Rational Numbers

Category : 7th Class




  • Natural numbers (N): 1, 2, 3, 4, 5..... ..etc., are called natural numbers.
  • Whole numbers (W): 0, 1, 2, 3, 4, etc.., are called whole numbers.
  • Integers (Z): 1.......\[-4,-3,-2,-1,\,\,0,\,\,1,\,\,2,\,\,3,\,\,4\]........ etc.., are called integers. (denoted by I or Z) 1, 2, 3, 4, .. ...etc., are called positive integers denoted by \[{{Z}^{+}}\]or \[{{I}^{+}}\].

\[-1,\,\,-2,\,\,-3,\,\,-4,\].......etc.., are called negative integers denoted by \[{{Z}^{-}}\] or \[{{I}^{-}}\].


Note: 0 is neither positive nor negative numbers.

  • Fractions: The numbers of the form \[\frac{x}{y}\], where \[x\] and \[y\]c are natural numbers, are known as fractions. e.g., \[\frac{2}{5},\,\,\frac{3}{1},\,\,\frac{1}{122},.....\]etc.

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 \[0=\frac{0}{1}.\]

  • A rational number \[\frac{p}{q}\] is positive if p and q are either both positive or both negative.

e.g. \[\frac{6}{11},\frac{-8}{-16}\]

  • A rational number \[\frac{p}{q}\] is negative if either of p and q is positive and the other term (q or p) is negative.

e.g., \[\frac{-4}{7},\frac{8}{-23}\]


Note: 0 is neither a positive nor a negative rational number.

  • Representation of Rational numbers on a number line:

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.

  • In general, any rational number is either of the following two types.

(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.

  • Standard form of a rational number:

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, \[x+\frac{5}{6}=0\]

\[\therefore \,\,\,x=\left( -\frac{5}{6} \right)\]




Other Topics

Notes - Rational Numbers
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