Problem 40 - Introductory and Intermediate Algebra for College Students 4th

Question

Give an example of a rational number that is not a natural number.

Step-by-Step Solution

Answer

-2 or 1/2 are examples of rational numbers that are not natural numbers.

Step 1: Understanding definitions

Let's first clarify what a rational number is and what a natural number is. A rational number is a number that can be written as a fraction \(p/q\), where \(p\) and \(q\) are integers and \(q\) ≠ 0. Natural numbers, on the other hand, are the set of positive integers (1, 2, 3, ...).

Step 2: Consider a negative integer

One way to find a rational number that is not a natural number is to consider negative integers. Any negative integer is a rational number, as it can be expressed as a ratio, eg. \(-2\) is a rational number because it can be written as \(-2/1\). But \(-2\) is not a natural number since even though it is an integer, natural numbers include only positive integers.

Step 3: Consider a fraction

Another valid solution is a fraction where the numerator is smaller than the denominator such as \(1/2\). \(1/2\) is a rational number as it can be expressed as the ratio of two integers, but it is not a natural number because natural numbers are positive integers and not fractions.