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A number that can be expressed in the form p q , where p and q are integers and q ¹ 0, is called a rational number.

Lowest form of a rational number – A rational number p q is said to be in the lowest form or simplest form if p and q have no common factor other than 1 and q ¹ 0. Addition, subtraction, multiplication and division of ational numbers are done in the same way as we do for fractions. Rational numbers are closed under the operations of addition, subtraction and multiplication. The operations of addition and multiplication for rational numbers are (i) commutative, (ii) associative……………… **READ MORE**

An algebraic equation is an equality involving variables. It has an equality sign. The expression on the left of the equality sign is the Left Hand Side (LHS) and the expression on the right of the equality sign is the Right Hand Side (RHS). In an equation the values of the expressions on the LHS and RHS are equal for certain values of the variables.

These values are the solutions of the equation. Equations where the expressions which form the equation contain only one variable and the highest power of the variable appearing in the equation is 1, are called linear equations in one variable………………. **READ MORE**

A simple closed curve made up of only line segments is called a polygon. A diagonal of a polygon is a line segment connecting two nonconsecutive vertices. A convex polygon is a polygon in which no portion of its any diagonal is in its exterior. A quadrilateral is a polygon having only four sides. A regular polygon is a polygon whose all sides are equal and also all angles are equal. The sum of interior angles of a……………… **READ MORE**

A simple closed curve made up of only line segments is called a polygon. A diagonal of a polygon is a line segment connecting two nonconsecutive vertices. A convex polygon is a polygon in which no portion of its any diagonal is in its exterior. A quadrilateral is a polygon having only four sides. A regular polygon is a polygon whose all sides are equal and also all angles are equal. The sum of interior angles of a ……………… **READ MORE**

The information collected in term of numbers is called data. Data are represented graphically to have a quick glance on them. Data available in an unorganised form are called raw data. The number of times a particular observation occurs in a given data is called its frequency. When the data are large, they can be arranged in groups and each group is known as Class Interval or Class. A table showing the frequencies of various observations or class intervals of a given data is called a Frequency Distribution table………………. **READ MORE**

A natural number is called a perfect square if it is the square of some natural number.i.e., if m = n2, then m is a perfect square where m and n are natural numbers. A natural number is called a perfect cube if it is the cube of some natural number.i.e., if m = n3, then m is a perfect cube where m and n are natural numbers. Number obtained when a number is multiplied by itself is called the square of the number. Number obtained when a number is multiplied by itself three times are called cube number cube number………………. **READ MORE**

Squares and cubes of even numbers are even.

A natural number is called a perfect square if it is the square of some natural number.i.e., if m = n2, then m is a perfect square where m and n are natural numbers. A natural number is called a perfect cube if it is the cube of some natural number.i.e., if m = n3, then m is a perfect cube where m and n are natural numbers. Number obtained when a number is multiplied by itself is called the square of the number. Number obtained when a number is multiplied by itself three times are called cube number ……………… **READ MORE**

Discount is a reduction given on marked price.Discount = Marked Price – Sale Price (S.P.) Discount can be calculated when discount percentage is given. Discount = Discount % of Marked Price Additional expenses made after buying an article are included in the cost price and are known as overhead expenses. cost price = buying price + overhead expenses Sales Tax is charged on the sale of an item by the government and is added to the Bill Amount.

Sales tax = Tax% of sale amount……………… **READ MORE**

Algebraic Expression Terms are formed by the product of variables and constants, e.g. –3xy, 2xyz, 5×2, etc.

Terms are added to form expressions, e.g. –2xy + 5×2. Expressions that contain exactly one, two and three terms are called monomials, binomials and trinomials, respectively. In general, any expression containing one or more terms with nonzero coefficients (and with variables having……………… **READ MORE**

3D shapes/objects are those which do not lie completely in a plane.3D objects have different views from different positions. A solid is a polyhedron if it is made up of only polygonal faces, the faces meet at edges which are line segments and the edges meet at a point called vertex. Euler’s formula for any polyhedron is, F + V – E = 2 Where F stands for number of faces, V for number of vertices and E for number of edges. Types of polyhedrons: (a) Convex polyhedron A convex polyhedron is one in which all faces make it convex. A polyhedron is regular if its faces are congruent regular polygons and the same number of……………… **READ MORE**

Length of boundary of a simple closed figure is known as perimeter. Area is the measure of region enclosed in a simple closed curve. Perimeter of a rectangle = 2 (length + breadth). Area of a rectangle = length × breadth.

Perimeter of a square = 4 × side. Area of a square = side × side. Area of a triangle ……………… **READ MORE**

Exponential notation is a powerful way to express repeated multiplication of the same number. Specifically, powers of 10 express very large and very small numbers in a manner which is convenient to read, write and compare.

For any non-zero integer a, a–m 1 m a = Laws of exponents are ……………… **READ MORE**

Two quantities x and y are said to be in direct proportion if they increase or decrease together in such a manner that the ratio of their corresponding values remains constant. That is, x y = k where k is a positive number, if x and y are in direct proportion or vary directly. In case of direct proportion, if y1, y2 are the values of y corresponding to the values x1, x2 of x respectively, then 1 2 1 2 x x y y. Two quantities x and y are said to be in inverse proportion if an increase in x causes a proportional decrease in y and vice-versa, in such a manner that the product of their corresponding values remains constant………………. **READ MORE**

Representation of an algebraic expression as the product of two or more expressions is called factorisation. Each such expression is called a factor of the given algebraic expression. When we factorise an expression, we write it as a product of its factors. These factors may be numbers, algebraic (or literal) variables or algebraic expressions.

A formula is an equation stating a relationship between two or more variables. For example, the number of square units in the area (A) of a rectangle is equal to the number of ……………… **READ MORE**

Graphical representation of data is easier to understand. A bar graph, a pie chart and histogram are graphical representations of data. A line graph displays data that changes continuously over periods of time. A line graph in which all the line segments form a part of a single line is called a linear graph. For fixing a point on the graph sheet, we need two mutually perpendicular lines (in which horizontal line is called x axis and the vertical line as y axis) alongwith, x coordinate (abscissa) and y coordinate (ordinate) of the point………………. **READ MORE**

Numbers can be written in general form. For example, a two digit number ab is written as ab = 10a + b; a three digit number abc is written as abc = 100a + 10b + c. The general form of numbers are helpful in solving various problems related to numbers. Rationale for the divisibility of numbers by 11, 10, 5, 2, 9 or 3 can be explained by writing the numbers in general form. Many number puzzles involving different letters for different digits are solved using rules of number operations………………. **READ MORE**