Problem 57 - Calculus Early Transcendentals: Pearson New International Edition

Question

. Show that between any two different real numbers there is a rational number. (Hint: If \(a0\), so there is a natural number \(n\) such that \(1 / nb\\}\) and use the fact that a set of integers that is bounded from below contains a least element.) Show that between any two different real numbers there are infinitely many rational numbers.

Step-by-Step Solution

Answer

For any real numbers \(a < b\), we can always find a rational number between them, specifically \(\frac{k_0}{n}\). For infinitely many, consider intervals \(a_m = a + \frac{1}{m}\), yielding infinite rational numbers between \(a\) and \(b\).

Step 1: Setting Up the Problem

We need to show that for any two distinct real numbers, say \(a\) and \(b\), there exists at least one rational number between them. Without loss of generality, we assume \(a < b\).

Step 2: Understanding the Hint

From the hint, we know that since \(a < b\), \(b - a > 0\). According to properties of rational numbers, we can choose a natural number \(n\) such that \(\frac{1}{n} < b - a\). This means \(n\) is large enough so that the distance \(\frac{1}{n}\) is within \(b - a\).

Step 3: Constructing a Rational Number

To find a rational number that lies between \(a\) and \(b\), consider the integer part of \(an\), denoted as \(\lfloor an \rfloor\), which is the greatest integer less than or equal to \(an\). Thus, \(\lfloor an \rfloor \le an < \lfloor an \rfloor + 1\).

Step 4: Defining the Set and Applying the Least Element Principle

Let's construct a set \(S = \{ k \in \mathbb{Z} : k > an \}\). Since \(n\) is such that \(\frac{1}{n} < b-a\), \(an < bn\), ensuring \(S\) is non-empty. According to the properties of integers, any non-empty set of integers bounded below has a least element, say \(k_0\).

Step 5: Checking the Rational Number

The rational number we are looking for is \(\frac{k_0}{n}\). By definition of \(k_0\), \(k_0 - 1 \le an < k_0\), implying \(an < k_0\) and therefore \(a < \frac{k_0}{n}\). Also, since \(k_0\) is the smallest integer greater than \(an\), it must be that \(\frac{k_0}{n} < b\) because choosing \(\frac{1}{n} < b - a\) ensures this. Thus, \(\frac{k_0}{n}\) is a rational number between \(a\) and \(b\).

Step 6: Infinitely Many Rational Numbers

To find infinitely many rational numbers between \(a\) and \(b\), consider the intervals \(a_m = a + \frac{1}{m}\) and \(b_m = b - \frac{1}{m}\) for natural numbers \(m > n\). This setup maintains \(a_m < b_m\), and within each such interval, there exists a rational number as per the previous construction. Hence, as \(m\) varies, infinitely many such rational numbers can be found.

Key Concepts

Real Numbers

Real numbers are numbers that include both rational and irrational numbers. They can be represented on a continuous number line. Examples include numbers like 3, -5, 0.5, and even \(\pi\). Real numbers can be positive, negative, or zero.

• Rational numbers: These are numbers that can be expressed as a fraction \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q eq 0\). Examples include 2/3 or -4.
• Irrational numbers: These cannot be expressed as a simple fraction. Examples include the square root of 2 or \(\pi\).

These numbers are essential in understanding more complex mathematical concepts, and they form the basis of many calculations in mathematics and science.

Integers

Integers are a set of numbers that include all the whole numbers and their negative counterparts, along with zero. They are represented by the set \(\{..., -3, -2, -1, 0, 1, 2, 3, ...\}\). Integers do not include fractions or decimals.

• Positive integers: Numbers like 1, 2, 3...
• Negative integers: Numbers like -1, -2, -3...
• The integer zero: Zero is neutral and is considered neither positive nor negative.

Integers are used in various calculations and are essential in problem-solving, such as when determining least elements or working with sequences of numbers.

Natural Numbers

Natural numbers are the very basic numbers used for counting. They start from 1 and increase by 1 each time, like 1, 2, 3, 4, and so on. Some definitions include zero in this set, but traditionally, natural numbers start from 1.

• Whole positive numbers: These numbers are used in everyday counting, like counting apples or books.

Natural numbers are fundamental in arithmetic and are the building blocks for all other types of numbers. They are critical in understanding rational numbers because when used as the denominator in a fraction, they help define rational numbers.

Least Element Principle

The Least Element Principle is a fundamental concept in mathematics, particularly in set theory and number theory. It states that any non-empty set of integers that is bounded below has a smallest element. This principle is invaluable when dealing with integers.
For example, consider a set of integers \(S = \{ k \in \mathbb{Z} : k > an \}\). Even though this set contains possibly infinitely many elements, it will have a smallest element. This concept is applied in proving that between any two real numbers, a rational number can be found by establishing an appropriate set and identifying its least element.
The principle helps solve various mathematical problems, including those requiring the construction of specific types of numbers or ensuring that solutions exist within certain bounds.