combine these two fractions: \(r + s = \frac{ad + bc}{bd}\) Now we can divide this expression by 2: \(\frac{r+s}{2} = \frac{\frac{ad + bc}{bd}}{2}\)
04

Show that the expression is a rational number

Finally, we can multiply the numerator and denominator of the fraction by 2 to cancel out the division by 2: \(\frac{r+s}{2} = \frac{2(ad + bc)}{2(bd)}\) Now we can simplify the expression: \(\frac{r+s}{2} = \frac{ad + bc}{bd}\) The expression \(\frac{ad + bc}{bd}\) is the quotient of two integers (since a, b, c, and d are integers) with a non-zero denominator (as b and d are both non-zero). Therefore, the expression \(\frac{r+s}{2}\) is a rational number, as we wanted to show.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Discrete Mathematics
Discrete mathematics is a branch of mathematics that deals with discrete elements that are distinct and separate. This field of study involves a wide range of topics including logic, set theory, graph theory, and number theory. Rational numbers are a critical component of discrete mathematics, especially because they can be enumerated, which is central to combinatorics, another area within discrete mathematics. Understanding how rational numbers work, their properties, and their implications aligns with discrete mathematics鈥� focus of studying structures that are fundamentally discrete rather than continuous.
Arithmetics of Rational Numbers
In the realm of arithmetic, rational numbers play a pivotal role due to their structure and versatility. The operations of addition, subtraction, multiplication, and division 鈥� the core of arithmetics 鈥� are defined for rational numbers. When we add or multiply two rational numbers, the outcome is also rational, which speaks to the closed nature of rational numbers with respect to these operations. The step-by-step solution illustrates this property by showing how the average of two rational numbers is also rational, reinforcing the notion that arithmetic with rational numbers adheres to the same foundational rules as arithmetic with integers.
Proofs in Mathematics
Proofs are the essence of mathematics; they validate the truths within the mathematical realm. When we provide a mathematical proof, we are rigorously demonstrating that a proposition or theorem is universally true under the axioms and rules of the field. The solution to the exercise is an example of a direct proof, where we start with the known information 鈥� that rational numbers can be expressed as a fraction of integers 鈥� and through a series of logical arithmetic steps, we deduce that the average of any two rational numbers is rational, thus confirming a general property of these numbers.
Properties of Rational Numbers
Rational numbers have several intriguing properties that make them a fundamental object of study in arithmetic and number theory. They include closure under addition, subtraction, and multiplication, existence of additive and multiplicative inverses, and the density property, which states that between any two rational numbers there is another rational number. The exercise highlighted another property: the result of performing a binary operation on two rational numbers (e.g., taking the average) yields another rational number. Understanding these properties is essential as they form the basis of more advanced mathematical concepts and proofs.

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Most popular questions from this chapter

Prove those that are true and disprove those that are false.\(6-7 \sqrt{2}\) is irrational.

A calculator display shows that \(\sqrt{2}=1.414213562\), and \(1.414213562=\frac{1414213562}{1000000000}\). This suggests that \(\sqrt{2}\) is a rational number, which contradicts Theorem 3.7.1. Explain the discrepancy.

Suppose \(a\) is an integer and \(p\) is a prime number such that \(p \mid a\) and \(p \mid(a+3)\). What can you deduce about \(p\) ? Why?

There is an integer \(n>5\) such that \(2^{n}-1\) is prime.

For all real numbers \(x\) and \(y_{,}|x+y| \leq|x|+|y| .\) This result is called the triangle inequality. (Hint: Use 51 and 52 above.)
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