Category : 7th Class
Rational Numbers
1, 2, 3, 4, etc., are called positive integers denoted by 7. \[{{Z}^{+}}\].
-1, -2, - 3, - 4,...... etc., are called negative integers denoted by Z-
Note: 0 is neither positive nor negative.
The numbers of the form \[\frac{x}{y}\], where x and y are natural numbers, are known as fractions.
e.g.\[\frac{3}{5},\frac{2}{1},\frac{1}{125},\]......... etc.
A number of the form \[\frac{p}{q}\](q\[\ne \]O), where p and q are integers is called a rational number.
e.g.,\[\frac{-3}{17},\frac{5}{-19},\frac{10}{1},\frac{-11}{-23},\]…. etc.
Note: 0 is a rational number, since 0=\[\frac{\mathbf{0}}{\mathbf{1}}\].
e.g., \[\frac{3}{5},\frac{-2}{-7}\]
e.g., \[\frac{-5}{3},\frac{7}{-23}\]
Note: 0 is neither a positive nor a negative rational number.
We can mark rational numbers on a number line just as we do integers.
The negative rational numbers are marked to the left of 0 and the positive rational numbers are marked to the right of 0.
Thus,,\[\frac{1}{3}and-\frac{1}{3}\]would be at an equal distance from 0 but on its either side.
Similarly, other rational numbers with different denominators can also be represented on the number line.
Thus, in general, any rational number is either of the following two types.
(i)\[\frac{m}{n}\] where m < n (ii)\[\frac{m}{n}\] where m > n
e.g., 4, 6 \[\frac{1}{2},\frac{3}{4},\frac{5}{6}\]etc.,
e.g.,\[\frac{7}{6},\frac{3}{2},\frac{15}{6}\] etc.
Representation of\[\frac{\mathbf{m}}{\mathbf{n}}\] on the number line where m < n:
The rational number - (5 < 6) is represented on the number line as shown.
Consider the rational number \[\frac{17}{5}\].
First convert the rational number\[\frac{17}{5}\]into a mixed fraction and then mark it on the number line i.e.
A rational number'\[\frac{p}{q}\]is said to be in standard form if q is a positive integer and the integers p and q have no common factor other than 1.
Step 1: Express each of the two given rational numbers with a positive denominator.
Step 2: Take the L.C.M. of these positive denominators.
Step 3: Express each rational number with this L.C.M. as common denominator.
Step 4: The number having a greater numerator is greater.
(i) Addition: To add two rational numbers with the same denominator, we simply add their numerators and divide by the common denominator.
e.g., \[\frac{-15}{7}+\frac{1}{7}=\frac{-5+1}{7}=\frac{-4}{7}\]
When denominators of given rational numbers are different find their L.C.M and express each one of the given rational numbers with this L.C.M as the common denominator. Then add as usual.
Additive Inverse:
\[\frac{-p}{q}\]is the additive inverse of\[\frac{p}{q}and\frac{p}{q}\] and\[\frac{-p}{q}\]is the additive inverse of
e.g.,\[\frac{-2}{5}+\frac{2}{5}=0=\frac{2}{5}+\left( \frac{-2}{5} \right).\]
Subtraction: While subtracting two rational numbers, we add the additive inverse of the rational number to be subtracted to the other rational number.
Thus, \[\frac{-2}{5}+\frac{2}{5}=0=\frac{2}{5}+\left( \frac{-2}{5} \right).\]additive inverse of
\[\frac{2}{3}=\frac{7}{8}+\frac{\left( -2 \right)}{3}=\frac{21+\left( -16 \right)}{24}=\frac{5}{24}\]
Product of the numerators denominators separately and write the product as
\[\frac{Product\,of\,the\,numbers}{Product of the denominators}\]
Reciprocal of a rational number: If the product of two rational numbers is 1 then each rational number is called the reciprocal of the other.
Clearly, (a) reciprocal of 0 does not exist.
(b) reciprocal of 1 does not exist.
(c) reciprocal of-1 is-1.
Thus, if\[\frac{a}{b}\,and\,\frac{c}{d}\]are two rational numbers such that\[\frac{c}{d}\ne 0\].
then\[\left( \frac{a}{b}\div \frac{c}{d} \right)=\frac{a}{b}\times \left( reciprocal\,\frac{c}{d} \right)=\left( \frac{a}{b}\times \frac{d}{c} \right)\]
e.g.,\[\frac{-7}{2}\div \frac{4}{3}=\frac{-7}{2}\times \left( Reciprocal\,of\,\frac{4}{3} \right)=\frac{-7}{2}\times \frac{3}{4}=\frac{-21}{8}\]
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