Is -1/2 A Rational Number

Rational Numbers: Definition, Types, Solved Examples

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Rational Numbers: Rational Numbers are the numbers that can be expressed in the form of p/q or in betwixt two integers where q is not equal to zero (q ≠ 0). The set of rational numbers besides contains the gear up of integers, fractions, decimals, and more. All the numbers that can exist expressed in the course of a ratio where the denominator is not one are referred to as rational numbers. For example, 9 is a rational number since it can be expressed as 9/1. Fifty-fifty 0 is a rational number. Some of the other examples are 2/3, v/two, 5.907, and more than.

Students will learn the different types of rational numbers and their related data and calculation in this article. This article also discusses the concept of Rational Numbers in detail, including their definition, classifications, and solved examples. Read on to know more nearly this topic.

What is a Rational Number?

Every rational number is defined as a number that tin be expressed in the grade of \(\frac{{\rm{p}}}{{\rm{q}}},\) where \(p\) and \(q\) are co-prime integers and \({\rm{q}} \ne 0.\)

Numerator and Denominator: In the given form \(\frac{{\rm{p}}}{{\rm{q}}},\) the integer \(p\) is the numerator and the integer \({\rm{q\;}}\left( { \ne 0} \right)\) is the denominator. So, in \(\frac{{ – 3}}{7}\) the numerator is \( – 3\) and the denominator is \(7.\)

Rational Numbers Examples

Some Examples of Rational Numbers are:

\(p\)	\(q\)	\(\frac{p}{q}\)	Rational
\(1\)	\(ii\)	\(\frac{one}{2}\)	Aye
\( – 3\)	\(4\)	\(\frac{{ – 3}}{4}\)	Yes
\(\;0.3\)	\(1\)	\(\frac{3}{{10}}\)	Yes
\( – 0.vii\)	\(1\)	\(\frac{{ – seven}}{{10}}\)	Yeah
\(0.141414 \ldots .\)	\(i\)	\(\frac{{14}}{{99}}\)	Aye

How to Place a Rational Number?

We know that a rational number can be expressed as a fraction or an integer. Every integer is a rational number. Each of these numbers is considered a rational number. At present, to identify whether the given number is a rational number or not we demand to cheque with the following conditions:

1. We can represent the number as a fraction of integers like \(\frac{p}{q},\) where \(q \ne 0.\)
2. The ratio \(\frac{p}{q}\) can exist simplified and represented in the decimal course which is either terminating or not-terminating recurring.

Types of Rational Numbers

1. Positive rational numbers: \(\frac{2}{5},\;0.two,\;6,\) are some examples of positive rational numbers. Here \(0.2\) can be written as \(\frac{1}{v}\) and \(6\) can be written as \(\frac{half-dozen}{1}.\)
2. Negative rational numbers: \( – \frac{two}{7},\; – 0.5,\; – 8,\) are some examples of negative rational numbers. Here \( – 0.5\) tin be written as \(\frac{one}{two}\) and \( – 8\) can be written as \( – \frac{8}{ane}.\)
3. Integer form of rational number: As we know that all integers are rational numbers because we can write them in the form of \(\frac{{\rm{p}}}{{\rm{q}}},\) where \(p\;\) and \(q\) are co-prime integers and \({\rm{q}} \ne 0.\) Example, \(6\) can be written as \(\frac{6}{one}.\)
4. Decimal class of rational number: Terminating and not-terminating recurring decimal numbers are rational numbers. Instance, \(0.3\) is terminating decimal number which tin be written equally \(\frac{3}{{10}}\) and \(0.33333 \ldots \) is a non-terminating recurring decimal number which can be written every bit \(\frac{1}{3}.\)

Standard Class of Rational Numbers

Let us observe the rational numbers: \(\frac{iii}{5},{\rm{\;}}\frac{{ – 5}}{eight},{\rm{\;}}\frac{2}{7},{\rm{\;}}\frac{{ – vii}}{{11}}.\)

The denominators of those rational numbers are positive integers and \(1\) is the just common cistron between the numerators and denominators. Farther, the negative sign occurs only in the numerator. These rational numbers are said to be in standard form or simplest class or lowest form.

Definition: A rational number is claimed to exist in its standard class if its denominator is a positive integer and therefore, the numerator and denominator do not accept whatever common gene other than \(1.\)

If yous remember the method of reducing the fractions to their lowest forms, we split the numerator and the denominator of the fraction by the same nonzero positive integer. We shall use an equivalent method for reducing the rational numbers to their standard form.

Case: Reduce to standard form \(\frac{{36}}{{ – 24}}.\)

Solution: So, the HCF of the numbers \(36\) and \(24\) is \(12.\). So, the standard form tin can be obtained by dividing the given fraction by \( – 12.\)

\(\frac{{36}}{{ – 24}} = \frac{{36 \div \left( { – 12} \right)}}{{ – 24 \div \left( { – 12} \right)}} = \frac{{ – 3}}{2}\)

Positive and Negative Rational Numbers

Rational numbers tin can exist differentiated as positive and negative rational numbers. When the numerator and the denominator are both positive or negative, it's referred to as a positive rational number. When i of the numerators or the denominator is a positive integer, and the other is a negative integer, information technology is called a negative rational number.

Positive Rational Numbers	Negative Rational Numbers
When both the numerators and the denominators are of the same sign so it is called a positive rational number. Example: \(\frac{three}{viii}\) is a positive rational number.	When both the numerator and the denominator are of different signs then they are known equally negative rational numbers. Example: \(\frac{{ – 8}}{9}\) is a negative rational number.
All the numbers are greater than zero.	All the numbers are less than zero.

Properties of Rational Numbers

A rational number is the subset of the real number which volition obey all the backdrop of the real number system. A few of the important properties are as follows:

Whenever we multiply, add, subtract, or split any 2 rational numbers the consequence is in e'er a rational number.
Rational number remains the same when nosotros divide or multiply the numerator and the denominator with same number.
When nosotros add together nothing to whatsoever rational number, we go the same number every bit the result.
Rational numbers are closed under the subtraction, addition, and multiplication.

Are Integers Too Rational Numbers?

What are Equivalent Rational Numbers

A rational number tin be written using different numerators and denominators. For example: we will take the rational number \(\frac{{ – 2}}{3}.\)
\(\frac{{ – two}}{iii} = \frac{{ – 2 \times two}}{{iii \times 2}} = \frac{{ – 4}}{vi}.\) We can see that \(\frac{{ – two}}{3}\) is same as \(\frac{{ – 4}}{6}.\)

As well, \(\frac{{ – 2}}{three} = \frac{{\left( { – ii} \right) \times \left( { – 5} \right)}}{{3 \times \left( { – 5} \right)}} = \frac{{x}}{{ – 15}}.\) So, \(\frac{{ – 2}}{3}\) is besides same as \(\frac{{10}}{{ – 15}}.\)
Thus, \(\frac{{ – two}}{3} = \frac{{ – four}}{half-dozen} = \frac{{10}}{{ – 15}},\)
Then such rational numbers that are equal are known as equivalent rational numbers.

Arithmetics Operations on Rational Numbers

As y'all already know how to add, decrease, multiply or divide the integers as well equally fractions. Now, allow us run across these basic operations on rational numbers.

Addition: When calculation the rational numbers with the same denominator, add merely the numerators by keeping the denominator the aforementioned. Ii rational numbers with the different denominators are added by taking out the LCM of the 2 denominators and and then convert both the rational numbers to their equivalent forms to have the LCM as the denominator.

Instance: \(\frac{{ – two}}{3} + \frac{three}{8} = \frac{{ – 16}}{{24}} + \frac{9}{{24}} = \frac{{ – sixteen + 9}}{{24}} = \frac{{ – 7}}{{24}}.\) So, the LCM of the numbers \(iii\) and \(8\) is \(24.\)

Subtraction: While subtracting two rational numbers, we add together the additive changed of the rational number that's being subtracted from the other rational number.

Example: \(\frac{7}{8} – \frac{2}{3} = \frac{7}{8} + \) additive inverse of \(\frac{two}{3} = \frac{7}{8} + \frac{{\left( { – 2} \right)}}{3} = \frac{{21 + \left( { – sixteen} \right)}}{{24}} = \frac{5}{{24}}.\)

Multiplication: To multiply the two rational numbers, multiply their numerators and denominators separately and write the product as \(\frac{{{\rm{product\;of\;numerators}}}}{{{\rm{product\;of\;denominators}}}}.\)

When you lot multiply the rational number with a positive integer, then multiply the numerator by that integer, by keeping the denominator unchanged.

Here, multiply a rational number past the negative integer:

\(\frac{{ – ii}}{ix} \times \left( { – 5} \right) = \frac{{ – 2 \times \left( { – v} \correct)}}{9} = \frac{{ten}}{nine}\)

Division: To dissever ane rational number past the other non-zero rational number, nosotros multiply the rational number by the reciprocal of the other rational number.

Thus \(\frac{{ – 7}}{2} \div \frac{4}{3} = \frac{{ – 7}}{2} \times \left( {{\rm{reciprocal\;of}}\;\frac{4}{3}} \right) = \frac{{ – 7}}{two} \times \frac{three}{4} = \frac{{ – 21}}{8}.\)

Rational Numbers on a Number Line

Here, we will learn how to represent numbers on the number line, so let us draw a number line.

Then, hither the points to the correct of the \(0\) are denoted by \( + \) sign and are positive numbers. The point to the left of \(0\) are denoted by \( – \) sign and are negative numbers.

So, correspond the number \( – \frac{i}{2}\) on the number line.

As rational number \( – \frac{1}{2}\) is negative it will be marked on the left side of the \(0.\)

And then, while marking the integers on the number line, successive integers are marked at equal intervals. Too, from \(0\) the pair \(1\) and \( – 1\) is at same altitude. And so, are the pairs \(2\) and \( – ii,3\) and \( – 3\) and so on.

Similarly, the rational numbers \(\frac{ane}{2}\) and \( – \frac{ane}{ii}\) would be at equal distance from \(0.\) Now, yous know how to mark the rational number \(\frac{1}{2}.\) This is marked at a betoken which is one-half the distance between \(0\) and \(1\) So, \( – \frac{one}{ii}\) is marked at a point one-half the distance between \(0\) and \( – 1.\)

Now, try to marker \( – \frac{three}{2}\) on the number line. It lies on the left side of \(0\) and is the same altitude as \(\frac{three}{2}\) from \(0.\) In decreasing order, we have \(\frac{{ – 1}}{2},{\rm{\;\;}}\frac{{ – 2}}{two}\left( { = – 1} \correct),{\rm{\;\;}}\frac{{ – 3}}{2},{\rm{\;\;}}\frac{{ – 4}}{ii}\left( { = – 2} \right),\) which shows that \(\frac{{ – 3}}{2}\) lies between \( – one\) and \( – 2.\) Thus, \(\frac{{ – 3}}{2}\) lies halfway between \( – 1\) and \( – ii.\)

Rational numbers with different denominators

All the other rational numbers which are with different denominators tin can be represented in the same manner.

Relation Between Rational Number and Irrational Number

We know that the numbers which are not rational are known as irrational numbers. The comparison between the rational and the irrational number has been given below:

Rational Numbers	Irrational Numbers
Rational numbers are the numbers that can be expressed equally fractions of integers. Examples: \(0.75,{\rm{\;\;}}\frac{{ – 31}}{5}\)	Irrational numbers are numbers that cannot be expressed as fractions of integers. Example: \(\surd 2,\pi .\)
These numbers tin be terminating decimals.	These numbers tin can never exist terminating decimals.
Rational numbers can be non-terminating decimals with repetitive patterns of decimals.	Irrational numbers e'er have non-terminating decimal expansions with no repetitive patterns of decimals.
The set of rational numbers contains natural numbers, whole numbers, and integers.	The gear up of irrational numbers is a separate set up that does not comprise any of the other sets of numbers.

Look at the given diagram for a better understanding.

Multiplicative Inverse of Rational Number

A rational number \(\frac{q}{p}\) is known as multiplicative changed or reciprocal of \(\frac{p}{q}\) and is denoted by \({\left( {\frac{p}{q}} \right)^{ – 1}}….\) The numbers \(1\) and \( – one\) are the simply rational numbers that are their ain reciprocals. No other rational number is its ain reciprocal. Rational number \(0\) has no multiplicative changed.
Example: Number is \(\frac{ii}{eight}\) its multiplicative inverse is \(\frac{8}{2}.\)

How to Identify the Rational Number Between Two Rational Numbers?

The number of integers between two integers is express (finite). Will the same happen in the case of rational numbers besides? We volition see that below.

He converted them into rational numbers with the same denominators.
So, \(\frac{{ – iii}}{5} = \frac{{ – ix}}{{xv}}\) and \(\frac{{ – one}}{3} = \frac{{ – 5}}{{15}}\)

We take \(\frac{{ – 9}}{{15}} < \frac{{ – 8}}{{15}} < \frac{{ – 7}}{{fifteen}} < \frac{{ – 6}}{{15}} < \frac{{ – 5}}{{xv}}\) or \(\frac{{ – 3}}{5} < \frac{{ – viii}}{{15}} < \frac{{ – seven}}{{15}} < \frac{{ – 6}}{{xv}} < \frac{{ – 1}}{three}\)

He could observe rational numbers \(\frac{{ – 8}}{{fifteen}} < \frac{{ – 7}}{{15}} < \frac{{ – 6}}{{fifteen}}\) between \(\frac{{ – 3}}{5}\) and \(\frac{{ – 1}}{3}.\)

Are the numbers \(\frac{{ – 8}}{{fifteen}},{\rm{\;}}\frac{{ – seven}}{{15}},{\rm{\;}}\frac{{ – vi}}{{fifteen}}\) the only rational numbers between \( – \frac{three}{5}\) and \( – \frac{one}{3}?\)

Nosotros have \(\frac{{ – 3}}{5} = \frac{{ – xviii}}{{30}}\) and \(\frac{{ – eight}}{{15}} = \frac{{ – 16}}{{xxx}}\)

And \(\frac{{ – 18}}{{thirty}} < \frac{{ – 17}}{{30}} < \frac{{ – 16}}{{30}}.\) i.east., \(\frac{{ – 3}}{5} < \frac{{ – 17}}{{30}} < \frac{{ – 8}}{{xv}}\)

Hence, \(\frac{{ – 3}}{v} < \frac{{ – 17}}{{30}} < \frac{{ – eight}}{{xv}} < \frac{{ – 7}}{{15}} < \frac{{ – half dozen}}{{15}} < \frac{{ – 1}}{3}\)

And then, he could find one more rational number betwixt \(\frac{{ – iii}}{5}\) and \(\frac{{ – 1}}{3}.\)

By using the above method, we tin can insert as many rational numbers every bit we want between two different rational numbers. We tin observe an unlimited number of rational numbers between any two rational numbers.

Solved Examples

one. Notice iv rational numbers equivalent to the given rational number: \(\frac{3}{four}\)
Solution: We have\(\frac{3}{4} = \frac{{three \times 2}}{{4 \times 2}} = \frac{{three \times 3}}{{4 \times 3}} = \frac{{three \times 4}}{{four \times 4}} = \frac{{3 \times 5}}{{4 \times five}}\)
Therefore, \(\frac{3}{4} = \frac{half-dozen}{8} = \frac{ix}{{12}} = \frac{{12}}{{16}} = \frac{{15}}{{20}}\)
Thus, the four rational numbers equivalent to \(\frac{3}{4}\) are \(\frac{6}{8},\;\frac{9}{{12}},\;\frac{{12}}{{sixteen}}\) and \(\frac{{15}}{{20}}.\)

2. Write each of the following rational numbers with positive denominators:
\(\frac{3}{{ – viii}},\;\frac{7}{{ – 12}},\;\frac{{ – v}}{{ – two}},\;\frac{{ – 13}}{{ – 8}}\)
Solution: We have \(\frac{3}{{ – eight}} = \frac{{iii \times \left( { – 1} \right)}}{{\left( { – 8} \right) \times \left( { – 1} \right)}} = \frac{{ – 3}}{eight};\),
\(\frac{7}{{ – 12}} = \frac{{7 \times \left( { – 1} \right)}}{{\left( { – 12} \correct) \times \left( { – one} \right)}} = \frac{{ – 7}}{{12}};\)
\(\frac{{ – 5}}{{ – 2}} = \frac{{\left( { – five} \correct) \times \left( { – one} \right)}}{{\left( { – 2} \right) \times \left( { – 1} \right)}} = \frac{5}{2};\)
\(\frac{{ – 13}}{{ – eight}} = \frac{{\left( { – thirteen} \right) \times \left( { – one} \correct)}}{{\left( { – 8} \right) \times \left( { – one} \correct)}} = \frac{{13}}{8}\)
3. Express \(\frac{{ – five}}{{13}}\) as a rational number with positive numerator.
Solution: We have:
\(\frac{{ – 5}}{{13}} = \frac{{\left( { – v} \right) \times \left( { – i} \right)}}{{xiii \times \left( { – 1} \correct)}} = \frac{5}{{ – 13}}\)
iv. Limited \(\frac{{ – four}}{seven}\) equally a rational number with (i) numerator \(\; = – 12\) and (ii) numerator \( = xx.\)
Solution: (i) Numerator of \(\frac{{ – 4}}{7}\) is \( – 4\)
By what number should nosotros multiply \(\left( { – 4} \right)\) to get \(\left( { – 12} \right)\)?
Clearly such number is \(\left( { – 12} \right) \div \left( { – iv} \right) = 3\)
Then, we multiply its numerator and denominator by \(3.\)
Therefore, \(\frac{{ – 4}}{7} = \frac{{\left( { – 4} \right) \times 3}}{{7 \times 3}} = \frac{{ – 12}}{{21}}.\)
Hence, \(\frac{{ – 4}}{7} = \frac{{ – 12}}{{21}}\)
(ii) Numerator of \(\frac{{ – 4}}{7}\) is \( – 4.\)
By what number should we multiply \(\left( { – 4} \right)\) to get \(20\)?
Conspicuously, such number is \(\left( {20} \correct) \div \left( { – four} \right) = – five.\)
Therefore, \(\frac{{ – 4}}{7} = \frac{{\left( { – 4} \right) \times \left( { – 5} \right)}}{{vii \times \left( { – 5} \right)}} = \frac{{20}}{{ – 35}}.\)
Hence, \(\frac{{ – 4}}{seven} = \frac{{20}}{{ – 35}}.\)
5. Find \(x\) such that \(\frac{{ – three}}{viii}\) and \(\frac{x}{{ – 24}}\) are equivalent rational numbers.
Solution: It is given that \(\frac{{ – 3}}{eight} = \frac{x}{{ – 24}}\)
Therefore, \(\frac{{ – 3}}{8} = \frac{x}{{ – 24}}\)
So, \(8 \times x = \left( { – 3} \right) \times \left( { – 24} \correct)\)
So, \(8 \times x = 72\)
So, \(x = \frac{{72}}{viii} = nine\)
Hence, \(ten = 9\)

FAQs on Rational Numbers

Nosotros take answered the most often asked questions on Rational Numbers below:

Question ane: Is \(7\) a rational number?
Answer: \(vii\) is a rational number as it can be written equally \(\frac{7}{1}.\)

Question 2: Is \(0\) a rational number?
Answer: A rational number is defined every bit the number that can be expressed in the form of \(\frac{{\rm{p}}}{{\rm{q}}},\) where \(p\) and \(q\) are integers and \({\rm{q}} \ne 0.\) We can limited \(0\) every bit \(\frac{0}{1}\) which is in the course of \(\frac{p}{q}\) where \(p\) is equal to zero and \(q\) is \(i\) or whatever integer.

Question 3: What is an irrational number?
Reply: An irrational number is a real number which cannot be written in the course of \(\frac{{\rm{p}}}{{\rm{q}}},\) where \(p\) and \(q\) are co-prime integers and \({\rm{q}} \ne 0.\) Example of irrational numbers are \(\pi\), ratio of a circumvolve's circumference to its bore, Euler'due south number due east, the golden ratio \({\rm{\varphi }},\) and the foursquare root of two. All square roots of natural numbers are, other than perfect squares, are irrational.

Question 5: Is \(3.14\) a Rational number?
Respond: Aye, the number \(3.14 = \frac{{314}}{{100}}\) is a rational number.

Question 6: Is \(\frac{1}{3}\) a rational or irrational number?
Respond: \(\frac{1}{3}\) is a rational number.

Question 7: How to identify the rational numbers?
Answer: To identify whether the given number is a rational number or not we need to check if we can represent the number in the form of \(\frac{{\rm{p}}}{{\rm{q}}},\) where \(p\) and \(q\) are co-prime integers and \({\rm{q}} \ne 0.\)

\(p\)	\(q\)	\(\frac{p}{q}\)	Rational
\(one\)	\(2\)	\(\frac{i}{2}\)	Yes
\( – 3\)	\(4\)	\(\frac{{ – three}}{4}\)	Yes
\(\;0.3\)	\(one\)	\(\frac{iii}{{10}}\)	Yep
\( – 0.7\)	\(one\)	\(\frac{{ – 7}}{{10}}\)	Yes
\(0.141414 \ldots .\)	\(1\)	\(\frac{{xiv}}{{99}}\)	Yes

Besides Bank check,

Now yous have detailed data on Rational Numbers. When you prepare for exams, ensure that you lot understand all the concepts, topics, and chapters on time. Embibe provides NCERT Solutions For Class 8 Maths Chapter 1 and NCERT Books For Class 8 Maths, both of which will provide ample understanding and do for Rational Numbers. Yous can also take Form 8 Maths Mock Test to amend your score in Mathematics.

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