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
\(\begin{align} & W=\left\{ 1,2,3,...... \right\} \\ & N\cup \{0\}=W \\ \end{align} \)
Integers : Integers are defined as the set of all whole numbers with a negative set of natural numbers. The integer set is represented by the symbol “Z”.
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 .
\(Q=\left\{ \frac{p}{q};p,q\in Z\And q\ne 0 \right\}\)
Ex :
1) \(\frac{4}{7},\frac{-3}{10},\frac{0}{1},\sqrt{1},\sqrt{4},\sqrt{9}\)
2) \(1\over 0\) is not defined & it is not rational number
Equivalent rational numbers: If p/q is a rational number and ‘n’ is non-zero integer,then \({p \over q}={p\times n \over q\times n}\)
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
\( {6 \over 4}> {-6 \over 4}\)
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 \( {-5 \over 2}< {-3 \over 2}, {-6 \over 7}< {-1 \over 7}\)
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 different denominators , convert them into equivalent rational numbers with the same denominator, which is equal to the L.C. M of their denominators
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
‘0’ is additive identity for any rational number
v) Inverse : a+(-a) =-a+a=0
‘-a’ is additive inverse of a.
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
‘1’ is multiplicative identity in rational numbers.
v) Inverse : \(a\times \left( \frac{1}{a} \right)=\left( \frac{1}{a} \right)\times a=1\)
‘\(1 \over a\)’ is multiplicative inverse of a.
vi) Distributive Law: \(a\times \left( b+c \right)=\left( a\times b \right)+\left( a\times c \right)(or)\left( a+b \right)\times c=\left( a\times c \right)+\left( b\times c \right)\)
Note:
1. The results are always a rational number if we multiply, add or subtract any two rational numbers.
2. A rational number remains the same if we divide or multiply both the numerator and denominator with the same factor.
Additional Properties of Rational Numbers :
1) Law of Tricotomy: For every a,b, 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 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
ii) If a,b \(\in\) Q and c is negative integer if a>b then a.c
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:
Ex : Find rational number between \(\frac{1}{4}\text{ and }\frac{1}{2}\text{ }\)
Method 2:
Mean value for the given two rational numbers lies between them which is required rational number.
If we want to find more numbers repeat the same process with the old & newly obtained rational number
Ex : Find rational number between \(\frac{3}{4}\And \frac{1}{2}\)
\(\begin{align} & \text{Solution:Rational number }\frac{3}{4}\And \frac{1}{2}~\text{is} \\ & \Rightarrow \frac{\frac{3}{4}+\frac{1}{2}~}{2}=\frac{\frac{3+2}{4}}{2}=\frac{5}{8} \\ & \frac{5}{8}\text{ lies between}~\frac{3}{4}\And \frac{1}{2} \\ \end{align} \)
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:
\(\begin{align} & \text{1}\text{. Arrange the rational numbers }\frac{-7}{10}\text{,}\frac{5}{-8}\text{and }\frac{2}{-3}\text{ in ascending order:} \\ & \frac{-7}{10}\text{,}\frac{-5}{8}\text{, }\frac{2}{-3}\text{ } \\ & \text{Now LCM of 10, 8, 3 is 120} \\ & \Rightarrow \frac{-7\times 12}{10\times 12}\text{,}\frac{-5\times 15}{8\times 15}\text{, }\frac{-2\times 40}{3\times 40} \\ & \Rightarrow \frac{-84}{120}\text{,}\frac{-75}{120}\text{, }\frac{-80}{120}\text{ } \\ & \text{Among the above} \\ & \frac{-84}{120}<\frac{-75}{120}\text{,}\frac{-80}{120}\text{ } \\ & \text{Ascending order is }\frac{-7}{10}\text{,}\frac{2}{-3},\frac{5}{-8} \\ \end{align} \)
In arranging the rational numbers in descending order. After expressing denominator as LCM as above case.
The number having the greater numerator is greater that should be written first and go from greater to smaller.
Descending order of \(\frac{-7}{10}\text{,}\frac{5}{-8}\text{and }\frac{2}{-3}\) is \(\text{}\frac{5}{-8},\frac{2}{-3} ,\frac{-7}{10}\)
Operations on rational numbers :
Addition : There are two possibilities
1. Rational numbers with same denominator
2. Rational numbers with different denominators.
Case1: Rational numbers having same denominator.
If \(p\over q\) and \({r \over q}(q>0)\) are two rational numbers then
\(\begin{align} & \mathbf{Ex}:\text{ }\mathbf{Add}\text{ }~\frac{7}{-11}\mathbf{and}~\frac{3}{11} \\ \\ \end{align} \)
\(\therefore \frac{7}{-11}+\frac{3}{11}=\frac{-7}{11}+\frac{3}{11}=\frac{\left( -7 \right)+3}{11}=\frac{-4}{11}\)
Case 2 : Rational numbers having different denominators:
1. Convert the denominator as common denominator.
2. Find LCM of their denominators and express each rational number with this LCM as their denominator, then add them as in case 1.
\(\begin{align} & \mathbf{Ex}:\text{ }\mathbf{Add}\text{ }~\frac{-3}{5}\mathbf{and}~\frac{2}{3} \\ & \therefore \frac{-3\times 3}{5\times 3}+~\frac{2\times 5}{3\times 5}\text{(LCM of 5 }\!\!\And\!\!\text{ 3 is 15)} \\ & \frac{-9}{15}+\frac{10}{15}=\frac{-9+10}{15}=\frac{1}{15} \\ \end{align}\)
Subtraction :
\(\begin{align} & \text{Ex 1: Subtract }\frac{5}{7}\text{ from }\frac{-3}{14} \\ & \frac{-3}{14}-\left( \frac{5}{7} \right)=\frac{-3}{14}+\left( \frac{-5}{7} \right) \\ & =\frac{-3}{14}+\left( \frac{-5\times 2}{7\times 2} \right)\text{ }\!\![\!\!\text{ LCM of 14, 7 is 14 }\!\!]\!\!\text{ } \\ & \text{=}\frac{-3}{14}+\frac{-10}{14}=\frac{-13}{14} \\ \end{align} \)
Multiplication of rational numbers
Product of two rational numbers is equal to the product of numerators divided by the product of denominators.
i.e., \({a \over b}\times{c \over d}\)=\(ac\over bd\)
Multiplication and division have an identitiy element; when we multiply or divide a number by one, the number doesn't change
Multiplicative inverse of a number is nothing but reciprocal of a number :
Note:
i) Reciprocal of a number x is \(1 \over x\)
ii) Multiplicative inverse of \({a \over b}\) is \({b \over a}\)
\(\begin{align} & \text{Ex : Multiply }\frac{7}{3}\And \frac{5}{2} \\ & \text{Solution :}\frac{7}{3}\times \frac{5}{2}\text{=}\frac{7\times 5}{3\times 2}\text{ =}\frac{35}{6} \\ \end{align} \)
Division of rational nummbers:
To divide a rational number by a rational number, just multiply by the reciprocal of denominator
\(\begin{align} & \frac{a}{b}\div \frac{c}{d}=\frac{a}{b}\times \frac{d}{c} \\ & Ex:1.\frac{1}{2}\div \frac{1}{4}=\frac{1}{2}\times \frac{4}{1}=2 \\ & 2.\frac{7}{3}\div \frac{2}{3}=\frac{7}{3}\times \frac{3}{2}=\frac{7}{2} \\ \end{align} \)
Note :
i) Reciprocal of a rational number is called multiplicative inverse of rational number.
ii) \(\frac{\text{0}}{\text{any rational number}}\text{= 0}\)
iii) Where rational number (except 0) is divided by another rational number (except 0) the quotient is always a rational number.
\(\frac{a}{b}\div \frac{c}{d}=\frac{ad}{bc}\)
iv) Division of any rational number by itself gives the quotient 1.
\(\frac{a}{b}\div \frac{a}{b}=1\)
v) When a rational number is divided by 1, the quotient is a rational number itself.
\(\frac{a}{b}\div 1=\frac{a}{b}\)