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
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