Hello Aspirants,\r\nMany of you will find this article to be little lengthy. We have tried to make this article as short as possible. But, since,\r\nthe number system is the one of the most fundamental concept in mathematics or even we can call number system\r\nas the base of mathematics, every concept relating to number system is important.
\r\nAll of us know something about number system. So, it\'s time to go little deep. Because thats the only way to\r\nget more information. Let\'s think on an important \r\nquestion
"What is number System?"\r\nConsider three comparable entities a,b,c. Now, we have found the relation between $a$,$b$&$c$ as $ a>b>c $ which\r\nalso implies $c$ is not smaller than $b$ and $b$ is not smaller than $a$. But, can this representation tell us that by how\r\nmuch $b$ is greater than $a$ and $c$ is greater than b.
Can you tell whether $ b-a = c-b $??\r\nBut, 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\r\n1,2,5 then $ b-a \neq c-b $.
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So, number system is basically a technique which help us to compare different entities and to establish exact\r\nrelation(< ,=,>) between these entities.\r\nLet\'s take a look at some popular number systems:(this portion you can skip)
\r\nRoman number system(Europian number System):
\r\nThis number system is almost extinct. But, still used in some europian countries at small scale.\r\nIn this system basically three symbols(I,V,X) were used together to represent a number. 1-10 numbers were represented\r\nin this system as I,II,III,IV,V,VI,VII,VIII,IX,X. This number system is not suitable for scientific calculations.
\r\nHindu - Arabian number system:
\r\nThis number system was invented by indians, later arabs adopted this number system and arab merchants spread this in other parts of world.\r\nThis 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.\r\nThe 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\r\nindian astrologist were doing calculations involving a number as high as 10^7. These indian astrologist later invented indices which is now integral part of\r\nour modern number system. Our modern number system is hindu-arabic number system.
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Natural numbers:\r\nThe numbers starting from 1 to infinity together are called as natural numbers. It does not involve fractions,zero,decimals,negative numbers and imaginary numbers.
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Whole numbers:\r\nAll natural numbers together with 0 are called as whole numbers. Zero is the only whole number which is not\r\nnatural number.\r\n
Integers:\r\nAll numbers including negative numbers, fractions,zero,decimals and natural numbers are called integers.\r\n
Some important characteristics of number system:\r\n1. The value of the highest symbol in any modern number system is less by one or by 2nd smallest number\r\nin number system than total number of symbols used to represent numbers in number system.
\r\nFor example, modern number system involves 10 numbers starting from 0 to nine. The value of highest symbol\r\nthat is 9 = 10 - 1(2nd smallest symbol in number system).
\r\n\r\n2. The difference between the values of successive symbols is equal to \r\n the smallest 2nd symbol in the number system.
\r\nFor ex. 3-2=1 or 5-4 = 1 and also for binary system 1-0 = 1.
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(Based on these two properties anyone can design their own number systems involving n number of different symbols.\r\nFor Ex. binary,hexadecimal number systems etc.)\r\n3. Every digit(symbol) in number has two values. (I)face value and (II) place value
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(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\r\nin that number system in comparison to the values of other symbols from that system.
\r\nFor ex. In number 123 , the face value of 2nd digit from right is 2 and face value of 1st digit from right is 3
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(II)Place value: \r\nConsider, a number 3333. As, we traverse from rightmost 3 to left most 3, the value of 3 incremented. If we observe\r\ncarefully, then the value of each 3 maintains a constant ratio of 10 (total number of symbols in decimal system) with its preceding term. \r\n
Important: That means the place value of digits in a number of any number system are in geometric progression and common ratio of\r\nthis GP is always equal to the total number of symbols in that number system. This important property will help us \r\nwhile converting from binary(number system having less symbols) to decimal(number system having more symbols). \r\n
We have disscussed about GP below in this article\r\nEx. 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 $.
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Conversion of number from decimal to binary(from more symbol system to less symbol system):\r\nConsider a number from m symbol system which we want to convert to n symbol system where (m>n).
\r\nIn such case we devide that no. by n & write remainder obtained by division in reverse order to represent that number\r\ninto n number system.
\r\nFor example lets take m=10 & n=2
\r\nthen lets convert number 12 into binary
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\r\n\r\n\r\n| Divisor | \r\nNumber | \r\nRemainder | \r\n
\r\n\r\n\r\n\r\n| 2 | \r\n12 | \r\n0 | \r\n
\r\n\r\n| 2 | \r\n6 | \r\n0 | \r\n
\r\n\r\n| 2 | \r\n3 | \r\n1 | \r\n
\r\n\r\n| 2 | \r\n1 | \r\n1 | \r\n
\r\n\r\n
\r\nNow, we arrange the obtained remainder in reverse order to represent that number in binary form.
\r\nSo, 12 in binary becomes 1100.\r\n
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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\r\nabout this in next section\r\n
Conversion of a number from Binary to Decimal(less symbol system to more symbol system):\r\nConsider a number from m symbol system which we want to convert to n symbol system where (m
\r\nIn 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\r\nthe number digits from right to left.
\r\nFor ex. let\'s assume m=2 and n=10
\r\nLet\'s convert 1100(m=2) into decimal.
\r\nSo, $ 1100(2) = 1*2^3 + 1*2^2 + 0*2^1 + 0*2^0.(10) $
\r\n 1101(2) = 8+4+0+0 = 12.
\r\nHexadecimal number System
\r\nHexadecimal number system is widely used number system by computer programmers especially system programmers.\r\nConverting a number from binary to hexadecimal and vice versa is easier.
\r\nThe hexadecimal number system has total 16 symbols starting from 0. The digits 10,11,12,13,14,15 are represented in \r\nhexadecimal 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 \r\nwe done while converting from decimal to binary.
\r\nConversion from Hexadecimal to binary
\r\nDo you remember sequence 8,4,2,1? We are going to use this sequence while converting from hexadecimal to binary.\r\nThe 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,\r\nhow we will do that? $ 1*2^3 + 1* 2^2 + 1* 2^1 + 1* 2^0 = 8+4+2+1 = 15 $.
\r\nwhile converting a number from hexadecimal to binary we just replace the indivisual hex number by its equivalent binary value.
\r\nBelow is the table of binary values of hexdecimal numbers.
\r\n\r\n \r\n\r\n| Decimal | \r\n Hexadecimal | \r\n Binary | \r\n
\r\n\r\n\r\n\r\n| 0 | \r\n 0 | \r\n 0000 | \r\n
\r\n\r\n| 1 | \r\n 1 | \r\n 0001 | \r\n
\r\n\r\n| 2 | \r\n2 | \r\n0010 | \r\n
\r\n\r\n| 3 | \r\n3 | \r\n0011 | \r\n
\r\n\r\n| 4 | \r\n4 | \r\n0100 | \r\n
\r\n\r\n| 5 | \r\n5 | \r\n0101 | \r\n
\r\n\r\n| 6 | \r\n6 | \r\n0110 | \r\n
\r\n\r\n| 7 | \r\n7 | \r\n0111 | \r\n
\r\n\r\n| 8 | \r\n8 | \r\n1000 | \r\n
\r\n\r\n| 9 | \r\n9 | \r\n1001 | \r\n
\r\n\r\n| 10 | \r\nA | \r\n1010 | \r\n
\r\n\r\n| 11 | \r\nB | \r\n1011 | \r\n
\r\n\r\n| 12 | \r\nC | \r\n1100 | \r\n
\r\n\r\n| 13 | \r\nD | \r\n1101 | \r\n
\r\n\r\n| 14 | \r\nE | \r\n1110 | \r\n
\r\n\r\n| 15 | \r\nF | \r\n1111 | \r\n
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\r\nProgression
\r\nA succession of numbers formed and arranged in a definite order according to certain definite rule, is called progression.
\r\n1. Airthmatic progression:
\r\n If each term of the progression differs from it\'s preceding terms by a constant, then such a progression is called\r\nairthmatic progression.
\r\nThe 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.
\r\nThe sum of nth term of AP $ = n/2[1st term + last term] $
\r\n\r\n2. Geometric Progression:
\r\nThe progression in which every number bears a contant ratio r with its preceding term is called Geometric progession.
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EX. $ a,ar,ar^2,ar^3,..... $
\r\nThe nth term of an AP is given by $ g(n) = ar^(n-1) $ .
\r\nThe sum of the 1st nth terms of GP are given by $ a(r^n - 1)/ (r-1) $.
\r\nSome division tips:
\r\n$ x^n - a^n $ is divisible by $ (x-a) $ for all values of n.
\r\n$ x^n - a^n $ is divisible by $ (x+a) $ for all even values of n.
\r\n$ x^n + a^n $ is divisible by $ (x+a) $ for all odd values of n.
\r\n\r\nTest of divisibility
\r\nDivision by 2
\r\nIf the units digit of number are even then the number is divisible by 2.
\r\nDivision by 3
\r\nIf the sum of digits of number are divisible by 3, then number is divisible by 3.
\r\nDivision by 5
\r\nIf the unit digits of number are either 5 or 0, then number is divisible by 5.
\r\nDivision by 11
\r\nIf 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.
\r\nEx. 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.
\r\nDivision by 4
\r\nThe number is divisible by 4, if the last two digits of a number are divisible by 4.
\r\n Division by 6
\r\nThe number is divisible by 6, if it is divisible by both 2 and 3.
\r\nDivision by 9
\r\nThe number is divisible by 9, if the sum of digits of number is divisible by 9
\r\nEx. 3942 is divisible by 9, since, the all digits of number i.e. 18 is divisible by 9.
\r\nDivision by 10
\r\nIf the digit in unit place of a number is 0, the number is divisible by 10
\r\nDivision by 12
\r\nIf the number is divisible by both 4 and 3, then number is divisible by 12.
\r\nDivision by 14
\r\nIf the number is divisible by both 7 and 2, then number is divisible by 14.
\r\nDivision by 15
\r\nIf the number is divisible by both 5 and 3, then number is divisible by 15.
\r\nDivision by 16
\r\nThe number is divisible by 16, if the number formed by last 4 digits is divisible by 16.
\r\nDivision by 22
\r\nIf the number is divisible by both 11 and 2, then number is divisible by 22.
\r\nSome basic formulae
\r\n1. $ (a + b)^2 = a^2 + b^2 + 2ab $
\r\n2. $ (a - b)^2 = a^2 + b^2 - 2ab $
\r\n3. $ (a + b)^2 - (a - b)^2 = 4ab $
\r\n4. $ (a + b)^2 + (a - b)^2 = 2(a^2 + b^2) $
\r\n5. $ (a^2 - b^2) = (a + b) (a - b) $
\r\n6. $ (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ac) $
\r\n7. $ (a^3 + b^3) = (a + b) (a^2 - ab + b^2) $
\r\n8. $ (a^3 - b^3) = (a - b) (a^2 + ab + b^2) $
\r\n9. $ (a^3 + b^3 + c^3) = (a + b + c) (a^2 + b^2 + c^2 - ab - bc - ca) $
\r\n10. If $ a + b + c = 0 $ , then $ a^3 + b^3 + c^3 = 3abc $
\r\nThank you for reading the article. Our next articles will not be this much lengthy. Keep visiting us.'),