mithilesh kumar singh

1. NUMBERS SYSTEM

 

 

IMP ORTANT FACTS AND F ORMULAE
 

I..N ume ral : In Hindu Arabic system, we use ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 called di git s to represent any number.

A group of digits, denoting a number is called a numeral .

We represent a number, say 689745132 as shown below :

 

6            8          9              7          4           5          1          3      2

We read it as : 'Sixty- eight crores, ninety- seven lacs, forty- five thousand, one hundred and thirty-two'.

 

II  Place Value or Local Value of a D igit in a N ume ral :

In the above numeral :

Place value of 2 is (2 x 1) = 2; Place value of 3 is (3 x 10) = 30;

Place value of 1 is (1 x 100) = 100 and so on. Place value of 6 is 6 x 108 = 600000000

 

III.  Face Value : The face value of a digit in a numeral is the value of the digit itself

at whatever place it may be. In the above numeral, the face value of 2 is 2; the face value of 3 is 3 and so on.

 

IV.TYPES OF N UMBERS

1.             N atural N umbe rs : Count ing numbers 1, 2, 3, 4, 5,..... are called nat ural

numbers.

2.Whole N umbe rs : All counting numbers together with zero form the set of whole numbers. Thus,

(i)  0 is the only who le number which is not a natural number.

(ii)  Every natural number is a whole number.

 

3.Inte ge rs  : A ll  na tura l  numbers,  0  a nd  nega t ives  o f co unt ing  numbers  i.e.,

{…, - 3, - 2, - 1,  0, 1, 2, 3,…..} together form the set of integers.

(i)  Pos itive Inte ge rs : {1, 2, 3, 4, …..} is the set of all posit ive integers.

(ii)  Ne gative Inte ge rs : {- 1, - 2, - 3,…..} is the set of all negative integers.

(iii)  N on-Pos itive and N on-N egative Inte ge rs : 0 is ne it her pos it ive nor negative. So, {0, 1, 2, 3,….} represents the set of non- negative integers, while

{0, - 1, - 2, - 3, …..} represents the set of non- positive integers.

 

4.            Even N umbers : A number divisible by 2 is called an even number, e.g., 2, 4, 6, 8,

10, etc.

 

5.          Odd Numbe rs : A number not divisible by 2 is called an odd number. e.g., 1, 3, 5, 7, 9, 11, etc.

 

6.            Prime Numbers : A number greater than 1 is called a prime number, if it has exactly two factors, namely 1 and the number itself.

Prime numbers upto 100 are : 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43,

47,  53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

Prime numbers Greater than 100 : Let p be a given number greater than 100. To find out whether it is prime or not, we use the following method :

Find a whole number nearly greater than the square root of p. Let k > *jp. Test whether p is divisible by any prime number less tha n k. If yes, then p is not prime. Otherwise, p is prime.

e.g,,We have to find whether 191 is a prime number or not. Now, 14 > V191. Prime numbers less than 14 are 2, 3, 5, 7, 11, 13.

191 is not divisible by any of them. So, 191 is a prime number.

 

7.  Composite Numbe rs : Numbers greater than 1 which are not prime, are known as

composite numbers, e.g., 4, 6, 8, 9, 10, 12.

Note :   (i) 1 is neither prime nor composite.

(ii) 2 is the only even number which is prime.

(iii) There are 25 prime numbers between 1 and 100.

 

8.            Co-primes : Two numbers a and b are said to be co-primes, if their H.C.F. is 1. e.g.,

(2, 3), (4, 5), (7, 9), (8, 11), etc. are co-primes,

 

V. TESTS OF DIVISIBILITY

1.          Divisibility By 2 : A number is divisible by 2, if its unit's digit is any of 0, 2, 4, 6, 8.

Ex. 84932 is divisible by 2, while 65935 is not.

 

2.          Divisibility By 3 : A number is divisible by 3, if the sum of its digits is divisible by 3.

Ex.592482 is divisible by 3, since sum of its digits = (5 + 9 + 2 + 4 + 8 + 2) = 30, which is divisible by 3.

But, 864329 is not divisible by 3, since sum of its digits =(8 + 6 + 4 + 3 + 2 + 9) = 32,

which is not divisible by 3.

 

3.            Divisibility By 4 : A number is divisible by 4, if the number formed by the last two

digits is divisible by 4.

Ex. 892648 is divisible by 4, since the number formed by the last two digits is 48, which is divisible by 4.

But, 749282 is not divisible by 4, since the number formed by the last tv/o digits is 82, which is not divisible by 4.

 

4.            Divisibility By 5 : A number is divisible by 5, if its unit's digit is either 0 or 5. Thus, 20820 and 50345 are divisible by 5, while 30934 and 40946 are not.

 

5.            Divisibility By 6 : A number is divisible by 6, if it is divisible by both 2 and 3. Ex.

The number 35256 is clearly divisible by 2.

Sum of its digits = (3 + 5 + 2 + 5 + 6) = 21, which is divisible by 3. Thus, 35256 is

divisible by 2 as well as 3. Hence, 35256 is divisible by 6.

 

6.              Divisibility By 8 : A number is divisible by 8, if the number formed by the last three digits of the given number is divisible by 8.

Ex. 953360 is divisible by 8, since the number formed by last three digits is 360, which is divisible by 8.

But, 529418 is not divisible by 8, since the number formed by last three digits is 418, which is not divisible by 8.

7.              Divisibility By 9 : A number is divisible by 9, if the sum of its digits is divisible

by 9.

Ex. 60732 is divisible by 9, since sum of digits * (6 + 0 + 7 + 3 + 2) = 18, which is

divisible by 9.

But, 68956 is not divisible by 9, since sum of digits = (6 + 8 + 9 + 5 + 6) = 34, which is

 

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