RATIONAL NUMBERS
FUNDAMENTALS
Rational Number:-
- A number which can be expressed as\[\frac{x}{y}\], where x and y are Integers and \[y\ne 0\] is called a rational number.
e.g., \[\frac{1}{2},\frac{2}{2},\frac{-1}{2},0,\frac{3}{-\,2}\] etc.
- Set of rational number is denoted by Z.
- A Rational number may be positive, zero or negative
- If \[\frac{x}{y}\] is a rational number and \[\frac{x}{y}>0\], then\[\frac{x}{y}\] is called a positive Rational Number.
e.g., \[\frac{1}{2},\frac{2}{5},\frac{-3}{-2},-\left( -\frac{1}{2} \right)\]etc.
Negative Rational Numbers:-
- If \[\frac{x}{y}\] is a rational number and \[\frac{x}{y}<0\], then \[\frac{x}{y}\]is called a Negative Rational Number.
e.g., \[\frac{-1}{2}.\frac{3}{-2},\frac{-7}{11}......\]etc.
Standard form of Rational Number:-
- A Rational number \[\frac{x}{y}\] is said to be m standard form, if x and y are integers having no common divisor other than one, where \[y\ne 0\].
e.g., \[\frac{-1}{2},\frac{5}{6},\frac{8}{11}\]……etc.
Note:- There are infinite rational numbers between any two rational numbers.
Property of Rational Number
- Let x and y are two rational number and y > x, then the rational number between x and y is\[\frac{1}{2}\left( x+y \right).\]
e.g., find 2 rational number between \[\frac{1}{3}\]and \[\frac{1}{2}\]
Solution:- Let \[x=\frac{1}{3}\] and \[y=\frac{1}{3}\] and y > x.
Then, Rational no. between\[\frac{1}{3}\]and\[\frac{1}{2}\]is
\[\frac{1}{2}\left( \frac{1}{3}+\frac{1}{2} \right)=\frac{1}{2}\left( \frac{2+3}{6} \right)=\frac{5}{12}\]
Again Let \[x=\frac{5}{12}\] and \[y=\frac{1}{2}\] and y > x. then
Rational no. between \[\frac{5}{12}\] and \[\frac{1}{2}\] is
\[\frac{1}{2}\left( \frac{5}{12}+\frac{1}{2} \right)=\frac{1}{2}\left( \frac{5+6}{12} \right)=\frac{1}{2}\times \frac{11}{12}=\frac{11}{24}\]
Hence the Rational Numbers between \[\frac{1}{3}\] and \[\frac{1}{2}\] are \[\frac{5}{12}\] and \[\frac{11}{24}\].
- Let x and y are two rational number and y > x. Consider to find n rational numbers between x and y. Let d = \[\frac{y-x}{n+1}\]
Then 'n' rational number lying between x and y are \[\left( x+d \right),\left( x+2d \right),\left( x+3d \right),\_\_\_\left( x+nd \right).\]
Example:- Find 9 rational number between 2 and 3.
Solution:- Let x = 2 and y = 3 then y > x
Now \[\mathbf{d}=\frac{y-x}{n+1}=\frac{3-2}{9+1}=\frac{1}{10}\]
Then, rational number are, 2 + 0.1, 2 + 0.2, 2 + 0.3, 2 + 0.4, 2 + 0.5, 2 + 0.6, 2 + 0.7, 2 + 0.8, 2 + 0.9 = 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8 and 2.9.
Representation of Rational Number on the Number line
- To represent - on the number line first we draw a number line
Let O represent 0 (zero) and A represent 1. So divide OA into 4 equal parts, each point in the middle representing P, Q and R. Point R represent\[\frac{3}{4}\].
Operations on Rational Numbers
- Addition of Rational Numbers:
Example: Find the sum of the rational numbers \[\frac{-4}{9},\frac{15}{12}\] and \[\frac{-7}{18}\].
Solution: \[\frac{-4}{9}+\frac{15}{12}+\frac{-7}{18}=\frac{-16+45-14}{36}=\frac{15}{36}=\frac{5}{12}\]
Properties of Addition of Rational Number
- Closure Property:- If a and b are two rational numbers, then a + b is always a rational number.
E.g., Let \[a=3\], \[b=-2,\] then \[a+b=3+\left( -2 \right)-1\]
- Commutative Property:- If a and b are two rational number then a + b = b + a.
E.g., Let \[a=\frac{1}{2}\]and \[b=\frac{1}{3}\] then
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Convex polygon
Concave Polygon
In a concave polygon, a line segment between two points on the boundary goes outside the polygon.
or
In a concave polygon atleast one of the interior angle is more than\[180{}^\circ \].
Concave polygon
Some important Formulae
(i) Sum all the angles in a convex polygon is\[(2n-4)90{}^\circ \].
(ii)Exterior angle of a regular polygon is\[\frac{360{}^\circ }{n}\].
(iii)Interior angle of a regular polygon is\[\left( 180{}^\circ -\frac{360{}^\circ }{n} \right),\] where n is number of sides of the polygon
(iv) Number of diagonals of a convex polygon of n sides is\[\frac{n\left( n-3 \right)}{2}\].
Quadrilaterals
A plane closed figured bounded by four segments is called quadrilateral.
Parallelogram
A quadrilateral having opposite sides are parallel is called a parallelogram. In a parallelogram,
(i) opposite sides are equal.
(ii) opposite angles are equal.
(iii) each diagonal divides the parallelogram into two congruent triangles.
(iv) sum of any two adjacent angles is\[180{}^\circ \].
(v) the diagonals bisect each other.
Rhombus
A parallelogram is a rhombus is which every pair of adjacent sides are equal (all four sides of a rhombus are equal).
Since, a parallelogram is a rhombus, all the properties of a parallelogram apply to a rhombus. Further, in a rhombus, the diagonals are perpendicular to each other.
Rectangle
A parallelogram is a rectangle in which each of the angles is equal to\[90{}^\circ \]. The diagonals of a rectangle are equal.
A rectangle is also a special type of parallelogram and hence all properties of parallelogram apply to rectangles also.
Square
A rectangle is a square in which all four sides are equal (a rhombus in which all more... 
(a) \[4080\,c{{m}^{2}}\] (b) \[5050\,c{{m}^{2}}\]
(c) \[6060\,c{{m}^{2}}\] (d) \[7070\,c{{m}^{2}}\]
(e) None of these
Answer (b)
Explanation: Area of the given figure
\[=\frac{1}{2}\times 30\times 10+\frac{1}{2}\times 20\times 20+\frac{1}{2}\times 40(30+40)+\frac{1}{2}\times 40\]\[(20+60)+\frac{1}{2}\times 40\times 40\times \frac{1}{2}\times 60\times 30c{{m}^{2}}\]
\[=(150+200+1400+1600+800+900)c{{m}^{2}}\]
\[=5050c{{m}^{2}}\]
Cuboid and Cube
A cuboid is a closed rectangular solid bounded by six rectangular faces. A cuboid has 12 edges and 8 vertices. The length, breadth and height of a cuboid is generally denoted by I, b and h respectively. A cuboid whose length, breadth and height are equal is called a cube and each equal side is called edge of the cube.
Co-ordinates of a Point
The pair of points which is used to describe the location of a point in two dimensional system are called co-ordinates of a point. The x-coordinate of a point is horizontal distance of the point from origin and y-coordinate of the point is the vertical distance from the origin.
Line Graph
A line graph is very useful for displaying data or information which changer continuously over a certain period of time. A line graph compares two variables. One variable is plotted along x-axis while another variable is plotted along y-axis.
Linear Graph
A linear graph is a graph which is used to represent the linear relationship between two variables. To draw a linear graph we use co-ordinates along x and y axis. The difference between a line graph Or linear graph is that a line graph display information as a series of points joined by line segments while a linear graph is always a straight line.
According to another basis, polygons are classified as;
(i) Regular Polygons: Polygons in which all sides and all angles are equal, are called Regular Polygons. Specific names are given to each of them depending upon no of sides.
(a) In Triangles:
Equilateral \[\Delta \,1e\]
(b) In quadrilaterals: Square
(c) In Pentagons: Regular pentagon
(d) In Hexagons: Regular hexagon
\[\angle A=\angle B=\angle C=\angle D=\angle E=\angle F={{120}^{{}^\circ }}\]& so on.
(ii) Irregular polygons: Polygons which do not nave equal sides or equal angles, are called Irregular Polygons.
Thus, RHOMBUS, which has equal sides but unequal angles, is irregular polygon.
Properties of Regular polygons:-
If n = no. of sides of a regular polygon, then,
(i) Exterior angle, \['\theta '=\left( \frac{{{360}^{{}^\circ }}}{n} \right)\]
(ii) Interior angle, = \[\left( {{180}^{{}^\circ }}-\theta \right)\]
(iii) No. of diagonals of polygons of 'n’ sides \[=\frac{n(n-2)}{2}\]
Adjacent sides of a more... 
