Rational Numbers
Category : 7th Class
RATIONAL NUMBERS
FUNDAMENTALS
\[-1,\,\,-2,\,\,-3,\,\,-4,\].......etc.., are called negative integers denoted by \[{{Z}^{-}}\] or \[{{I}^{-}}\].
Note: 0 is neither positive nor negative numbers.
Elementary questions:
Identify which of the following number is a whole number as well as a fraction?
(a) \[\frac{3}{36}\] (b) \[\frac{36}{3}\] (c) \[\frac{20}{8}\] (d) \[\frac{8}{20}\]
Ans. (b) \[\frac{36}{3}=12\] which can be expressed as a fraction \[\left( \frac{12}{1} \right)\] as well as a whole number (=12).
Rational numbers (Q):
A number of the form \[\frac{p}{q}(q\ne 0).\] where p and q are integers is called a rational number.
e.g., \[\frac{-3}{6},-\frac{1}{12},\frac{10}{13},\frac{12}{17},\ldots \ldots ..\]etc.
Note: 0 is rational number, since \[0=\frac{0}{1}.\]
e.g. \[\frac{6}{11},\frac{-8}{-16}\]
e.g., \[\frac{-4}{7},\frac{8}{-23}\]
Note: 0 is neither a positive nor a negative rational number.
We can mark rational numbers on a number line just as we do for integer. The negative rational numbers are marked to the left of 0 and the positive rational numbers are marked to the right of 0.
Thus, \[\frac{1}{6}\] and \[-\,\,\frac{1}{6}\] would be at an equal distance from 0 but on its either side of zero.
Similarly, other rational numbers with different denominators can also be represented on the number line.
(a) \[\frac{p}{q}\] where p < q (b) \[\frac{p}{q}\] where p > q
e.g., \[\frac{1}{8},\frac{2}{9},\frac{16}{17}\] etc. e.g.,\[\frac{8}{1},\frac{9}{2},\frac{17}{16}\]etc.
Representation of \[\frac{p}{q}\] on the number line where p < q:
The rational number \[\frac{4}{6}\](4<6) is represented on the number line as shown.
Representation of \[\frac{p}{q}\]on the number line where p > q:
Consider the rational number \[\frac{13}{6}\]
Let us convert the rational number \[\frac{13}{6}\] into a mixed fraction \[=2\frac{1}{6}\]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 integer p and q have no common factor other than 1.
Additive Inverse:
\[\frac{-\,p}{q}\]is the additive inverse of \[\frac{p}{q}\] and \[\frac{p}{q}\] is the additive inverse of \[\frac{-p}{q}.\]
e.g., \[\frac{-13}{6}+\frac{13}{6}=0=\frac{13}{6}+\left( \frac{-13}{6} \right).\]
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.
Thus, the reciprocal of \[\frac{a}{b}\] is \[\frac{b}{a}\]and we write,\[{{\left( \frac{a}{b} \right)}^{-1}}=\frac{b}{a}\]
Note (a); Clearly, (a) Reciprocal of 0 does not exist.
(b) Reciprocal of 1 is 1.
(c) Reciprocal of-lis-1.
Elementary question:
Find the reciprocal of \[\frac{5}{6}\] and also its additive increase.
Ans. Reciprocal of \[\frac{5}{6}\]
Let 'a' be reciprocal of 5/6
Then \[a\times \frac{5}{6}=1\Rightarrow a=\frac{6}{5}\]
Let 'x' be addition inverse
Then, \[x+\frac{5}{6}=0\]
\[\therefore \,\,\,x=\left( -\frac{5}{6} \right)\]
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