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Chapter 1: Problem 17

decide whether each statement is true or false. Every integer is a rational number.

Short Answer

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Step by step solution

01

Understand Definitions

First, review the definitions of 'integer' and 'rational number'. An integer is any whole number, including negative numbers, zero, and positive numbers (e.g., -3, 0, 2). A rational number is any number that can be expressed as the quotient or fraction \(\frac{a}{b}\) where \(a\) and \(b\) are integers and \(b eq 0\).
02

Relate Integers to Rational Numbers

Express an integer as a fraction. Any integer \(n\) can be written as \(\frac{n}{1}\), where \(n\) is an integer and 1 is a non-zero integer. This fits the form of a rational number.
03

Make the Determination

Since any integer can be expressed as \(\frac{n}{1}\), which is a rational number, every integer is indeed a rational number. Therefore, the statement is true.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

integer definition
An integer is a fundamental concept in mathematics. Integers are the whole numbers that you are most familiar with: they can be positive, negative, or zero. For example, -3, 0, and 5 are all integers. One of the key features of integers is that they do not have fractional or decimal parts. This makes them very straightforward to work with.
rational number definition
In mathematics, a rational number is any number that can be expressed as the quotient or fraction of two integers, denoted as \(\frac{a}{b}\), where \(a\) (the numerator) and \(b\) (the denominator) are integers and \(b\) is not zero. This means rational numbers include all integers (since any integer \(n\) can be written as \(\frac{n}{1}\)), as well as any number that can be written as a fraction, like \(\frac{1}{2}\) or \(\frac{3}{4}\).
relationship between integers and rational numbers
The relationship between integers and rational numbers is quite straightforward. All integers are rational numbers. This is because any integer \(n\) can be expressed as \(\frac{n}{1}\). Since all rational numbers are defined as \(\frac{a}{b}\) where \(a\) and \(b\) are integers and \(b\) is not zero, it's clear that integers fit this definition perfectly. So, whenever you see an integer, remember it is also a rational number.
fraction representation of integers
One way to understand why integers are rational numbers is by representing them as fractions. For example, if you have the integer 5, you can represent it as \(\frac{5}{1}\). Here, 5 is the numerator, and 1 is the denominator. This fraction form confirms that 5 is a rational number. You can do this with any integer: \(\frac{-3}{1}\), \(\frac{0}{1}\), and so on. Representing integers as fractions is a helpful method for understanding their place within the broader category of rational numbers.

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Most popular questions from this chapter

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