Problem 46 - Calculus Early Transcendentals: Pearson New International Edition

Question

What is the smallest positive integer? The smallest positive rational number? The smallest positive irrational number?

Step-by-Step Solution

Answer

Smallest positive integer is 1; no smallest positive rational or irrational number.

Step 1: Understanding Positive Integers

Positive integers are all the whole numbers greater than zero. They start from 1 and increase without limit (1, 2, 3, ...). Since 1 is the first number we encounter in this sequence, it is the smallest positive integer.

Step 2: Understanding Positive Rational Numbers

A rational number is any number that can be expressed as the quotient or fraction of two integers, where the denominator is not zero. For positive rational numbers, both the numerator and denominator must be positive. Given any small fraction like \( \frac{1}{n} \) where \( n \) is a large positive integer, we can always find a smaller fraction by increasing \( n \). Thus, there is no smallest positive rational number, as they get infinitely close to zero.

Step 3: Understanding Positive Irrational Numbers

An irrational number cannot be expressed as a simple fraction and is not periodic. Positive irrational numbers are greater than zero. Like rational numbers, you can have values that are arbitrarily close to zero, such as very tiny segments of irrational numbers. Therefore, there is no smallest positive irrational number because, similar to rational numbers, they approach zero.

Key Concepts

Positive Integers

Positive integers are the simplest type of numbers most people learn. These include all whole numbers greater than zero, such as 1, 2, 3, and so on. Positive integers do not include zero or any negative numbers.
They are often referred to as natural numbers in many contexts. Imagine a counting method starting from one and going upwards without any end—infinity lies in the continuous progression of positive integers.
What makes them especially interesting in mathematics is their simplicity and their foundational role in number theory.

• The sequence of positive integers starts at 1.
• They increase without any limit.
• Each step in the sequence is a uniform increment by one.

In particular, when considering a set of positive integers, the number 1 is the smallest. It serves as the base for operations and understanding other sets of more complex numbers.

Positive Rational Numbers

Positive rational numbers include fractions and whole numbers greater than zero. These numbers can be expressed as the division of two positive integers, with the common notation being \( \frac{a}{b} \), where both \(a\) and \(b\) are positive integers and \(b eq 0\).
Positive rational numbers add a layer of versatility to early mathematics, as they allow expressions of quantities not only by wholes but also by parts.
What is fascinating about positive rational numbers is the idea that they can become very small, yet they can never reach zero.
This characteristic sets them apart from positive integers.

• Rational numbers include all numbers that have a precise fractional representation.
• Values can be positioned anywhere, even very close to zero, without being zero.
• The sequence or set of these numbers is dense, meaning between any two rational numbers, there is another rational number.

Due to this density, there is no 'smallest' positive rational number, as you can always find another positive rational number between zero and any given positive rational number. This infinite divisibility is a fundamental property of rational numbers.

Positive Irrational Numbers

Positive irrational numbers are those numbers greater than zero that cannot be accurately written as a fraction. Common examples include \(\sqrt{2}\), \(\pi\), and \(e\).
Unlike rational numbers, they are non-repeating and non-terminating when expressed as a decimal, which gives them a unique place in mathematics.
This lack of periodicity in their decimal expansion is one of the key differences from rational numbers.

• These numbers can't be expressed as a finite or repeating decimal.
• Just like rational numbers, they can be infinitely close to zero without ever being zero.
• They offer continuity and completion to the real number line, filling in gaps left by rational numbers.

You might wonder, "Why isn't there a smallest positive irrational number?" The answer lies in their continuous nature: for every positive irrational number, there exists another positive irrational number that is smaller yet still greater than zero.
Therefore, much like rational numbers, they never settle on an endpoint or smallest value, reflecting the boundless nature of the number line.