Mathematics: Algebra and Geometry Exercises, Study notes of Mathematics

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Mathematics

Classes 9-

Natural Number

1 , 2 , 3 , 4 ......... etc. numbers are called natural number or positive whole numbers.

2 , 3 , 5 , 7 .........etc. are prime numbers and 4 , 6 , 8 , 9 ,......... etc. are composite numbers.

Integers

All numbers (both positive and negative) with zero (0) are called integers i.e. .......

 3 ,  2 , 1 , 0 , 1 , 2 , 3 .........etc. are integers.

Fractional Number

If p , q are co-prime numbers ; q z 0 and q z 1 , numbers expressed in

q

p

form are

called fractional number.

Example :

, , etc. are fractional numbers.

If p  q , then it is a proper fraction and if p! q then it is an improper fraction :

Example , , , ,.........

etc. proper and , , , ,....

etc. improper fraction.

Rational Number

If p and q are integers and q z 0 , number expressed in the form

q

p

is called rational

number. For example : 1. 666 ...

3

etc. are rational numbers.

Rational numbers can be expressed as the ratio of two integers. So, all integers and all

fractional numbers are rational numbers.

Irrational Number

Numbers which cannot be expressed in

q

p

form, where p , q are integers and q z 0 are

called Irrational Numbers. Square root of a number which is not perfect square, is an

irrational number. For example: 2 1. 414213 ....., 3 1. 732 ....., 1. 58113 ....

etc. are irrational numbers. Irrational number cannot be expressed as the ratio of two

integers.

Decimal Fractional Number

If rational and irrational numbers are expressed in decimal, they are known as decimal

fractional numbers. As for instance, 33333 ......., 3 1732 .........

3

10

25 ,

2

5

3 3 ˜ 0 , ˜ ˜ ˜

etc.

are decimal fractional numbers. After the decimal, if the number of digits are finite, it is

terminating decimals and if it is infinite it is known as non-terminating decimal number.

For example, 0.52, 3.4152 etc. are terminating decimals and

1 ˜ 333 ......., 2 ˜ 123512367 ...........etc. are non-terminating decimals. Again, if the digits

after the decimal of numbers are repeated among themselves, they are known recurring

decimals and if they are not repeated, they are called non-recurring decimals. For

example : 1 ˜ 2323 ........, 5 654

˜ etc. are the the recurring decimals and

0 ˜ 523050056 ........, 2 ˜ 12340314 ........etc. are non-recurring decimals.

Real Number

All rational and irrational numbers are known as real numbers. For example :

0 , r 1 ,r 2 ,r 3 ,.......... ,........

3

r r r 2 , 3 , 5 , 6 ......

etc. are real numbers.

Positive Number

All real numbers greater than zero are called positive numbers. As for instance

etc. are positive numbers.

Negative Number

All real numbers less than zero are called negative numbers. For example,

etc. are

negative numbers.

Non-Negative Number

All positive numbers including zero are called non-negative numbers. For example,

etc. are non-negative numbers.

Classification of real Number.

Real

Rational

Integer Fraction

Positive 0 Negative Simple Decimal Irrational

Fractional 1 Composite Proper Improper Mixed Terminating Recurring Non-

recurring

? 2 is not an integer.

? 2 is either a rational number or a irrational number. If 2 is a rational number

let, ;

q

p

2 where p and q are natural numbers and co-prime to each other and q! 1

or, ; 2

2

q

p

squaring

or, ;

q

p

q

2

2 multiplying both sides by q.

Clearly 2 q is an integer but

q

p

2

is not an integer because p and q are co-prime natural

numbers and q! 1

? 2 q and

q

p

2

cannot be equal, i.e.,

q

p

q

2

2 z

? Value of 2 cannot be equal to any number with the form

q

p

i.e.,

q

p

2 z

? 2 is an irrational number.

Example 2. Prove that, sum of adding of 1 with the product of four consecutive natural

numbers becomes a perfect square number.

Solution : Let four consecutive natural numbers be x , x  1 , x  2 , x  3 respectively.

By adding 1 with their product we get,

2 2    

x x x x

xx x x xx x x

2 1 ; [ 3 ]

2 a a   x  x a

a ( a  2 ) 1 ;

2 2

a  2 a  1 a  1 3 1 ;

2 2

x  x 

which is a perfect square number.

? If we add 1 with the product of four consecutive numbers, we get a perfect square

number.

Activity : Proof that, 3 is an irrational number

Classification of Decimal Fractions

Each real number can be expressed in the form of a decimal fraction.

For example, 2 2 ˜ 0 , 0. 4 ,

5

etc. There are three types of decimal

fractions :terminating decimals, recurring decimals and non-terminating decimals.

Terminating decimals : In terminating decimals, the finite numbers of digits are in the

right side of a decimal points. For example, (^0). 12 , 1. 023 , 7. 832 , 54. 67 ,.......... etc. are

terminating decimals.

Recurring decimals : In recurring decimals, the digits or the part of the digits in the

right side of the decimal points will occar repeatedly. For example,

3. 333 ....., 2. 454545 ......, 5. 12765765 ǤǤǤǤǤǤǤǤǤǤetc. are recurring decimals.

Non-terminating decimals : In non-terminating decimals, the digits in the right side of

a decimal point never terminate, i.e., the number of digits in the right side of decimal

point will not be finite neither will the part occur repeatedly. For

example. 1. 4142135 ......, 2. 8284271 .......etc. are non-terminating decimals.

Terminating decimals and recurring decimals are rational numbers and non-terminating

decimals are irrational numbers. The value of an irrational number can be determined

upto the required number after the decimal point. If the numerator and denominator of a

fraction can be expressed in natural numbers, that fraction is a rational number.

Activity : Classify the decimal fractions stating reasons :

1. 723 , 5. 2333 ........, 0. 0025 , 2. 1356124 ......., 0. 0105105 ........ and

Recurring decimal fraction :

Expressing the fraction

into decimal fractions, we get,

Required decimal fraction = 0. 2727 ...... Required decimal fraction =

Conversion of Recurring Decimal into Simple Fraction

Determining the value of recurring fraction :

Example 5. Express 0 3

. into simple fraction.

Solution : 0. 3 0. 3333 ........

1. 3 u 10 0. 333 .......u 10 3. 333 ......

and 0. 3 u 1 0. 333 .......u 1 0. 333 ......

subtracting, 0 3 u 10  03 u 1 3

or, 0 3 u 10  1 3

. or 0 3 u 9 3

Therefore,

Required fraction is

Example 6. Express 0 24

. into simple fraction.

Solution : 0 24

So, 0. 24 u 100 0. 242424 .......u 100 24. 2424 ......

and 0. 24 u 1 0. 242424 .......u 1 0. 242424 ......

Subtracting 0 24100  1 24

or, 0. 24 u 99 24

or,

Required fraction is

Example 7. Express 5 1345

. into simple fraction.

Solution : 5 1345

So, 5. 1345 u 10000 5. 1345345 ......u 10000 51345. 345 ......

and 5. 1345 u 10 5. 1345345 .......u 10 51. 345 ......

Subtracting, 5. 1345 u 9990 51345  51

So, 5 1345

Required fraction is

1665

224

5

Example 8. Express 42. 3478

into simple fraction.

Solution : 42. 3478

So, 42 3478

. u (^10000 42). 347878 .........u 10000 42348. 7878

and 42 3478

. u 100 = 42. 347878 ........u 100 = (^4234). 7878

Subtracting, 42. 3478

u 9900 = 423478  4234

Therefore, 42 3478

Required fraction is

825

Explanation : From the examples 5, 6, 7 and 8 , it appears that,

x The recurring decimal has been multiplied by the number formed by putting at the

right side of 1 the number of zeros equal to the number of digits in the right side of

decimal point in the recurring decimal.

x The recurring decimal has been multiplied by the number formed by putting at the

right side of 1 the number of zeros equal to the number of digits which are non-

recurring after decimal point of the recurring decimal.

x the second product has been subtracted from the first product. By subtracting the

second product from the first product the whole number has been obtained at the

right side. Here it is observed that, the number of non-recurring part has been

subtracted from the number obtained by removing the decimal and recurring points

of recurring decimal fraction.

x The result of subtraction has been divided by the number formed by writing the same

number of 9equal to the number of digits of recurring part at the left and number of

zeros equal to the number of digits of non-recurring part at the right.

x In the recurring decimals, converting into fractions the denominator is the number of

9equal to the number of digits in the rec urring part and in right side of all 9’s

number of zeros equal to the number of digits in the non-recurring part. And the

numerator in the result that is obtained by subtracting the number of the digits

formed by omitting the digits of recurring part from the number formed by removing

the decimal and recurring points of recurring decimal.

Remark : Any recurring decimal can also be converted into a fraction. All recurring

decimals are rational numbers.

Example 9. Express 5 23457

. into simple fraction.

Similar recurring decimals and Non-similar Recurring decimals :

If the numbers of digits in non-recurring part of recurring decimals are equal and also

numbers of digits in the recurring parts are equal, those are called similar recurring

decimals. Other recurring decimals are called non-similar recurring decimals. For

example : 12 45

. and 6 32

. and 125 897

. are similar recurring decimals. Again,

. and 7. 45789 ;

. and 2 89345

. are none-similar recurring decimals.

The Rules of Changing Non-Similar Recurring Decimals into Similar Recurring

Decimals

The value of any recurring decimals is not changed, if the digits of its recurring part are

written again and again, For Example, 6. 4537 6. 453737

Ǥ

Here each one is a recurring decimal, 6. 45373737 .........is a non-terminating decimal.

It will be seen that each recurring decimal if converted into a simple fraction has the

same value.

9900

63892

999900

6453092

999900

6453737 645

6. 453737

9900

63892

9900

4537 645

6. 4537



 

 



In order to make the recurring decimals similar, number of digits in the non-recurring

part of each recurring decimal is to be made equal to the number of digits of non-

recurring part of that recurring decimal in which greatest number of digits in the non-

recurring part exists and the number of digits in the recurring part of each recurring

decimal is also to be made equal to the lowest common multiple of the numbers of digits

of recurring parts of recurring decimals.

Example 12. Convert 5. 6 , 7. 345

and (^10 )

˜ into similar recurring decimals.

Solution : The number of digits of non-recurring part of 5 6 , 7345

˜ ˜ and 10 78423

. are

0 , 1 and^2 respectively. Here the number of dig^ its in the non-recurring part occurs in

. and that number is 2. Therefore to make the recurring decimals similar the

number of digits in the non-recurring part of each recurring decimal is to be made 2.

Again, the numbers of digits to recurring parts of 5. 6 , 7. 345

and 10. 78423

are 1 , 2

and 3 respectively. The lowest common multiple of 1 , 2 and 3 is 6. So the number of

digits in the recurring part of each recurring decimal would be 6 in order to make them

similar.

So, 5. 6 5. 66666666 , 7. 345 7. 34545454

and 10. 78423 10. 78423423

Required similar recurring decimals are 5. 66666666 , 7. 34545454

,

respectively.

Example 13. Convert 1. 7643 , 3 24

. and 2. 78346

into similar recurring decimals.

Solution : In (^1). 7643 the number of digits in the non-recurring part means 4 digits after

decimal point and here there is no recurring part.

In 3 24

. the number of digits in the recurring and non-recurring parts are respectively 0

and 2.

In 2 78346

. the number of digits in the recurring and non-recurring parts are

respectively 2 and 3.

The highest number of digits in the nonrecurring parts is 4 and the L.C.M. of the

numbers of digits in the recurring parts i.e. 2 and 3 is 6. The numbers of digits is the

recurring and nonrecurring parts of each decimal will be respectively 4 and 6.

Required recurring similar decimals are 1 7643000000

and

Remark : In order to make the terminating fraction similar, the required number of

zeros is placed after the digits at the extreme right of decimal point of each decimal

fraction. The number of non-recurring decimals and the numbers of digits of non-

recurring part of decimals after the decimal points are made equal using recurring digits.

After non-recurring part the recurring part can be started from any digit.

Activity : Express 3. 467 , 2. 01243

and 7. 5256

into similar recurring fractions.

Addition and Subtraction of Recurring Decimals

In the process of addition or subtraction of recurring decimals, the recurring decimals are

to be converted into similar recurring decimals. Then the process of addition or

subtraction as that of terminating decimals is followed. If addition or subtraction of

terminating decimals and recurring decimals together are done, in order to make

recurring decimals similar, the number of digits of non-recurring part of each recurring

should be equal to the number of digits between the numbers of digits after the decimal

points of terminating decimals and that of the non-recurring parts of recurring decimals.

The number of digits of recurring part of each recurring decimal will be equal to L.C.M.

as obtained by applying the rules stated earlier and in case of terminating decimals,

necessary numbers of zeros are to be used in its recurring parts. Then the same process

of addition and subtraction is to be done following the rules of terminating decimals. The

sum or the difference obtained in this way will not be the actual one. It should be

˜ =^5 89798798 |^79

Here the number is extended upto 2 more digits after the completion of the recurring

part. The additional digits are separated by drawing a vertical line. Then 2 has been

carried over from the sum of the digits at the right side of the vertical line and this 2

is added to the sum of the digit at the left side of the vertical line. The digit in the

right side of the vertical line are the same and the recurring point. Therefore both the

sums are the same.

Example 15. Add : 8 ˜ 9478 , 2 ˜ 346

and 4 71

Solution : To make the decimals similar, the number of digits of non-recurring parts

would be 3 and that of recurring parts would be 6 which is L.C.M. of 3 and 2.

[ 8  0  1  1 10 , Here the digit in the

second place on the left is 1 which is to be

carried over. Therefore 1 of 10 is added.]

The required sum is 16 011019565

Activity : Add :1. 2 097

˜ and 5 12768

˜ ˜ and 8 05678

Example 16. Subtract 5 24673

˜ from 8 243

Solution : Here the number of digits in the non-recurring part would be 2 and that of

recurring part is 6 which is L.C.M. of 2 and 3. Now making two decimal numbers

similar, subtraction is done.

[Subtracting 6 from 3, 1 is to be

carried over.]

The required sum is 2 99669760

Remark : If the digit at the beginning place of recurring point in the number from which

deduction to be made is smaller than that of the digit in the number 1 is to be subtracted

from the extreme right hand digit of the result of subtraction.

Note : In order to make the conception clear why 1 is subtracted, subtraction is done in

another method as shown below :

The required difference is 2 99669760 | 67

Here both the differences are the same.

Example 17. Subtract 16

˜ from^24

Solution :

[7 is subtracted from 6.1 is to be carried

over.]

The required difference is 8. 01901

Note :

Activity : Subtract :1. 10 ˜418 from 13 12784

˜ from 23 0394

Multiplication and Division of Recurring Decimals :

onverting recurring decimals into simple fraction and completing the process of their C

multiplication or division, the simple fraction thus obtained when expressed into a decimal

fraction will be the product or quotient of the recurring decimals. In the process of

multiplication or division amongst terminating and recurring decimals the same method is to

be applied. But in case of making division easier if both the divident and the divisior are of

recurring decimals, we should convert them into similar recurring decimals.

Example 18. Multiply 4 3

˜ by 5 7