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Number Systems

Hello MBA Aspirants, Many of you will find this article to be little lengthy. We have tried to make this article as short as possible. But, since, the number system is the one of the most fundamental concept in mathematics or even we can call number system as the base of mathematics, every concept relating to number system is important and people who are going to face one of the toughest exam should not skip them. People who are preparing for other examinations can skip reading some portion and to make navigating easier for them we have used bold letters and underlined letters which will help you to focus on important sections immediately.

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 "What is number System?"

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. Can you tell whether $ b-a = c-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 $.
So, number system is basically a technique which help us to compare different entities and to establish exact relation(< ,=,>) between these entities.

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.

Natural numbers:

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.

Whole numbers:

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

Integers:

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

Some important characteristics of number system:

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.
(Based on these two properties anyone can design their own number systems involving n number of different symbols. For Ex. binary,hexadecimal number systems etc.)

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

(I)Face value: The face value of a digit(symbol) in a number is the value of digit itself. The face value of a digit gives its weightage in that number system in comparison to the values of other symbols from that system.
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

(II)Place value:
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.

Important: That means the place value of digits in a number of any number system are in geometric progression and common ratio of this GP is always equal to the total number of symbols in that number system. This important property will help us while converting from binary(number system having less symbols) to decimal(number system having more symbols). We have disscussed about GP below in this article

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 $.

Conversion of number from decimal to binary(from more symbol system to less symbol system):

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.

Shortcut method: group the given number in groups of 4 and use 8,4,2,1 sequence to represent that number. You will understood clearly about this in next section

Conversion of a number from Binary to Decimal(less symbol system to more symbol system):

Consider a number from m symbol system which we want to convert to n symbol system where (m In such case we multiply each digit of the given number by m^a such that a takes values from 0 to n as we traverse the number digits from right to left.

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
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.

Conversion from Hexadecimal 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
Progression

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

1. Airthmatic 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 nth term of this AP is given by $ T(n) = a(n-1)d $ where a = 1st term of AP, n= nth term of AP & d is the constant difference.

The sum of nth term of AP $ = n/2[1st term + last term] $

2. Geometric 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,..... $

The nth term of an AP is given by $ g(n) = ar^(n-1) $ .

The sum of the 1st nth terms of GP are given by $ a(r^n - 1)/ (r-1) $.

Some division tips:
$ 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.

Test of divisibility

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

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

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

Division by 11
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.

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

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

Division by 9
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.

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

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

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

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

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

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

Some basic formulae

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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