# Mathematics Class 8 CBSE (NCERT) | Integer | Division (Mathematics)

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Contents Forewordiii Prefacev  Chapter 1 Rational Numbers1 Chapter 2 Linear Equations in One Variable21 Chapter 3 Understanding Quadrilaterals37 Chapter 4 Practical Geometry57 Chapter 5 Data Handling69 Chapter 6 Squares and Square Roots89 Chapter 7 Cubes and Cube Roots109 Chapter 8 Comparing Quantities117 Chapter 9 Algebraic Expressions and Identities137 Chapter 10 Visualising Solid Shapes153 Chapter 11 Mensuration169 Chapter 12 Exponents and Powers193 Chapter 13 Direct and Inverse Proportions201 Chapter 14 Factorisation217 Chapter 15 Introduction to Graphs231 Chapter 16 Playing with Numbers249 Answers  261 Just for Fun  275  R ATIONAL  N UMBERS   1 1.1 Introduction In Mathematics, we frequently come across simple equations to be solved. For example,the equation  x  + 2 =13(1)is solved when  x  = 11, because this value of  x  satisfies the given equation. The solution11 is a natural number . On the other hand, for the equation  x  + 5 =5(2)the solution gives the whole number 0 (zero). If we consider only natural numbers,equation (2) cannot be solved. To solve equations like (2), we added the number zero tothe collection of natural numbers and obtained the whole numbers. Even whole numberswill not be sufficient to solve equations of type  x  + 18 =5(3)Do you see ‘why’? We require the number –13 which is not a whole number. Thisled us to think of integers, (positive and negative) . Note that the positive integerscorrespond to natural numbers. One may think that we have enough numbers to solve allsimple equations with the available list of integers. Consider the equations2  x  =3(4)5  x  + 7 =0(5)for which we cannot find a solution from the integers. (Check this)We need the numbers 32  to solve equation (4) and 75 −  to solveequation (5). This leads us to the collection of rational numbers .We have already seen basic operations on rationalnumbers. We now try to explore some properties of operationson the different types of numbers seen so far.   Rational Numbers CHAPTER 1  2 M ATHEMATICS 1.2 Properties of Rational Numbers 1.2.1 Closure (i) Whole numbers Let us revisit the closure property for all the operations on whole numbers in brief. Operation Numbers Remarks Addition0 + 5 = 5, a whole numberWhole numbers are closed4 + 7 = ... . Is it a whole number?under addition.In general, a +  b  is a wholenumber for any two wholenumbers a and b .Subtraction5 – 7 = – 2, which is not aWhole numbers are not  closedwhole number.under subtraction.Multiplication0 × 3 = 0, a whole numberWhole numbers are closed3 × 7 = ... . Is it a whole number?under multiplication.In general, if a  and b  are any twowhole numbers, their product ab is a whole number.Division5 ÷  8 = 58 , which is not awhole number.Check for closure property under all the four operations for natural numbers. (ii) Integers Let us now recall the operations under which integers are closed. Operation Numbers Remarks Addition– 6 + 5 = – 1, an integerIntegers are closed underIs – 7 + (–5) an integer?addition.Is 8 + 5 an integer?In general, a  + b  is an integerfor any two integers a  and b .Subtraction7 – 5 = 2, an integerIntegers are closed underIs 5 – 7 an integer?subtraction.– 6 – 8 = – 14, an integerWhole numbers are not  closedunder division.
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