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Writing Integers as Rational Numbers

Write each of the following as a rational number.

a7\text{\textcircled{a}} \enspace 7 \quad b0\text{\textcircled{b}} \enspace 0 \quad c8\text{\textcircled{c}} \enspace -8

Solution

Write a fraction with the integer in the numerator and 1 in the denominator.

a7=71\text{\textcircled{a}} \enspace 7=\frac{7}{1} \quad b0=01\text{\textcircled{b}} \enspace 0=\frac{0}{1} \quad c8=81\text{\textcircled{c}} \enspace -8=-\frac{8}{1}

Case

The above case, including the title and solution, is attributed to College Algebra 2e. Page 8.

Q&A

What are natural numbers?

Page 8, College Algebra 2e

The numbers we use for counting, or enumerating items, are the natural numbers: 1,2,3,4,5,1, 2, 3, 4, 5, and so on. We describe them in set notation as {1,2,3,}\{1,2,3,\mathellipsis\} where the ellipsis ()(\mathellipsis) indicates that the numbers continue to infinity. The natural numbers are, of course, also called the counting numbers. Any time we enumerate the members of a team, count the coins in a collection, or tally the trees in a grove, we are using the set of natural numbers.

What are the set of whole numbers?

Page 8, College Algebra 2e

The set of whole numbers is the set of natural numbers plus zero: {0,1,2,3,}\{0,1,2,3,\mathellipsis\} .

What are the set of integers?

Page 8, College Algebra 2e

The set of integers adds the opposites of the natural numbers to the set of whole numbers: {,3,2,1,0,1,2,3,}\{\mathellipsis,-3,-2,-1,0,1,2,3,\mathellipsis\} . It is useful to note that the set of integers is made up of three distinct subsets: negative integers, zero, and positive integers. In this sense, the positive integers are just the natural numbers. Another way to think about it is that the natural numbers are a subset of the integers.

,3,2,1negative integers0zero1,2,3,positive integers\stackrel{\text{negative integers}}{\mathellipsis,-3,-2,-1} \quad \stackrel{\text{zero}}{0} \quad \stackrel{\text{positive integers}}{1,2,3,\mathellipsis}

What are the set of rational numbers?

Page 8, College Algebra 2e

The set of rational numbers is written as {mn | m and n are integers and n0}\{\frac{m}{n} \text{ | } m \text{ and } n \text{ are integers and } n \not = 0 \} . Notice from the definition that rational numbers are fractions (or quotients) containing integers in both the numerator and the denominator, and the denominator is never 0. We can also see that every natural number, whole number, and integer is a rational number with a denominator of 1.

Because they are fractions, any rational number can also be expressed in decimal form. Any rational number can be represented as either:

  • a terminating decimal:158=1.875\frac{15}{8} = 1.875, or
  • a repeating decimal: 411=0.36363636=0.36\frac{4}{11} = 0.36363636\mathellipsis=0.\overline{36}

We use a line drawn over the repeating block of numbers instead of writing the group multiple times.

Glossary

Denominator

https://en.wiktionary.org/wiki/denominator

The number or expression written below the line in a fraction (such as 2 in ½).

Last accessed

December 26, 2023

Fraction

https://en.wikipedia.org/wiki/Fraction

A fraction represents a part of a whole or, more generally, any number of equal parts.

Last accessed

December 26, 2023

Integer

https://en.wikipedia.org/wiki/Integer

An integer is the number zero (0), a positive natural number (1, 2, 3, etc.) or a negative integer (−1, −2, −3, etc.).

Last accessed

December 26, 2023

Numerator

https://en.wiktionary.org/wiki/numerator

The number or expression written above the line in a fraction (such as 1 in ½).

Last accessed

December 26, 2023

Rational Number

https://commons.wikimedia.org/wiki/Category:Rational_numbers

A rational number is any number that can be expressed as the quotient or fraction pq\frac{p}{q} of two integers, with the denominator qq not equal to zero. Since qq may be equal to 1, every integer is a rational number. This category represents all rational numbers, that is, those real numbers which can be represented in the form: a b ab\frac{a}{b} ...where aa and bb are integers and bb is not equal to zero. All integers are rational, including zero.

Last accessed

December 26, 2023

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