Learning Objectives:
Know about the properties of Rational numbers.
Understand the Decimal representation of rational numbers.
Understand the terminating decimals and non terminating decimals.
Real time applications:
The word ‘real’ was introduced in the 17th century by Descartes.
In physical sciences, most of the physical constants are modeled using real numbers.
Real numbers satisfies the usual rules of arithmetic.
In our daily lives we interact with numbers when making calculations and timing.
Natural Numbers :
The numbers 1, 2, 3 ........... which are used in counting are called Natural
numbers (or) positive integers.
Whole Numbers: Natural numbers together with zero are called whole numbers
\( W = \{ 0,1,2,3, - - - - \} \)
\( N \cup \{ 0\} = w\)
Integers : Z = { ............, -3, -2, -1, 0, 1, 2, 3, 4, ........…
Rational Numbers: The numbers, that can be expressed in the form of p/q, where p and q are integers and are called rational numbers
\(Q = \left\{ {\frac{p} {q};p,q \in Z\& q \ne 0} \right\} \)
Ex:\( 1.\frac{4} {7},\frac{{ - 3}} {{10}},\frac{7} {5} \) etc are rational numbers.
2. The square root of every perfect square number is rational.
\( \sqrt 1 ,\sqrt 4 ,\sqrt 9 ,\sqrt {16} ,\sqrt {25} \)etc are all rational numbers.
3. 0 can be written as \( \frac{0} {1} \), which is rational.
4. \( \frac{1} {0} \) is not defined and therefore it is not a rational number.
Irrational number:
Every number which when expressed in decimal form is expressible as a non terminating and non-repeating decimal is called an irrational number.
1. Square root of every non perfect square natural number is irrational number etc.
2. If m is a positive integer which is not a perfect cube, then is an irrational number. Thus etc are all irrational numbers.
3. Every non-terminating and non-repeating decimal is an irrational number 0.10110111011110------ and 0.434434443---etc are irrational numbers
4. The value of is 3.1416----- which is a non-terminating and non repeating decimal.so is irrational.
Real Numbers :
All rational and all irrational numbers together form the set of all real numbers.
Equivalent rational numbers: If p/q is a rational number and ‘n’ is non-zero integer,then
Representing Real numbers on number line:
Absolute Value : The absolute value of a rational number is the numerical value of the number regard to its sign.
\( \left| x \right| = x, \)if x > 0
\( \left| x \right| = 0, \)if x = 0
\( \left| x \right| = x, \)if x < 0
Eg: \( \left| { - 3} \right| = 3,\left| 5 \right| = 5,\left| 0 \right| = 0 \)
Standard form of a rational number :
A rational number is said to be in standard form it is in its lowest terms.
Comparision of Rational numbers :
1. While comparing positive rational numbers, with the same denominator, the number with the greatest numerator is the largest \( \frac{{36}} {{20}} > \frac{{30}} {{20}} > \frac{{26}} {{20}} \)
2. A positive rational number is always greater than a negative rational number \( \frac{6} {4} > \frac{{ - 6}} {4} \)
3. While comparing negative rational numbers with the same denominator compare there numerators ignoring the minus sign. The number with the greatest numerator is the smallest \( \frac{{ - 5}} {2} < \frac{{ - 3}} {2},\frac{{ - 6}} {7} < \frac{{ - 1}} {7} \)
4. Positive Rational numbers lie to the right of ‘0’ while negative rational numbers lie to the left of ‘0’ on the number line
5. To compare rational numbers with diffrent denominators , convert them into equalant rational numbers with the same denominator, which is equal to the L.C. M of there denominators
Difference between fraction and a rational number :
A fraction is a number of the form p/q where ‘p’ and ‘q’ are natural numbers.A Rational number is a number of the form p/q where ‘p’ and ‘q’ are integers and \( q \ne 0 \)
Properties of Rational numbers :
Under addition:
i) Closure : For any two rational numbers a and b , (a+b) is also a rational number
ii) Commutative : For any two rational numbers a and b a+b=b+a this property is called commutative.
iii) Associative : For any three rational numbers a,b,and c (a+b)+c =a+(b+c)
iv) Identity : For any rational number a, a+0=0+a=a
v) Inverse : a+(-a) =-a+a=0
Under multiplication:
i) Closure : For any two rational numbers a and b , (axb) is also a rational number.
ii) Commutative : For any two rational numbers a and b axb=bxa this property is called commutative.
iii) Associative : For any three rational numbers a,b,and c (axb)xc =ax(bxc)
iv) Identity : For any rational number a, ax1=1xa=a
v) Inverse : \( ax\left( {\frac{1} {a}} \right) = \left( {\frac{1} {a}} \right)Xa = 1 \)
vi) Distributive Law: \( a \times \left( {b + c} \right) = \left( {a \times b} \right) + \left( {a \times c} \right) \) or \( \left( {a + b} \right) \times c = \left( {a \times c} \right) + \left( {b \times c} \right) \)
Rational Numbers Properties
Since a rational number is a subset of the real numbers, the rational number will obey all the properties of the real number system. Some of the important properties of the rational numbers are as follows:
The results are always a rational number if we multiply, add or subtract any two rational numbers.
A rational number remains the same if we divide or multiply both the numerator and denominator with the same factor.
If we add zero to a rational number then we will get the same number itself.
Rational numbers are closed under addition, subtraction, and multiplication.
Additional Properties of Rational Numbers :
1) Law of Tricotomy: For every a,b, \( \in \) Q any one of the following is hold.
i) ab iii) a=b
This property of Rational numbers is called 'Law of Tricotomy'
2) Transitive property: For every a,b,c \( \in \) Q and if a>b and b>c then a>c
This property of rational numbers is called 'Transitive property'.
3) Density property: Between two rational numbers infinite number of rational numbers are there. This property of rational numbers is called Density property.
4) i) If a,b\( \in \)Q and c is positive integer and if a>b then (i) a+c > b+c
ii) If a,b\( \in \)Q and c is negative integer and if a>b then (i) a - c > b - c
5) i) If a,b\( \in \)Q and c is the positive integer if a>b then a.c>b.c also \( \frac{a} {c} > \frac{b} {c} \)
ii) If a,b\( \in \)Q and c is negative integer if a>b then a.c
Standard form of rational numbers
The standard form of a rational number can be defined if it’s no common factors aside from one between the dividend and divisor and therefore the divisor is positive.
For example, 12/36 is a rational number. But it can be simplified as 1/3; common factors between the divisor and dividend is only one. So we can say that rational number 1/3 is in standard form.
Positive and Negative Rational Numbers
As we know that the rational number is in the form of p/q, where p and q are integers. Also, q should be non-zero integer. The rational number can be either positive or negative. If the rational number is positive, both p and q are positive integers. If the rational number takes the form -(p/q) , then either p or q takes the negative value. It means that -(p/q)=(-p)/q=p/(-q).
Is 0 a rational number?
Yes, 0 is a rational number because it is an integer that can be written in any form such as 0/1, 0/2 where b is a non-zero integer. It can be written in the form : p/q = 0/1. Hence, we conclude that 0 is a rational number.
How to find the rational numbers between two rational numbers?
There are infinite numbers of rational numbers between two rational numbers. The rational numbers between two rational numbers can be found easily using two different methods. Now, let us have a look at the two different methods.
Method 1:
Find out the equivalent fraction for the given rational numbers and find out the rational numbers in between them. Those numbers should be the required rational numbers.
Ex : Find rational number between and .
Solution: Let \(
\frac{1}
{4}
\), \(
\frac{1}
{2}
\)
\(
\begin{gathered}
\Rightarrow \frac{{1 \times 2}}
{{4 \times 2}},\frac{{1 \times 4}}
{{2 \times 4}} \hfill \\
\Rightarrow \frac{2}
{8},\frac{4}
{8} \hfill \\
\end{gathered}
\)
\(
\frac{3}
{8}
\) is between \(
\frac{2}
{8}\& \frac{4}
{8}
\)
Method 2:
Find out the mean value for the two given rational numbers. The mean value should be the required rational number. In order to find more rational numbers, repeat the same process with the old and the newly obtained rational numbers.
Ex : Find rational number between \(
\frac{3}
{4}\& \frac{1}
{2}
\)
Solution : Rational number \(\frac{3}
{4}\& \frac{1}
{2}\) between is
\(
\frac{{\frac{3}
{4} + \frac{1}
{2}}}
{2} = \frac{{\frac{{3 + 2}}
{4}}}
{2} = \frac{5}
{8}
\)
\(
\therefore \frac{5}
{8}
\) lies between \(
\frac{3}
{4}\& \frac{1}
{2}
\)
Rational Numbers in Ascending Order
We will learn how to arrange the rational numbers in ascending order.
General method to arrange from smallest to largest rational numbers (increasing):
Step 1: Express the given rational numbers with positive denominator.
Step 2: Take the least common multiple (L.C.M.) of these positive denominator.
Step 3: Express each rational number (obtained in step 1) with this least common multiple (LCM) as the common denominator.
Step 4: The number having the smaller numerator is smaller.
Solved examples on rational numbers in ascending order:
1. Arrange the rational numbers in ascending order:
Solution:
We first write the given rational numbers so that their denominators are positive.
We have, \( \frac{5} {{ - 8}} = \frac{{5 \times \left( { - 1} \right)}} {{\left( { - 8} \right) \times \left( { - 1} \right)}} = \frac{{ - 5}} {8}and\frac{2} {{ - 3}} = \frac{{2 \times \left( { - 1} \right)}} {{\left( { - 3} \right) \times \left( { - 1} \right)}} = \frac{{ - 2}} {3} \)
Thus, the given rational numbers with positive denominators are \( \frac{{ - 7}} {{10}},\frac{{ - 5}} {8},\frac{{ - 2}} {3} \)
Now, LCM of the denominators 10, 8 and 3 is 120
We now write the numerators so that they have a common denominator 120 as follows:
\(
\frac{{ - 7}}
{{10}} = \frac{{\left( { - 7} \right) \times 12}}
{{10 \times 12}} = \frac{{ - 84}}
{{120}},
\)
\(
\frac{{ - 5}}
{8} = \frac{{\left( { - 5} \right) \times 15}}
{{8 \times 15}} = \frac{{ - 75}}
{{120}}
\) and \(
\frac{{ - 2}}
{3} = \frac{{\left( { - 2} \right) \times 40}}
{{3 \times 40}} = \frac{{ - 80}}
{{120}}
\)
Comparing the numerators of these numbers, we get, - 84 < -80 < -75
\(
\frac{{ - 84}}
{{120}} < \frac{{ - 80}}
{{120}} < \frac{{ - 75}}
{{120}} \Rightarrow \frac{{ - 7}}
{{10}} < \frac{{ - 2}}
{3} < \frac{{ - 5}}
{8} \Rightarrow \frac{{ - 7}}
{{10}} < \frac{2}
{{ - 3}} < \frac{5}
{{ - 8}}
\)
Hence, the given numbers when arranged in ascending order are: \(
\frac{{ - 7}}
{{10}},\frac{2}
{{ - 3}},\frac{5}
{{ - 8}}
\)
Arrange rational numbers in Descending order :
We will learn how to arrange the rational numbers in descending order.
General method to arrange from largest to smallest rational numbers (decreasing):
Step 1: Express the given rational numbers with positive denominator.
Step 2: Take the least common multiple (L.C.M.) of these positive denominator.
Step 3: Express each rational number (obtained in step 1) with this least common multiple (LCM) as the common denominator.
Step 4: The number having the greater numerator is greater.
Solved examples on rational numbers in descending order:
1. Arrange the numbers \(
\frac{{ - 3}}
{5},\frac{7}
{{ - 10}}
\) and \(
\frac{{ - 5}}
{8}
\) in descending order.
Solution:
First we write each of the given numbers with positive denominator.
We have; \(
\frac{7}
{{ - 10}} = \frac{{7 \times \left( { - 1} \right)}}
{{\left( { - 10} \right) \times \left( { - 1} \right)}} = \frac{{ - 7}}
{{10}}
\).
Thus, the given number are \(
\frac{{ - 3}}
{5},\frac{7}
{{ - 10}}
\) and \(
\frac{{ - 5}}
{8}
\)
L.C.M of 5, 10, 8 is 40.
Now, \(
\frac{{ - 3}}
{5} = \frac{{\left( { - 3} \right) \times 8}}
{{5 \times 8}} = \frac{{ - 24}}
{{40}};
\)
\(
\frac{{ - 7}}
{{10}} = \frac{{\left( { - 7} \right) \times 4}}
{{10 \times 4}} = \frac{{ - 28}}
{{40}}
\)
\(
\begin{gathered}
and\text{ }\frac{{ - 5}}
{8} = \frac{{\left( { - 5} \right) \times 5}}
{{8 \times 5}} = \frac{{ - 25}}
{{40}} \hfill \\
clearly\text{ }\frac{{ - 24}}
{{40}} > \frac{{ - 25}}
{{40}} > \frac{{ - 28}}
{{40}} \hfill \\
\frac{{ - 3}}
{5} > \frac{{ - 5}}
{8} > \frac{{ - 7}}
{{10}},i.e.,\frac{{ - 3}}
{5} > \frac{{ - 5}}
{8} > \frac{7}
{{ - 10}} \hfill \\
\end{gathered}
\)
Hence, the given numbers when arranged in descending order are \(
\frac{{ - 3}}
{5} > \frac{{ - 5}}
{8} > \frac{7}
{{ - 10}}
\)