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All of us know something about number system. So, it's time to go little deep. Because thats the only way to get more information. Let's think on an important question

Consider three comparable entities a,b,c. Now, we have found the relation between $a$,$b$&$c$ as $ a>b>c $ which also implies $c$ is not smaller than $b$ and $b$ is not smaller than $a$. But, can this representation tell us that by how much $b$ is greater than $a$ and $c$ is greater than b.

But, now if I give $a$, $b$ & $c$ numbers as $ a=1,b=2,c=3 $ then you can tell $ b-a = c-b $. And if I give a,b,c numbers 1,2,5 then $ b-a \neq c-b $.

Let's take a look at some popular number systems:(this portion you can skip)

Roman number system(Europian number System):

This number system is almost extinct. But, still used in some europian countries at small scale. In this system basically three symbols(I,V,X) were used together to represent a number. 1-10 numbers were represented in this system as I,II,III,IV,V,VI,VII,VIII,IX,X. This number system is not suitable for scientific calculations.

Hindu - Arabian number system:

This number system was invented by indians, later arabs adopted this number system and arab merchants spread this in other parts of world. This system has some highly useful features like inclusion of magic no. 0, ten different symbol to represent no. etc. because of which this became the primary choice for scientific calculations. The importance of the right number system can be unsterstood from the fact that at the time when 10,000 was treated as very big number in europe indian astrologist were doing calculations involving a number as high as 10^7. These indian astrologist later invented indices which is now integral part of our modern number system. Our modern number system is hindu-arabic number system.

The numbers starting from 1 to infinity together are called as natural numbers. It does not involve fractions,zero,decimals,negative numbers and imaginary numbers.

All natural numbers together with 0 are called as whole numbers. Zero is the only whole number which is not natural number.

All numbers including negative numbers, fractions,zero,decimals and natural numbers are called integers.

1. The value of the highest symbol in any modern number system is less by one or by 2nd smallest number in number system than total number of symbols used to represent numbers in number system.

For example, modern number system involves 10 numbers starting from 0 to nine. The value of highest symbol that is 9 = 10 - 1(2nd smallest symbol in number system).

2. The difference between the values of successive symbols is equal to the smallest 2nd symbol in the number system.

For ex. 3-2=1 or 5-4 = 1 and also for binary system 1-0 = 1.

3. Every digit(symbol) in number has two values. (I)face value and (II) place value

For ex. In number 123 , the face value of 2nd digit from right is 2 and face value of 1st digit from right is 3

Consider, a number 3333. As, we traverse from rightmost 3 to left most 3, the value of 3 incremented. If we observe carefully, then the value of each 3 maintains a constant ratio of 10 (total number of symbols in decimal system) with its preceding term.

Ex. In number 123, the face value of 1st digit from left is $ 10^0 $ i.e 1, 2nd digit from left has value $ 10^1 $ and 3rd digit from left has value of $ 10^2 $.

Consider a number from m symbol system which we want to convert to n symbol system where (m>n).

In such case we devide that no. by n & write remainder obtained by division in reverse order to represent that number into n number system.

For example lets take m=10 & n=2

then lets convert number 12 into binary

Divisor | Number | Remainder |
---|---|---|

2 | 12 | 0 |

2 | 6 | 0 |

2 | 3 | 1 |

2 | 1 | 1 |

Now, we arrange the obtained remainder in reverse order to represent that number in binary form.

So, 12 in binary becomes 1100.

Consider a number from m symbol system which we want to convert to n symbol system where (m

For ex. let's assume m=2 and n=10

Let's convert 1100(m=2) into decimal.

So, $ 1100(2) = 1*2^3 + 1*2^2 + 0*2^1 + 0*2^0.(10) $

1101(2) = 8+4+0+0 = 12.

Hexadecimal number system is widely used number system by computer programmers especially system programmers. Converting a number from binary to hexadecimal and vice versa is easier.

The hexadecimal number system has total 16 symbols starting from 0. The digits 10,11,12,13,14,15 are represented in hexadecimal system by A,B,C,D,E & F respectively. while converting from decimal to hexadecimal system, we divide the number by 16 instead by 2 which we done while converting from decimal to binary.

Do you remember sequence 8,4,2,1? We are going to use this sequence while converting from hexadecimal to binary. The highest number in hexadecimal system is F which equals to 15 in decimal and 1111 in binary. Now, if we want to convert this 1111 into decimal, how we will do that? $ 1*2^3 + 1* 2^2 + 1* 2^1 + 1* 2^0 = 8+4+2+1 = 15 $.

while converting a number from hexadecimal to binary we just replace the indivisual hex number by its equivalent binary value.

Below is the table of binary values of hexdecimal numbers.

Decimal | Hexadecimal | Binary |
---|---|---|

0 | 0 | 0000 |

1 | 1 | 0001 |

2 | 2 | 0010 |

3 | 3 | 0011 |

4 | 4 | 0100 |

5 | 5 | 0101 |

6 | 6 | 0110 |

7 | 7 | 0111 |

8 | 8 | 1000 |

9 | 9 | 1001 |

10 | A | 1010 |

11 | B | 1011 |

12 | C | 1100 |

13 | D | 1101 |

14 | E | 1110 |

15 | F | 1111 |

A succession of numbers formed and arranged in a definite order according to certain definite rule, is called progression.

If each term of the progression differs from it's preceding terms by a constant, then such a progression is called airthmatic progression.

The progression in which every number bears a contant ratio r with its preceding term is called Geometric progession.

EX. $ a,ar,ar^2,ar^3,..... $

$ x^n - a^n $ is divisible by $ (x-a) $ for all values of n.

$ x^n - a^n $ is divisible by $ (x+a) $ for all even values of n.

$ x^n + a^n $ is divisible by $ (x+a) $ for all odd values of n.

If the units digit of number are even then the number is divisible by 2.

If the sum of digits of number are divisible by 3, then number is divisible by 3.

If the unit digits of number are either 5 or 0, then number is divisible by 5.

If the difference of the sum of digits at the odd places and the sum of digits at even places are equal, then the number is divisible by 11.

Ex. 14641 is divisible by 11, since, sum of digits at odd places i.e 1+6+1 is equal to the sum of digits at the even places i.e. 4+4.

The number is divisible by 4, if the last two digits of a number are divisible by 4.

The number is divisible by 6, if it is divisible by both 2 and 3.

The number is divisible by 9, if the sum of digits of number is divisible by 9

Ex. 3942 is divisible by 9, since, the all digits of number i.e. 18 is divisible by 9.

If the digit in unit place of a number is 0, the number is divisible by 10

If the number is divisible by both 4 and 3, then number is divisible by 12.

If the number is divisible by both 7 and 2, then number is divisible by 14.

If the number is divisible by both 5 and 3, then number is divisible by 15.

The number is divisible by 16, if the number formed by last 4 digits is divisible by 16.

If the number is divisible by both 11 and 2, then number is divisible by 22.

1. $ (a + b)^2 = a^2 + b^2 + 2ab $

2. $ (a - b)^2 = a^2 + b^2 - 2ab $

3. $ (a + b)^2 - (a - b)^2 = 4ab $

4. $ (a + b)^2 + (a - b)^2 = 2(a^2 + b^2) $

5. $ (a^2 - b^2) = (a + b) (a - b) $

6. $ (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ac) $

7. $ (a^3 + b^3) = (a + b) (a^2 - ab + b^2) $

8. $ (a^3 - b^3) = (a - b) (a^2 + ab + b^2) $

9. $ (a^3 + b^3 + c^3) = (a + b + c) (a^2 + b^2 + c^2 - ab - bc - ca) $

10. If $ a + b + c = 0 $ , then $ a^3 + b^3 + c^3 = 3abc $

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