RATIONAL NUMBERS, PROPERTIES, OPERATIONS
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
\(\begin{gathered} W = \{ 0,1,2,3, - - - - \} \hfill \\ N \cup \{ 0\} = W \hfill \\ \end{gathered} \)
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 \(q\ne0\) are called rational numbers .
\(Q = \left\{ {\frac{p}{q};p,q \in Z\& q \ne 0} \right\}\)
Ex:1)\(\frac{4}{7},\frac{{ - 3}}{{10}},\frac{0}{1},\sqrt 1 ,\sqrt 4 ,\sqrt 9 ....\)
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 \(\frac{p}{q} = \frac{{p \times n}}{{q \times n}}\)
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
\(\frac{6}{4} > \frac{{ - 6}}{4}\)
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 different denominators , convert them into equivalent rational numbers with the same denominator, which is equal to the L.C. M of their denominators