Class 8 – Number System

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Number System
  • Rational numbers
  • Decimal expansion of rational numbers
  • Square root of a decimal
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Number System
  • Irrational numbers
  • Introduction to surds
  • Operations on surds
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Introduction: A number which can be written in the form of , where  and  are integers and  is called a rational number.

http://images.clipartpanda.com/bloc-clipart-note-md.png In, , because division of an integer by zero is meaningless.

Examples for rational numbers are

Positive and Negative Rational Numbers: In a rational number , if both p and q are of the same sign i.e., both positive or both negative, then it is called as a positive rational number, If any one of p and q is of opposite sign, then rational number  is called as a negative rational number.

For example,  etc., are positive rational numbers

and  etc., are negative rational numbers

Rational Number 0 (i.e.,  etc.,) is neither positive nor negative.

Some facts about rational numbers:

1.    All natural numbers, whole numbers, integers and fractions are rational numbers.
Example:

i)     -24 can be written as  here .

ii)   12 can be written as  here. In particular ‘zero’ is also a rational number.

2.    Rational numbers do not have a unique representation in the form of  (where p and q are integers and ).
Example:  these are all equivalent rational numbers.

3.    Arational number , is said to be in standard form if (i) q is positive and (ii) p, q do not have any common factors other than 1 (that is p, q are co-prime).
Example: The standard form of the rational number  is  (G.C.Dof 72, 108 is 36).

4.    Terminating decimals and non-terminating repeating decimals are also rational numbers.
Example: Convert the following rational numbers to standard form:
i)                       ii)                         iii)               iv)
Solution:
i).
ii)In the rational number, the HCF of 16 and 20 is 4.
So, divide both numerator and denominator by 4.
 which is in standard form.
iii) In  denominatoris not positive. Multiply both denominator and numerator with (-1)
HCF of 10 and 8 is 2,
So,
 which is in standard form
iv)  which is in standard form.

Equivalent Rational Numbers: Two rational numbers are said to be equivalent rational numbers if they have same standard form.

Example:  and  are equivalent rational numbers because  and

i.e., they have the same standard form.
Similarly,

 are equivalent rational numbers because standard form of each of them is .

http://images.clipartpanda.com/bloc-clipart-note-md.png There can be infinitely many equivalent rational numbers to a given rational number.

Example: Write  in an equivalent rational number form so that.

i)    Its numerator is 60

ii)  Its denominator is -15

Solution:

i)    

ii)  

Operations on rational numbers: The operation of addition, subtraction, multiplication and division of rational numbers are similar to that of integers and fractions.

Addition:

a)  Addition of rational numbers with same denominator: In general, to add two rational numbers  and ,which are having same denominators.

The above rule can be extended to more than two rational numbers.

Example: Add  and

Solution: We write

        

b)  Addition of rational numbers with different denominators: It is always convenient to add two rational numbers with different denominators, by converting their denominators into LCM of two denominators.
In general, to add two (or more) rational numbers with different denominators, first convert them into equivalent rational numbers with common denominator equal to their LCM and then add as done before for adding rational numbers with same denominator.

Example: Add ,  and

Solution:

        

 

Subtraction of Rational Numbers:

a)  Subtraction of rational numbers with same denominator: In general, if  and  are two rational numbers

Example: Subtract from

Solution:

b)  Subtraction of rational numbers with different denominators: Here again, convert the given rational numbers into equivalent form with same denominator, preferably their LCM.

Example: Subtract  from

Solution:

http://images.clipartpanda.com/bloc-clipart-note-md.pngIf  and  are two rational numbers such that  then  is called as additive inverse of  and vice versa.

Multiplication of Rational Numbers: For two fractions

Product of fractions =

The above rule of multiplication of two fractions is same for rational numbers, and it can be extended to more than two rational numbers.

Example: Multiply -2, and

Solution: We have

                                      

Division of Rational Numbers: To divide a fraction by another fraction, multiply the first fraction by the reciprocal of the second fraction.

The same rule can be applied for dividing a rational number by a non-zero rational number.

Example: Divide

Solution:

            

             

 

 

Properties of rational numbers: If a, b, c are rational numbers:

1.  Closure property:

Addition

 is rational

Subtraction

  is rational

Multiplication

  is rational

Division

  need not be rational

 

2.  Commutative property:

Addition

Commutative

Subtraction

Not commutative

Multiplication

Commutative

Division

Not commutative

3.  Associative property:

Addition

Associative

Subtraction

Not associative

Multiplication

Associative

Division

Not associative

 

4.  Distributive property:

i)     (Multiplication over addition)

ii)   (Multiplication over subtraction)

5.  Additive identity (zero property):  ‘0’ is called the additive identity for the set of rational numbers.

6.  Multiplicative identity (property of 1):  is called multiplicative identity for the set of rational numbers.

7.  Additive inverse (negative of a number):  is called the additive inverse of ‘a’.

8.  Multiplicative inverse (Reciprocal):  is called the multiplicative inverse of ‘a’ or reciprocal of ‘a’.

 

To compare two rational numbers:

i)    First make their denominators positive.

ii)  If they are of opposite signs, positive rational number is always greater than the negative rational number.

iii)If both are positive, compare them as fractions.

iv) If both are negative, compare their corresponding fractions negative (ignoring the– sign). If one fraction is greater than the other, then the corresponding rational number is less than the other.

v)   Every positive rational number is greater than 0 and every negative rational number is less than 0.

Example: Which is smaller of the given rational numbers?

(i)             (ii)

Solution:

Here, denominators are same, compare the numerators 55 and42.

Clearly, 55 > 42.

So, Hence,  is smaller.

Here, compare numerators -55 and -42, because denominators are the same.

–55 < –42

So,

Hence,  is smaller.

Rational numbers between the rational numbers: Between two rational numbers there are infinitely many rational numbers.

Method 1: Let and b be two rational numbers
[r1 is the rational number between a and b].

r2 is rational number between and r1, then

r3 is the rational number between  and b.In this manner infinite rational numbers can be found between two given distinct rational numbers.

Example: Find 3 rational numbers lying between  and .

Solution: Let  are any three rational numbers lying between  and .

 ∴ 3 rational numbers between  and , are

Method 2: Let  and  are two rational numbers.

Step 1: Make denominators equal in both rational numbers, (by multiplying the numerator and denominator of each rational number by LCM of denominators)

Step 2: To insert ‘n’ rational numbers between the two rational numbers (obtained in step 1, then multiply the numerator and denominator by such a number so that the difference between the numerators is at least ‘n’. Let us understand these steps through this example.

Example: Insert three rational numbers lying between  and .

Solution: Make denominators equal by multiplying 2 and3.  and . To insert 3 rational numbers, we multiply the numerators and denominators by such a number so that difference between the numerators is at least 3. Multiplying the numerators and denominators of both numbers by 4, we get  and .

Hence the required 3 rational numbers are.

Applications of Rational Numbers in Problem solving:

Example: A person travels a distance of  km towards west and then a distance of  km towards east on the same road. Find the distance and direction of the person from the starting point.

Solution: Let the person start from a point, say, O towards west.

Distance travelled towards west =  km

Distance travelled towards east =  km

Position of the person is given by the expression

                                   

So, the person is at a distance of  km from his original point in west direction.

Example: As a part of a  km long marathon race,  km wasto be covered by cycle,  km by boat and rest on foot. How much is the distance travelled on foot?

Solution: Distance to be covered by cycle and boat =  km

Distance to be covered on foot

                          

Absolute value of a rational number:

For any rational number ‘x’, .

Example:

 http://www.aids.harvard.edu/img/news/spotlight/newsletter_layout/logo.gif

1.   Every integer is a rational number.

2.   Every fraction is a rational number.

3.   Every terminating decimal is a rational number.

4.   Every non-terminating repeating decimal is also a rational number.