Properties of Rational Numbers
Category : 8th Class
Introduction
In mathematics we frequently come across different types of numbers. The different types of the numbers are natural numbers, whole numbers, rational numbers, integers, irrational numbers, and real numbers.
The natural number starts form 1 and goes to infinity. Thus we can say that all the positive real numbers starting from 1 are called natural numbers. The whole numbers are all positive real numbers starting from zero.
The rational numbers are the numbers which can be written in the form of \[\frac{p}{q}\], where \[q\ne 0\].
Properties of Rational Numbers
Closure Property
When we add two rational numbers the result is also a rational number.
\[\frac{5}{6}+\frac{8}{9}=\frac{31}{18}\]
The difference between two rational numbers is also a rational number.
\[\frac{5}{6}-\frac{8}{9}=\frac{-1}{18}\]
Multiplication and division of two rational numbers are not necessarily a rational number.
\[\frac{16}{5}\times \frac{25}{4}=20,\frac{12}{5}\times \frac{4}{25}=\frac{48}{125}\]
\[\left( \frac{2}{3} \right)+\left( \frac{1}{6} \right)=\left( \frac{5}{6} \right)\in Q;\]
\[\left( \frac{-1}{8} \right)+\left( \frac{-1}{7} \right)=-\left( \frac{1}{5}+\frac{1}{7} \right)=\frac{-15}{56}\in Q;\]
\[\left( \frac{1}{2} \right)+\left( \frac{-1}{8} \right)=\left( \frac{1}{2}-\frac{1}{8} \right)=\frac{3}{8}\in Q;\]
For any two numbers \[\left( \frac{a}{b} and\,\frac{c}{d} \right)\in Q\],
\[\Rightarrow \] \[\left( \frac{a}{b}+\frac{c}{d} \right)\in Q\] is also a rational number.
This is called the closure property of addition on the set Q.
Commutative Property
The two rational numbers can be added in any order, the result in both cases will be same. Hence we can say that addition of two rational numbers is commutative.
\[\frac{2}{3}+\frac{5}{6}=\frac{5}{6}+\frac{2}{3}=\frac{9}{6},\frac{9}{3}\times \frac{6}{3}=\frac{6}{3}\times \frac{9}{3}=6\]
For any two rational numbers \[\frac{a}{b}\] and \[\frac{c}{d}\in Q\],
\[\Rightarrow \]\[\frac{a}{b}+\frac{c}{d}=\frac{c}{d}+\frac{a}{b}\in Q\]
This is called the commutative property of addition on the set Q.
\[\left( \frac{7}{12} \right)+\left( \frac{-1}{9} \right)=\left( \frac{-1}{9} \right)+\left( \frac{7}{12} \right)\]
\[\left( \frac{-1}{136} \right)+\left( \frac{5}{96} \right)=\left( \frac{5}{96} \right)+\left( \frac{-1}{136} \right)\]
Associative Property
The addition of rational numbers is associative.
\[\left[ \frac{2}{3}+\frac{5}{6} \right]+\frac{3}{5}=\frac{2}{3}+\left[ \frac{5}{6}+\frac{3}{5} \right]=\frac{21}{10},\]\[\left[ \frac{2}{3}\times \frac{5}{6} \right]\times \frac{3}{5}=\frac{2}{3}\times \left[ \frac{5}{6}\times \frac{3}{5} \right]=\frac{1}{3}\]
For any three rational numbers \[\frac{a}{b},\frac{c}{d}\] and \[\frac{e}{f}\in Q\],
\[\Rightarrow \]\[\frac{a}{b}+\left( \frac{c}{d}+\frac{e}{f} \right)=\left( \frac{a}{b}+\frac{c}{d} \right)+\frac{e}{f}\in Q\]
This is called the associative property of addition.
Distributive of Multiplication over Addition
According to this property for any three rational numbers x, y, and z we can say that,
\[x(y+z)=xy+xz;xz(x+y)z=xz+yz\in Q\]
\[\frac{17}{18}\left( \frac{5}{6}+\frac{2}{9} \right)=\left( \frac{17}{18}\times \frac{5}{6} \right)+\left( \frac{17}{18}\times \frac{2}{9} \right),\left( \frac{17}{18}\times \frac{5}{6} \right)\frac{2}{9}=\]\[\left( \frac{17}{18}\times \frac{2}{9} \right)+\left( \frac{5}{6}\times \frac{2}{9} \right)\]
Additive Identity
The additive identity is that number which when added to any rational number gives the same result. Therefore, 0 is the additive identity. Look at the following Examples:
\[\frac{1}{9}+0=0+\frac{1}{9}=\frac{1}{9}\]
\[\frac{-13}{45}+0=0+\left( \frac{-13}{45} \right)=\frac{-13}{45}\]
Additive Inverse
Additive inverse is that number which when added to the rational number, the results is additive identity. For any rational number \[m\in Q\],-m is the additive inverse of the number m.
\[\Rightarrow \]\[m+(-m)=0\]
We have, \[\frac{1}{9}+\left( \frac{-1}{9} \right)=0,\frac{2}{3}+\left( \frac{-2}{3} \right)=0,\]\[\frac{3}{2}+\left( \frac{-3}{2} \right)=0\] and so on.
For every rational number \[\frac{a}{b}\], we have \[\frac{a}{b}+\left( \frac{-a}{b} \right)=0\].
Where \[\frac{a}{b}\] and \[\frac{-a}{b}\] are the additive inverse of each other.
Rational Numbers between any Two Rational Numbers
There are infinite number of rational numbers between any two rational numbers. There are different methods by which we can find the rational numbers between any two rational numbers. It can be done either by equating the denominator or by reducing it to tenth, hundredth, or thousandth. Thus we can find countless rational numbers between any two rational numbers. The other method is by taking the average value of the two given rational numbers.
Rational number between \[\frac{23}{25}\] and \[\frac{34}{125}\] is:
\[\left( \frac{23}{25}+\frac{34}{125} \right)\div 2\]
Or \[\left( \frac{23\times 5}{25\times 5} and \frac{34}{125} \right)=\left( \frac{115}{125} and \frac{34}{125} \right)\]
Operations on the Rational Numbers
Addition of Rational Numbers
We can add two rational numbers by equating their denominators or by taking the LCM of the denominators. Look at the following example:
\[\frac{56}{78}+\frac{45}{91}=\frac{392+270}{546}=\frac{331}{273}\]
Multiplication of Rational Numbers When we multiply the rational numbers we multiply numerators with numerators and denominators with the denominators \[\frac{a}{b}\times \frac{c}{d}=\]\[\frac{ac}{bd}\]. Look at the following example:
\[\frac{35}{78}\times \frac{91}{105}=\frac{7}{18}\]
Division of Rational Numbers
When we divide two rational numbers we normally take the reciprocal of the divisor and division sign changes into multiplications and hence we apply the rule of multiplication.
For any two rational numbers \[\frac{a}{b}\] and \[\frac{c}{d}\],
we have, \[\frac{a}{b}\div \frac{c}{d}=\frac{a}{b}\times \frac{d}{c}=\frac{ad}{bc}\]
\[\frac{35}{78}\div \frac{91}{105}=\frac{13}{15}\]
All the natural numbers, whole numbers and integers are rational numbers because all of them give same result or number when divided by 1, it means that they can be expressed in the form of \[\frac{p}{q}\] where \[q\ne 0\].
Points to keep in mind:
Reduction to Lowest Term
A rational number can be reduced to its lowest term, by simplifying both numerator and denominator by simplest factors.
\[\frac{91}{105}=\frac{13\times 7}{7\times 3\times 5}=\frac{13}{15}\]
Properties of Subtraction of Two Rational Numbers
The difference between two rational numbers is also a rational number. If \[\frac{a}{b}\] and \[\frac{c}{d}\] be any two rational numbers, then \[\Rightarrow \frac{a}{b}-\frac{c}{d}\] is also a rational number. Look at the following examples:
\[\frac{1}{3}-\frac{1}{2}=-\frac{1}{6}\in Q\] \[\frac{2}{3}-\frac{1}{4}=\frac{5}{12}\in Q\]
Subtraction is not commutative
For any two rational numbers \[\frac{a}{b}\] and \[\frac{c}{d}\],
\[\Rightarrow \]\[\frac{a}{b}-\frac{c}{d}\ne \frac{c}{d}-\frac{a}{b}\]
Hence, subtraction of two rational numbers is not commutative. Look at the following Examples:
\[\frac{2}{7}-\frac{1}{4}\ne \frac{1}{4}-\frac{2}{7};\]
\[\frac{7}{8}-\frac{6}{7}\ne \frac{6}{7}-\frac{7}{8}\]
Subtraction is not Associative
For any three rational numbers \[\frac{a}{b},\frac{c}{d},\] and \[\frac{e}{f}\in Q\],
We have, \[\left( \frac{a}{b}-\frac{c}{d} \right)-\frac{e}{f}\ne \frac{a}{b}-\left( \frac{a}{d}-\frac{e}{f} \right)\]
Look at the following examples:
\[\left( \frac{1}{8}-\frac{1}{5} \right)-\frac{1}{2}\ne \frac{1}{8}-\left( \frac{1}{5}-\frac{1}{2} \right);\]
\[\left( \frac{7}{8}-\frac{6}{7} \right)-\frac{1}{2}\ne \frac{7}{8}-\left( \frac{6}{7}-\frac{1}{2} \right)\]
Thus, we can say that subtractions of rational numbers is not associative.
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