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

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

Step 1: Understanding Positive Integers

Positive integers are the numbers starting from 1 and increasing by 1 indefinitely. These are the counting numbers that start from 1. Thus, the smallest positive integer is the number 1.

Step 2: Identifying Positive Rational Numbers

A positive rational number is any number that can be expressed as a fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers, \( b eq 0 \) and the fraction is positive. There is no smallest positive rational number, because you can always find a smaller positive rational number by choosing a smaller numerator or larger denominator, such as \( \frac{1}{1000} \), \( \frac{1}{1000000} \), etc.

Step 3: Considering Positive Irrational Numbers

Positive irrational numbers are numbers that cannot be expressed as a simple fraction and are greater than zero. Examples include \( \sqrt{2} \), \( \pi \), and \( e \). Just like rational numbers, there is no smallest positive irrational number, because between any two positive irrational numbers, you can always find another irrational number, similar to how you can always find smaller positive rational numbers.

Key Concepts

Positive Integers

Positive integers are the basic counting numbers that we're all familiar with. They are part of the number set typically known as natural numbers. In mathematics, these numbers start at 1 and continue indefinitely, increasing by 1 each time.

Essentially, positive integers are:

• Whole numbers, without any decimal or fractional part
• Always greater than zero
• Examples include 1, 2, 3, 4, and so on

When considering the smallest positive integer, it's important to focus on the idea of 'positive' starting from the number 1. Hence, the smallest amongst these positive integers is indeed 1. This is because there isn't any positive integer that exists below 1, as 0 does not count under positive integers and negative numbers are opposite since they lack positiveness.

Positive Rational Numbers

Positive rational numbers might sound complex at first, but they are numbers that can be written as a fraction, where both the numerator and denominator are integers (with the denominator differing from zero).

Key characteristics of positive rational numbers include:

• They can be written in the form \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b eq 0 \).
• These numbers need to be positive, meaning both \( a \) and \( b \) need to be either positive or the fraction effectively becomes positive overall.
• Examples include \( \frac{1}{2} \), \( \frac{5}{3} \), or any fraction where the division results in a positive number.

One intriguing aspect of positive rational numbers is that there isn't a 'smallest' one. This might seem counterintuitive, but no matter how small a fraction you have, like \( \frac{1}{1000} \), you can always construct another, smaller positive rational number (for example, \( \frac{1}{1001} \) or even \( \frac{1}{1000000} \)). This infinite divisibility ensures there's no smallest integer in this set.

Positive Irrational Numbers

When we delve into positive irrational numbers, we're entering a realm of numbers that cannot be expressed just as a simple fraction. These numbers are important in understanding the precision and vastness of real numbers.

Let's break them down:

• A numbers that cannot be written in the form of \( \frac{a}{b} \) with integers \( a \) and \( b \), defying simple fractional representation.
• These numbers are positive, meaning they are greater than zero.
• Examples include the square root of non-perfect squares, \( \sqrt{2} \), and transcendental numbers like \( \pi \) and \( e \).

Just like with rational numbers, there is no smallest positive irrational number. Between any two irrational numbers, you can always find another one, just reinforcing that infinite density and the absence of a smallest positive irrational number. This characteristic is deeply rooted in their non-repeating and non-terminating decimal nature, making them endlessly divisible between each other.