## Rational Numbers

Category : 7th Class

RATIONAL NUMBERS

FUNDAMENTALS

• 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.

$\frac{-\,p}{q}$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 $a\times \frac{5}{6}=1\Rightarrow a=\frac{6}{5}$

Then, $x+\frac{5}{6}=0$

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

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