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Identifying Rational Numbers

Write each of the following rational numbers as either a terminating or repeating decimal.

a◯−57\text{\textcircled{a}} \enspace -\frac{5}{7} \quad b◯155\text{\textcircled{b}} \enspace \frac{15}{5} \quad c◯1325\text{\textcircled{c}} \enspace \frac{13}{25}

Solution

Write each fraction as a decimal by dividing the numerator by the denominator.

a◯−57=−0.714285‾\text{\textcircled{a}} \enspace -\frac{5}{7}=-0.\overline{714285}, a repeating decimal

b◯155=3\text{\textcircled{b}} \enspace \frac{15}{5}=3 (or 3.0), a terminating decimal

c◯1325=0.52\text{\textcircled{c}} \enspace \frac{13}{25}=0.52, a terminating decimal

Case

The above case, including the title and solution, is attributed to College Algebra 2e. Page 9.

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 n≠0}\{\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​

Decimal​

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

A decimal numeral (also often just decimal or, less correctly, decimal number), refers generally to the notation of a number in the decimal numeral system. Decimals may sometimes be identified by a decimal separator (usually "." or "," as in 25.9703 or 3,1415).

The decimal numeral system (also called the base-ten positional numeral system and denary /ˈdiːnəri/[1] or decanary) is the standard system for denoting integer and non-integer numbers.

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

Repeating Decimal​

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

A repeating decimal or recurring decimal is a decimal representation of a number whose digits are periodic (repeating its values at regular intervals) and the infinitely repeated portion is not zero. It can be shown that a number is rational if and only if its decimal representation is repeating or terminating (i.e. all except finitely many digits are zero). For example, the decimal representation of 13\frac{1}{3} becomes periodic just after the decimal point, repeating the single digit "3" forever, i.e. 0.333.... A more complicated example is 3227555\frac{3227}{555}, whose decimal becomes periodic at the second digit following the decimal point and then repeats the sequence "144" forever, i.e. 5.8144144144.... At present, there is no single universally accepted notation or phrasing for repeating decimals. Another example of this is 59353\frac{593}{53}, which becomes periodic after the decimal point, repeating the 13-digit pattern "1886792452830" forever, i.e. 11.18867924528301886792452830....

Last accessed

January 01, 2024

Terminating Decimal​

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

A decimal that only has a finite number of digits after the decimal seperator.

Last accessed

January 01, 2024

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