1.Number System
- Numbers are the collection of certain symbols or figures called digits
- 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.
Example face value of 3 in 47326 = 3
Face value of 7 in 47326 = 7
- 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.
Ten Thousand | Thousand | Hundred | Tens | Unit Place |
---|---|---|---|---|
6 | 2 | 5 | 1 | 8 |
Place value of 5 = 5 x 100 = 500
Place value of 1 = 1×10 = 10
Eg: 276345
276345 | Face Value | Place Value |
---|---|---|
1) 4 | 4 | 4×10 = 40 |
2) 3 | 3 | 3×100 =300 |
3) 6 | 6 | 6×1000 =6000 |
4) 2 | 2 | 2×10000 = 20000 |
TYPES OF NUMBERS
1. Natural Numbers (N)
Counting numbers such as 1,2,3,4………… are called natural numbers.
Eg: N =1,2,3,4,……. ()infinity
2. Whole Numbers (W)
Includes all natural numbers and zero.
Eg. W = 0,1,2,3,……. ()infinity
3. Even Natural Numbers
These are the numbers which are completely divisible by 2.
Eg: 2,4,6,8……………
4. Odd Natural Numbers
Numbers which are not divisible by 2.
Eg: 1,3,5,7,………………
Note: Zero is an exception to the even – odd classification.
5. Integers (I)
Includes all whole numbers along with negative numbers.
Eg: I …….-2,-1,0,1,2,…………. .(infinity)
Positive Integers
This includes all natural numbers.
Eg: 1,2,3………. (infinity)
Negative Integers
This includes -1,-2……….. (infinity)
Note: Zero is neither positive nor negative.
6. Rational Numbers:
Real numbers which are expressed in the form of fractions like ‘a/b ‘ where a and b are integers and b not equal to Zero are called rational numbers.
Eg: 3/7, 8/5 , 3 (3/1), -9 (-9/1)
7. Irrational numbers
Real numbers which cannot be expressed in the form of fractions like a/b, where b not equal to 0 are called irrational numbers.
Eg: 7,3,√5
8. Prime Numbers
Numbers with only two factors → 1 and that number itself.
Eg: 2,3,5,7,11
2 = 1×2
3 =1×3
5 =1×5
CODE
4 | 1 to 10 |
4 | 10 to 20 |
2 | 20 to 30 |
2 | 30 to 40 |
3 | 40 to 50 |
2 | 50 to 60 |
2 | 60 to 70 |
3 | 70 to 80 |
2 | 80 to 90 |
1 | 90 to 100 |
1 to 25 – 9 prime numbers |
1 to 50 – 15 prime numbers |
1 to 100 – 25 prime numbers |
50 to 100 – 10 prime numbers |
Prime numbers between 1 and 100
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 Number
1 Digit | 2 Digits | 3 Digits | 4 Digits | |
Least | 2 | 11 | 101 | 1009 |
Greatest | 7 | 97 | 997 | 9973 |
9 .Composite Numbers
Numbers with more than 2 factors.
Eg: 4,6,8,9,10,12,14,……………..
4 = 1 x 4 3 factors = 1,2,4
= 2 x 2
9 = 1 x 9
= 3 x 3
10. Square Numbers
1,4,9,16,25,36
12 = 1
22 = 4
11. Cube Numbers
1,8,27,64,125
13 = 1
23 = 8
33 = 27
12. Perfect Numbers
Sum of all the factors of a number except the number, is the number itself is called perfect number.
Eg.1) 6 – Factor of 6 = 1,2,3,6
6 = 1 x6
6 = 3 x2
Sum of factors except 6 = 1+2+3 = 6 (The Number itself)
Eg:2) Factors of 28 = 1,2,4,7,14,28
28 = 1 x 28
28 = 14 x 2
28 = 7 x 4
Sum of factors except 28 = 1+2+4+7+14 = 28
13. Armstrong Numbers
153,370,371,407 are Armstrong Numbers.
153 = 13 + 53 + 33 = 1+125+27 = 153 |
370 = 33 + 73+ 03 = 9 + 343 + 0 = 370 |
371 = 33 + 73 + 13 = 9 + 343 + 1 = 371 |
407 = 43 + 03 + 73 = 64 + 0 + 343 = 407 |
14. Ramanujan Number
Ramanujan Number —– 1729
1729 —- It is the smallest number expressible as the sum of two (positive) cubes in two different ways.
1729 = 103 + 93 = 1000 + 729 = 1729
1729 = 123 + 13 = 1728 + 1 = 1729
The second number which is in the form of the Ramanujan number is 4104.
4104 = 153 + 93 = 3375 + 729 = 4104
4104 = 163 + 23 = 4096 + 8 = 4104
15. Numbers with Roman Representation
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|
I | II | III | IV | V | VI | VII | VIII | IX | X |
10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | 100 |
---|---|---|---|---|---|---|---|---|---|
X | XX | XXX | XL | L | LX | LXX | LXXX | XC | C |
100 | 200 | 300 | 400 | 500 | 600 | 700 | 800 | 900 | 1000 |
---|---|---|---|---|---|---|---|---|---|
C | CC | CCC | CD | D | DC | DCC | DCCC | CM | M |
I | 1 |
V | 5 |
X | 10 |
L | 50 |
C | 100 |
D | 500 |
M | 1000 |
V = 5 X 1000 = 5000 |
X = 10 X1000 = 10000 |
L = 50 X 1000 = 50000 |
C = 100 X 1000 = 100000 |
D = 500 X 1000 = 500000 |
M = 1000 X 1000 = 1000000 |
Eg: 15 = 10 + 5 =XV
64 = 50 + 10 + 4 = L X IV
167 = 100 + 60 + 7 = C LX VII
99 = 90 + 9 = XC IX
33 = 30 + 3 = XXX III
Million = 106 = 10 lakhs Billion = 109 = 100 Crores Trillion = 1012 = Lakh Crores |
Googole = 10100 |
Sum of ‘n’ Numbers
- Sum of first ‘n’ natural numbers = n(n+1)/2
ie; 1+2+3………… = n(n+1)/2 |
- Sum of first ‘n’ odd numbers = n2
n = no. of terms |
ie;
1+3+5+………… = n2 |
- Sum of first ‘n’ even numbers = n (n+1)
ie;
2 + 4 + 6 +…………… = n(n+1) |
- Sum of squares of first ‘n’ natural numbers = 1/6n (n + 1)(2n +1)
12+22+32 +……….. = 1/6n (n+1) (2n +1) |
ie ,
- Sum of cubes of first ‘n’ natural numbers = (n(n+1)/2)2
ie, 13 + 23 + 33 +…………….= (n(n+1)/2)2 = 1/4(n(n+1)/2)
2.Divisibilty
- Divisibility by 2
A number is divisible by 2 , if it’s unit digit is any of 0,2,4,6 and 8.
Eg; 362, 200, 124
- Divisibility by 3
A number is divisible by 3, it the sum of it’s digit is divisible by 3.
Eg; 321 = 3+2+1 =6 (divisible by3)
So 321 is divisible by 3.
Eg; 27356312 = 2+7+3+5+6+3+1+2 = 29
29 ( not divisible by 3)
So 27356312 is not divisible by 3.
- Divisiblity by 4
A number is divisible by 4 , if the number formed by the last two digits is
divisible by 4 or the last two digits are zeros.
Eg; 600 : Since last two digits are zeros
Therefore, 600 is divisible by 4.
Eg; 4528 : Since 28 is divisible by 4.
Therefore, 4528 is divisible by 4.
- Divisibility by 5
A number which is divisible by 5, if its unit’s digit ie either 5 or zero.
Eg; 3775, 4050
- Divisibility by 6
A number is divisible by 6, if it is divisible by both 2 and 3.
Eg; 612 : Divisible by both 2 and 3. 612 ÷ 2 = 306
So divisible by 6. 612 ÷ 3 = 204
Eg; 328 : Divisible by 2 but not by 3.
So not divisible by 6 328 ÷ 2 = 164
328 ÷ 3 = 109.333
- Divisible by 8
A number is divisible by 8, if the last 3 digits is divisible by 8 or
the last 3 digits are zeros.
Eg; 34000 : Last 3 digits are zeros 34000 ÷ 8 = 4250
So divisible by 8.
128 ÷ 8 = 16 |
62128 : Last 3 digits =128
So 62128 is divisible by 8.
- Divisibility by 9
A number is divisible by 9, if the sum of its digits is divisible by 9.
Eg; 864 = 8 + 6 + 4
= 18 ( 18 ÷ 9 = 2 )
2763 = 2 + 7 + 6 + 3
= 18 ( Divisible by 9)
324 = 3 + 2 + 4 =9
- Divisibility by 10
A number is divisible by 10 if its unit digit is 0.
Eg; 520 , 65320.
- 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; 231 -Sum of digits at odd places = 1 + 2
Sum of digits at even places = 3 (here only one digit).
Difference 3-3 =0
- Divisibility by 12
A number is divisible by 12 if it is divisible by both 3 & 4
Eg; 10224 , 672 10224 3 = 3408
10224 ÷ 4 = 2556
- Divisibilty by 14
A number is divisible by 14, if it is divisible by both 2 and 7.
Eg: 39200, 1988 39200 ÷ 2 = 19600
39200 ÷ 7 = 5600
- Divisibilty by 16
A number is divisible by 16, if the last 4 digits are divisible by 16 or it is
either ‘0000’ .
Eg; 10000, 60 5024 5024 ÷ 16 = 314
- Divisibilty by 18
A number is divisible by 18 if it is divisible by both 2 and 9.