Numbers form the basic units in Mathematics. After studying the basic formulae and operations you will be able to apply those in solving problems.Numbers are the collection of certain symbols or figures called digits. Different systems were evolved in different parts of the world for the purpose of numbering. The common number system in use is the Decimal number system. In this system, we use digits as 0,1,2,3,4,5,6,7,8 and 9. A combination of these digits representing a number is called a numeral.
FACE VALUE AND PLACE VALUE
Face value is equal to the value of the digit itself, irrespective of its place in the numeral.Place value is equal to the place of the given digit. We begin from the extreme right as unit's place, ten's place, hundred's place, thousand's place and so on. Eg:-276345 2763945 Face value Place value 1)5 5 5x10°= 5 2)4 4 4x10^1= 40 3)3 3 3xl0^3= 3000 4)6 6 6x10^4= 60000 5)7 7 7x10^5 =700000
TYPES OF NUMBERS
NATURAL NUMBERS (N)
Counting numbers such as 1,2,3,4,.... are called Natural Numbers Eg.- N = 1,2,3,4,..........8
WHOLE NUMBERS (W)
This includes all natural numbers and 0. Eg:- W = 0,1,2,3,....8
INTEGERS (I)
This includes all whole numbers along with negative numbers. Eg:- I = (,...,-2,-1,0,1,2,.....8)
POSITIVE INTEGERS:-
This include all natural numbers along with negative number Eg:- I=(8,....,-2,-1,0,1,2,.........8)
NEGATIVE INTEGERS:-
This include -1,-2,... 8 'Zero' is neither positive nor negative.
NON-NEGATIVE INTEGERS:-
This include all whole numbers. Eg:- 0,1,2,......8 Non-positive Integers:- This include 0,-1,-2,...8
EVEN NUMBERS
These are the numbers which are completely divisible by 2. Eg:- 2,4,6,8,
ODD NUMBERS
These include numbers which are not divisible by 2. Eg:- 1,3,5,... 'Zero' is an exception to the Even-Odd classification.
RATIONAL NUMBER
Real numbers which are expressed in the form of fractions like ' p/q , where p and q are integers and q not equal to 0 are called rational numbers. Eg:- 3/5,7/5,3,-9, etc
IRRATIONAL NUMBERS
Real numbers which cannot be expressed in the form of fractions like ' p/q where q ? are called Irrational Numbers. Eg :- 3,5,2, etc.
PRIME NUMBERS
Numbers which have no factors besides itself and unity are called prime numbers. Eg:-2,3,5,7,etc
'ZERO'
•Number which is neither ve nor -ve •Smallest whole number •Has no reciprocal •When multiplied by any number gives 0 itself. •Division by zero is not defined. •2 is the only even number which is prime •1 is neither prime nor composite •A composite number may be even or odd.
CO-PRIME NUMBERS
Two numbers are said to be co-prime if their HCF is 1. Eg:- (5,7), (13,15), etc
COMPOSITE NUMBERS
Numbers having other factors besides itself and unity are called Non-prime/ Composite Numbers. Eg- 6,7,16, etc
TEST FOR A PRIME NUMBER
To test whether any number is prime or not, take a number larger than the approximate square root of that number.Let it be 'x'. Test the divisibility of the given number by every prime number less than 'x'. If it is not divisible by any of them, it is a prime number. Eg;- (1) 293 Ans: 293 =17 Prime number less than 17 are 2,3,5,7,11 and 13 and 293 is not divisible by any of them .'.293 is a prime number (2)329 329=18 prime number less than 18 are 2,3,5,7 11,13 and 17. 329 is divisible by 7. .'.329 is not a prime number.
EXAMPLES
1.How many prime numbers are there below 100? There are 25 prime numbers below 100. 2.What is the difference in the place value and face value of 7 is 837635? The place value of 7 is 7000 The face value of 7 is 7 .'.Required difference = 7000 - 7 = 6993 3.Test whether the number 179 is prime or not. l79 =13 Prime numbers less than 13 = 2,3,5,7, 11 and 179 is not divisible by any of them. .'.179 is a prime number.
TEST OF DIVISIBILITY
1.DIVISIBILITY BY 2
: A number is divisible by 2 if it ends with 0,2,4,6 and 8. Eg: 262, 178, 200, etc
2.DIVISIBILITY BY 3 :
A number is divisible by 3 if the sum of the digits is divisible by 3. Eg:- (1)327= 3 2 7 = 12, divisible by 3. So 327 is divisible by 3 (2)27356312 = 2 7 3 5 6 3 1 2 = 29, not divisible by 3 So 27356312 is not divisible by 3.
3.DIVISIBILITY BY 4 :
A number is divisible by 4 if the number formed by its last two digits is divisible by 4 or the last two digits are 00. Eg:- (1)500 : Since last two digits are 00. 500 is divisible by 4. (2)2618 : Since last two digits = 18, not divisible by 4. 2618 is not divisible by 4.
4.DIVISIBILITY BY 5 :
If the last digit of a number is either 'o or 5', then the number is divisible by 5. Eg:- 73235, 2310
5.DIVISIBILITY BY 6 :
A number is divisible by 6, if it is divisible by both 2 and 3. Eg: - (1)312 : Divisible by both 2 and 3. So, divisible by 6. (2)410 : Divisible by 2 but not by 3. So, not divisible by 6
6.DIVISIBILITY BY 8 :
A number is divisible by 8, if the last 3 digits is divisible by 8 or the last 3 digits are 000. Eg:-(1) 27000; 57512
7.DIVISIBILITY BY 9 :
A number is divisible by 9, if the sum of the digits is divisible by 9. Eg:- 549, 2763
8.DIVISIBILITY BY 10 :
A number is divisible by 10 if its unit digit is 0. Eg:- 210, 75620
9.DIVISIBILITY BY 11 :
A number is divisible by 11 if the difference of the sum of its digits at odd places and the sum of its digits at even places is either 0 or a number divisible by 11. Eg:- (1)121: Sum of digits at odd places =11=2 2-2 = 0 .'.121 is divisible by 11 (2)7040: Sum of digits at even places = 7 4=11 .'.7040 is divisible by 11
10.DIVISIBILITY BY 12:
A number is divisible by 12 if it is divisible by both 3 and 4. Eg:- 672, 10224
11.DIVISIBILITY BY 14:
A number is divisible by 14 if it is divisible by both 2 and 7. Eg. 1456, 39200
12. DIVISIBILITY BY 16:
A numbed is divisible by 16 if the last 4 digits are divisible by 16 or it is either '0000' or it is divisible by 16. Eg:- 1088, 10000 13. DIVISIBILITY BY 18 : A number is divisible by 18 if it is divisible by both 2 and 9. Eg:-1422, 6570
EXAMPLES
1.A certain number consists of two digits whose sum is 8. If the order of the digits is reversed, the new number is 18 less than the original number. Find the original number. Let ten's digit be x and unit's digit be y. Then, x y =8 ------------(i) (10x y) - (10y x) = 18 9x-9y = 18 x-y =2 --------------(ii) Solving (i) and (ii) => 2y =6 .'. y =3 x y =8 .'.x =5 .'.The required number is 53 2.24 is divided into two parts such that 6 times the first part exceeds 4 times the second part by 4. Find the first part. Let the first part be x Then the second part = 24 - x Then, 6x - 4 (24 - x) = 4 6k - 96 4x =4 x = 10 .'.the first part is 10
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