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Differentiating Rational and Irrational Numbers

Determine whether each of the following numbers is rational or irrational. If it is rational, determine whether it is a terminating or repeating decimal.

a◯25\text{\textcircled{a}} \enspace \sqrt{25} \quad b◯339\text{\textcircled{b}} \enspace \frac{33}{9} \quad c◯11\text{\textcircled{c}} \enspace \sqrt{11} \quad d◯1734\text{\textcircled{d}} \enspace \frac{17}{34} \quad e◯0.3033033303333…\text{\textcircled{e}} \enspace 0.3033033303333\mathellipsis

Solution

a◯25\text{\textcircled{a}} \enspace \sqrt{25} : This can be simplified as 25=5\sqrt{25}=5. Therefore, 25\sqrt{25} is rational.

b◯339\text{\textcircled{b}} \enspace \frac{33}{9} : Because it is a fraction of integers, 339\frac{33}{9} is a rational number. Next, simplify and divide. 339=331193=113=3.6ˉ\frac{33}{9}=\frac{\overset{11}{{\cancel{33}}}}{\underset{3}{\cancel{9}}}=\frac{11}{3}=3.\bar{6} So, 339\frac{33}{9} is rational and a repeating decimal.

c◯11\text{\textcircled{c}} \enspace \sqrt{11} : This cannot be simplified any further. Therefore, 11\sqrt{11} is an irrational number.

d◯1734\text{\textcircled{d}} \enspace \frac{17}{34} : Because it is a fraction of integers, 1734\frac{17}34{} is a rational number. Simplify and divide. 1734=171342=12=0.5\frac{17}{34}=\frac{\overset{1}{{\cancel{17}}}}{\underset{2}{\cancel{34}}}=\frac{1}{2}=0.5 So, 1734\frac{17}{34} is rational and a terminating decimal.

e◯0.3033033303333…\text{\textcircled{e}} \enspace 0.3033033303333\mathellipsis is not a terminating decimal. Also note that there is no repeating pattern because the group of 3s increases each time. Therefore it is neither a terminating nor a repeating decimal and, hence, not a rational number. It is an irrational number.

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.

What are irrational numbers?​

Page 9, College Algebra 2e

At some point in the ancient past, someone discovered that not all numbers are rational numbers. A builder, for instance, may have found that the diagonal of a square with unit sides was not 2 or even 32\frac{3}{2} but was something else. Or a garment maker might have observed that the ratio of the circumference to the diameter of a roll of cloth was a little bit more than 3, but still not a rational number. Such numbers are said to be irrational because they cannot be written as fractions. These numbers make up the set of irrational numbers. Irrational numbers cannot be expressed as a fraction of two integers. It is impossible to describe this set of numbers by a single rule except to say that a number is irrational if it is not rational. So we write this as shown. {h | h is not a rational number}\{h \text{ | } h \text{ is not a rational number}\}

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

Irrational Number​

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

All the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers.

Among irrational numbers are the ratio π\pi of a circle's circumference to its diameter, Euler's number ee, the golden ratio ϕ\phi, and the square root of two.[1] In fact, all square roots of natural numbers, other than of perfect squares, are irrational.

Last accessed

January 01, 2024

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

note