Problem 49 - Calculus Early Transcendentals: Pearson New International Edition

Question

Is \(0.1234567891011121314 \ldots\) rational or irrational? (You should see a pattern in the given sequence of digits.)

Step-by-Step Solution

Answer

The number is irrational because it does not repeat or terminate.

Step 1: Understanding a Rational Number

Rational numbers are numbers that can be expressed as a fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b eq 0 \). If a decimal representation ends or repeats, it is a rational number.

Step 2: Identify the Pattern in the Sequence

Consider the sequence of digits: "1234567891011121314...". It represents the concatenation of natural numbers starting from 1. This sequence does not terminate or repeat because the numbers continue indefinitely without forming a repeating cycle.

Step 3: Determine the Nature of the Sequence

Since the sequence is formed by the non-terminating, non-repeating concatenation of natural numbers, it doesn't fit the form of a fraction where the decimal representation is either finite or cyclic. Therefore, it does not satisfy the condition of being a rational number.

Step 4: Conclusion on Rationality

Given that the decimal neither terminates nor has a repeating pattern, it is not possible to express this sequence of digits as a fraction of two integers. Hence, the number is irrational.

Key Concepts

Rational Numbers

Rational numbers are a fundamental concept in the realm of mathematics. They are numbers that can be expressed in the form of a fraction, specifically \( \frac{a}{b} \), where \( a \) and \( b \) are both integers, and \( b eq 0 \). This condition is crucial because division by zero is undefined in mathematics. Rational numbers encompass a wide variety of familiar numbers, including whole numbers, integers, and simple fractions like \( \frac{1}{2} \) or \( \frac{3}{4} \). To determine if a number is rational, you need to consider its decimal representation:

• If the decimal ends after a certain point, for example, 0.5 or 0.75, it is a rational number.
• If the decimal repeats a pattern, such as in 0.333... or 0.666..., it is also classified as a rational number.

All these cases imply that these numbers can be expressed as a simple fraction. For example, 0.5 is \( \frac{1}{2} \) and 0.333... is \( \frac{1}{3} \). If a number cannot be expressed in such a fractional form, it doesn't fall under the category of rational numbers.

Decimal Representation

The decimal representation of a number tells us how it appears as a series of digits after a decimal point. This representation can take different forms, which helps in classifying numbers as either rational or irrational. The main types of decimal representations are:

• *Terminating Decimals:* These are decimals that end after a finite number of digits. Examples include 0.25, 0.75, and 1.0.
• *Repeating Decimals:* These decimals showcase a periodic sequence of digits. Numbers like 0.333... or 0.142857142857... are examples where digits repeat infinitely.
• *Non-repeating, Non-terminating Decimals:* These decimals extend indefinitely without a repeating pattern, like the number \( \pi \) (3.14159...) or the one in your exercise (0.1234567891011121314...).

The first two types are related to rational numbers because they can be expressed in fraction form. Meanwhile, numbers with a non-repeating, non-terminating decimal sequence are classified as irrational, revealing the beauty and complexity of these extraordinary numbers.

Fraction Representation

The fraction representation of numbers is an essential method to express rational numbers. A fraction is written in the form \( \frac{a}{b} \) where:

• \( a \) is known as the numerator, representing a part of the whole.
• \( b \) is the denominator, representing the total sections the whole is divided into.

Fractions can be manipulated to reveal equivalent meanings or simplify numbers to aid understanding. For example, the fraction \( \frac{4}{8} \) simplifies to \( \frac{1}{2} \) because both the numerator and the denominator can be divided by the same number, 4, simplifying the concept without changing the value.In contrast, irrational numbers do not have fractional representations. This is because their decimal expansions do not repeat or terminate; hence, they can't be neatly expressed as fractions. Understanding this distinction is crucial to grasp the diverse nature of numbers within mathematics.