Here is a term paper on the ‘Number System’ for class 11 and 12. Find paragraphs, long and short term papers on the ‘Number System’ especially written for college and IT students.
Term Paper on the Number System
Term Paper Contents:
- Term Paper on the Introduction to Number System
- Term Paper on the Binary Number System
- Term Paper on the Hexadecimal Number System
- Term Paper on the Octal Number System
- Term Paper on the Equivalent Digits of Number System
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1. Term Paper on the Introduction to Number System:
In order to keep track of personal wealth like number of domesticated animals etc., out of sheer necessity, counting was invented by prehistoric scientists and number systems developed. But though it served their purpose it was inadequate for any useful arithmetic, being mostly centered around counting by fingers.
With the advent of civilisation, in the western hemisphere, the Romans developed a system called Roman Numbering System, which was again totally useless for scientific calculations, even for the simplest ones.
Moreover, in the Roman System, the number of symbols required to represent any numerical value becomes abnormally large and unwieldy even for values in the range of thousands, as each symbol used has a fixed value, irrespective of the position the symbol occupies in a group of symbols.
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For example, I represents one, and two I’s as II = 2, three I’s as III = 3. The other symbols used are V for 5, X for 10, L for 50, C for 100, and M for 1000. To write 1878 in Roman Numbering System, we will have to write MCCCCCCCCLXXVIII.
A big leap in the number system, totally revolutionizing it, came from our ancestors, who invented zero and defined its properties — a unique and spectacular concept by any standards. Unfortunately, in some literatures, the number system using zero is called Arabic System, because the concept of zero reached Europe from India via Arabs.
The concept of zero introduced the system of place or positional value of numeric digits, which means that any numeric symbol has a predetermined different value depending on its position in a group of numbers. Along with this a new number system using ten digits called Decimal Number System, to represent ten fingers of the two hands, was also developed — the ten digits being 1 to 9 and 0, to represent any number, large or small.
In the positional Decimal System, if we write two 9’s side by side as 99, it denotes ninety-nine — the first 9 having a value of ninety and the second 9 having a value of nine; because the first 9 from the left side is at ten’s place and the second nine is at unit’s place, holding different positions, as we have been taught.
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The place value is directly related to the base, also called radix, of the number system. The radix in the decimal system is 10 because ten digits are used, the highest one being (10-1) = 9, as 0 is also a digit. It must be noted that the base [radix] of a number system is always equal to the number of digits used by the system or vice versa.
With this basic rule, you can design as many number systems as you like with different bases. Generalizing, we can say that the highest number in a number system with base n will be (n-1). St) if you for some reason design a number system with base 6, its highest digit will be 5.
How is the positional value determined? The concept is that we take the rightmost digit of a group of digits, which is called the Least Significant Digit or LSD, and allot it a position of 0 which is the first digit of any number system. Then as we move from rightmost [LSD] position to the left in the string of digits, we go on increasing the position number by 1 at each step.
So, if we have four digits as 7654, LSD at position 0 is 4, 5 is at position 1, 6 is at position 2, and 7 is at position 3, which is the Most Significant Digit [MSD] here. Just remember, you have to start counting from the rightmost digit, starting with 0 and not 1.
2. Term Paper on the Binary Number System
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The computer being an inert electronic device, it’s input, output, and processing are ultimately done by operating different sets of switches, which range into millions in modern computers. In the earliest computers, in many cases, the switches had to be manually operated.
Since operating a switch manually is bound, to cause delay, a search was made to have electrically operated switches, which ultimately lead to using transistors as switches. Incidentally, a switch is a device which can be made either on or off and hence, it has only two positions. To represent the two positions of the switches, a number system was required having only two digits, so that it could be used in the computer systems to represent the position of the switches.
Having come a long way from those days, when switches were manually operated, now these switches are operated by giving data and instructions in human understandable languages using alphanumeric characters. Then an appropriate translating software converts these to switching operations, for the hardware to execute.
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As already said, all switches or similar devices have only two positions, it is either ON when the connection is made or, it is OFF when the connection is broken. When we make a light switch on, connection is made between the power supply and the bulb, making it to glow.
Switches, or the two-state devices being the basic unit for computer’s internal operation, the decimal number system was found unsuitable for representing hardware operations in terms of software codes, because it had 10 positions, when we needed only two positions.
In mid-1940s John von Neumann along with H.H. Goldstein and A.W. Burks proposed a new number system using two digits called Binary Number System and it readily became the basic number system for computer operations.
A number system using two digits, will have a base (radix) of two and its highest digit will be [2-1] = 1; the other digit being 0. Thus, the Binary Number System has two digits, 0 and 1, 0 representing the off-position and 1 representing the on-position of a two state device [switch] — the 0 and 1 are called Binary digits or Bits in short.
As far as the binary and the decimal systems are concerned the digits 0 and 1 are common in both the systems, but there after they differ, as in the binary system the highest digit is 1 and in the decimal system it is 9. To get the value of two in the decimal system, we add 1 with 1 to get 2. Similarly, in binary system we will have to add 1 with 1 — but we cannot write 2, as we do in the decimal system, because the digit 2 does not exist. What is the way out?
In the decimal system, when we add 1 to the highest digit of 9, we write 0 in unit’s place and carry over 1 to the ten’s place, the value of the digit changing to 10, thus we have:
Now after adding 1 to the highest digit 9, we have a 1 in the second place, whose value is 10.
By extending the same logic to the binary number system, if we add 1 with 1, we have:
Which is equivalent to 2 of the decimal system.
But is it really true? Let’s check by computing the place values under the binary system. The base value of the binary system being 2, the positional value will be 20 for LSD and there after 21 = 2, 22 = 4, 23 = 8, 24 = 16, and so on, as we move from right” to left in a group of digits; corresponding to 100,101,102,103 etc., of the decimal system.
Then:
Earlier, we found that 1101 in decimal system represents one-thousand-one- hundred-one.
In Binary Number System, the same group of digits will be evaluated as:
The subscript 10 indicates that the digit 13 is in decimal system. Thus we can convert any binary number, which the computer understands, to get the equivalent decimal number, which we easily understand.
There is another method, called Double Summing Method, which can also be used to get decimal equivalent of binary numbers. The procedure is that for each binary digit, a “sum” is obtained by adding it with a “double” obtained by doubling the “sum” of the previous digit immediately on left; starting with the MSD.
The last “sum” so obtained gives the answer. The logic followed is that when a binary digit is shifted one position to left from its existing position in a group of digits, its value is multiplied by two; which in the decimal system gets multiplied by ten — the bases in the binary number system being two and that in the decimal number system being ten.
Let us convert 11100101 to decimal equivalent.
This, procedure works because each digit, together with its own value is doubled n times, where n is its place number. For example the MSD is doubled 7 times whereas the LSD is not doubled at all [n = 0].
How about binary fractions, which are less than 1 in value? The same logic as was used in the decimal system is followed, but now, obviously, with a base of 2. In decimal system we used position numbers as negative power of base 10 to get 101, 102, 103, and so on. In binary system, correspondingly we have 21, 22, 23 etc., as we move further to the right from the binary- point, which we called decimal-point earlier.
Thus for 0.1101:
Corresponding to the double-summing method, for fractions we can use a method called half-summing method. In this procedure, the process of “halving” starts from extreme right and since it involves halving, the multiplier is 0.5 as against 2 used in the double-summing method.
Let us redo the previous example: 
The equivalent decimal value, as before is 0.812510.
i. Decimal to Binary Conversion:
As you saw, converting binary digits, or bits, to decimal numbers was a fairly straight forward process — we just added the place values ignoring it when the bit was 0. To convert decimal numbers to equivalent binary numbers, we follow a popular process called double-dabble method.
Under this procedure, we successively divide the decimal number by 2, noting down the remainder in each case serially, starting with the first remainder which is called LSD or more appropriately LSB — Least Significant Bit. The division stops when the quotient becomes 0. Let us find out the bits to represent 765410.
Therefore, the equivalent binary number is 11101111001102
Let us check it by reconverting to decimal number.
The above method is also called Remainder Method by some. The other alternative for conversion is called Subtraction Method, where the largest possible decimal value of a binary number is subtracted one after another from the decimal number till the decimal number becomes zero. The binary digits so obtained as the power of base 2 give the bits required.
ii. Binary Arithmetic:
Binary addition is extremely simple being identical to the decimal system, except, the carry over of 1 takes place whenever the result of summation is greater than 1 — in decimal system, the carry over of 1 is done whenever the sum is greater than 9 (the highest digit in the system). Let us take a few examples. In each case the decimal equivalent is also given.
The basic principle of binary addition is:
In case, addition is to be made for more than two rows of number, it is preferable to do so in steps, in each case the numbers being added being limited to only two rows.
To solve 110.11 + 11.101 + 1100.011, we do:
The binary digits or bits are always used in fixed combination: 4 bits is called Nybble, 8 bits together is called a Byte, and 16 bits is called a Word.
In case of binary subtraction, we can also follow the technique used in the decimal system, by using “borrow” from previous position.
Here we can write:
10 – 1 = 1 (with a borrow from next bit)
In the second example, when a borrow is required at position 5, there is no 1 at position 6. In such cases, the borrow is made from the first column where there is 1 [here position 8] and then the in between columns will have a value of 10 -1 = 1 after the borrowing.
Since in terms of electric circuitry, the process of addition is much simpler and faster than subtraction, in computer operations the process of subtraction is converted to a process of addition by converting the number to be subtracted to an equivalent negative number by obtaining its complement.
The complement of a number is the number which when added to that number becomes equal to the highest digit of the number system. For example, in the decimal system 6’s complement is 3, because 6 + 3 = 9. Similarly, 9’s 9’s complement is 0, as 9 + 0 = 9. This is called nine’s complement.
The process of subtraction using 9’s complement is as follows:
1. Get the 9’s complement of the number to be subtracted
2. Add the complemented number
3. Add the carry over number, if any.
It is of utmost importance that in the complement form of processing, both the numbers have same number of digit. If they are not, 0s are used to fill up the blank spaces. An example will make this clear. Suppose we want to subtract 3 from 80, then since 80 has two digits, we convert single digit 3 to double digit 03. The 9’s complement of 03 is 96, so we add 96 to 80.
The carry of 1 is added to 76 to give the final answer as 77. When the carry is 1, the result obtained is a positive number and when there is no carry, the answer is a negative number, but in 9’s complement form — so to get the final answer, again its 9’s complement is to be obtained and that will be the final answer. In our above example, the carry was 1, so the answer 77 is a positive number. Don’t lose patience, this is extremely useful.
Now let us subtract 84 from 36. 9’s complement of 84 is 15. Adding 36 to 15 we get 51. But since there was no carry, 51 is negative and it is in 9’s complement form. To get the final answer we get 9’s complement of 51, which is 48. Therefore the answer is -48; which is correct.
In a further modification of the system, instead of adding the carry as in 9’s complement, a 1 is added to 9’s complement to get 10’s complement, which is used in the computation. Let us subtract 3 from 80 using 10’s complement form. As before 9’s complement of 3 is 96, adding 1 we get 97 — which is the ten’s complement form of 3. Then 80 + 97 = 177. Ignoring the carry of 1 which indicates that the result is a positive number, the answer is 77.
The answer is +77, also obtained by 10’s complement, after discarding 1.
In the other case, 10’s complement of 84 is [15 + 1] = 16. Now;, 36 + 16 = 52. No carry, so the result is a negative number. 10’s complement of 52 is [47 + 1] = 48 — the final answer is -48.
You can justifiably question, why we use such a complex roundabout way to solve a case of simple subtraction! The answer will become obvious when we deal with binary numbers. In the binary system, the equivalent of 9’s complement is called 1’s complement and that of 10’s complement is termed 2’s complement.
How do we get a 1’s complement of a binary number? Extremely simple, just invert the bits — 0s become Is and Is become 0s. Adding 1 to it then gives 2’s complement form.
3. Term Paper on the Hexadecimal Number System
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Although the computers operate on binary system of 1s and 0s used in combinations, it was found that with increase in processing ability and storage capability, using so many 0s and 1s for each data/instruction was a tedious process, difficult to remember, and the possibility of mistake was very high, resulting in disastrous results. In a PC, 16 bits are used for normal working with PC AT operating at 32 bits level.
So any data or instruction would normally required to be represented by a combination of at least 16 or 32 bits of 0s and 1s. To find an easy way out as far as programmers are concerned, a short-hand substitute, another number system has been developed, called Hexadecimal System or Hex in short, which uses 16 digits. It has some special advantages.
Now, where do we get 16 numbers of single digits? 0 to 9 provides only ten digits. We cannot use 10, 11, etc., as these are combinations of basic digits of 0 and 1 and they occupy two places. The solution was found, selecting the first five alphabets, A to F — A being 10, B being 11, C being 12, D being 13, E being 14, and F being 15, the largest digit — the base of the system being 16.
The system has a unique advantage that exactly 4 binary digits (a nybble) is represented by a single hex digit, the F being represented by 11112 — a byte consisting of 8 bits is fully represented by 2 hex digits. If you peep into a computer’s primary memory using any software like Debug, PCTools, or Norton Utility, you will find everything being represented in hexadecimal numbers.
Note that two hex digits are required to represent a byte and very large decimal numbers can be represented by only a few hex digits. You should be proficient in interpreting hex digits, if you are serious about being a computer expert. It is a must.
i. Hexadecimal to Decimal Conversion:
Our basic number system being the decimal system, we have to know how to convert the hexadecimal numbers to decimal [and also binary system] and vice versa.
The process of conversion to decimal system is similar to that for the decimal to binary system, with the following exceptions:
1. The base here is obviously 16, instead of 2.
2. In binary system, the digits were either 0 or 1, so we either added or rejected the place values, depending on whether it was 1 or 0. But in hexadecimal system, each position may have any digit between 0 and F [15], so the respective digits have to be multiplied with corresponding place values to get the final values; as we did in the decimal system for expanded notation. For example, to convert 3A2F to decimal system.
In case of conversion of hexadecimal number to binary number, we simply write down the corresponding bits, as shown below:
It should be noted that since 4 bits make a hex digit, all the 4 bits need to be written, like 0001 for 1.
ii. Decimal to Hexadecimal Conversion:
This process is also identical with that for binary system, except instead of 2, we divide by 16 to get the remainders, which represent hex digits.
To convert 584386 to hexadecimal:
iii. Hexadecimal Arithmetic:
The addition of two hex digits is a straight forward case, with the value of carry being equal to 16. Thus — 3A6D + 7B4 = 4221. When we add D with 4, we get 17 and subtracting 16 for the value to be carried forward, we are left with 1, so the LSD of the result is 1.
We can verify the result by converting everything to the decimal system, as shown below:
The subtraction is also similarly performed with the value of the borrow being 16. For example: 3A6D – 7B4 = 32B916 = 1298510. If we do the above subtraction in 16’s complement form, we have 16’s complement of 7B4 as (F84B + 1) = F84C. And,
Although the programmers use the hex notations, the computer operates on binary system, so hex digits are converted to binary digits, 2’s complement taken, addition done, and result reconverted to hex digits.
Following this procedure, we get:
4. Term Paper on the Octal Number System
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Some of the earlier computer systems had adopted a configuration under which six binary digits were taken to form a byte (now it is 8 bits to a byte). Hence, they developed the octal number system, whereby two octal digits fully represented their bytes; as two hex digits now represent a byte.
Under the system, as the name suggests, there are eight digits, with the highest one being [8 – 1] = 7 and, naturally, the base is 8. Since three binary digits can fully represent 0 to 7, an octal digit can be represented by three bits.
i. Octal to Decimal and Back:
By now all of you should be fully proficient to tackle any number system, so two representative examples are given for symbolic reference.
a. Octal to Decimal Conversion:
b. Decimal to Octal Conversion:
5. Term Paper on the Equivalent Digits of Number Systems
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i. Binary Coded Decimal:
No matter how scientific and useful a number system is, we always feel a bit comfortable working with the decimal number system. Hence, a number of code systems have been developed to integrate the two number systems of decimal and binary — so that we operate in decimal system and the computer continues with the binary system, the codes facilitating conversion from one form to the other.
These codes are called Binary Coded Decimals [BCD]. A large number of such codes exist. Under these systems, individual decimal digits are converted to equivalent binary form, instead of the total decimal value being converted to its binary equivalent.
When the bit positions arranged in such codes are given positional weights in such a manner that the sum of the weighted digits corresponds to the respective code numbers then these BCD codes are called weighted codes. But to qualify as a weighted code the sum of the weight should not be more than 15 and it must be at least 9. For example, in case of the 8-4-2-1 code, the sum of the weight is 15, and so it is a weighted code.
In this system, each decimal number is expressed in binary form using its equivalent binary digit using 4 bits, like 0001 for 1, 0100 for 4, 1001 for 9. The weights given to the bit positions are 8, 4, 2, and 1, respectively and so it is called 8-4-2-1 code.
The decimal 386 in binary form using 8-4-2-1 code would be as:
By using this code the numbers of one system can be easily converted to the other system conveniently, but rules of binary arithmetic becomes invalid with such numbers.
Other Decimal BCD Codes:
It is a BCD code, where a special technique is used for conversion purpose, which is to add 3 to each decimal digit before it is converted to bits using the 8-4-2-1 code.
For example, to convert 386 to Excess 3 code we do as:
This system has been developed to counter the problem associated when adding 8-4-2-1 code digits.
Apart from theses 4 bit codes, there are also codes using 5 and more bits, which are supposed to be better in detecting errors. Such a code which is used extensively in communication system is called 2-out-of-5 code and this is an unweighted code, as you will see as it does not meet the necessary conditions.
The code numbers are:
It is a unweighted code which is used in input/output operations, analog to digital conversions, digital to analog conversions, etc. It can be derived from equivalent binary digits of decimal numbers.
The codes are:
ii. Character Codes:
A computer has to have codes to represent not only numeric data but also characters, like alphabets so that we can use English words and sentences for various purposes, including naming numeric variables. The codes used for such character representations are different for PC and non-PC groups, which are mainframe and minicomputers. The two common codes used in the non-PC groups are called EBCDIC and ASCII-8.
EBCDIC stands for Extended Binary Coded Decimal Interchange Code, which is pronounced as “ebb-see-dic”. It is used in IBM mainframe and compatible computers. It consists of 8 binary digits or bits, divided in two fields called Zone and Numeric, each having 4 bits, the Zone preceding the Numeric field. As you know, four bits can be represented by a hex-digit.
In addition to these 8 bits, an additional bit is used during storage, preceding the Zone field, called Parity Bit. It follows even parity system, that is, the number of 1s in all the bits including the parity bit is made even. For instance, if the number of Is used to represent a character by the 8 bits is odd, then the parity bit becomes 1 to make even number of 1s.
If the number of 1s in the other 8 bits is even, then the parity bit becomes 0. It is used for error checking during storage and retrieval. The 9 channel magnetic tapes generally use EBCDIC code.
Some of the EBCDIC codes are as given below:
ASCII-8 stands for American Standard Code for Information Interchange and is pronounced as “ask-ee”. It is also an 8 bit code used in non-PC class of computers.
It is different form the ASCII code used in the Personal Computers. It also uses 8 bits to represent a character divided into Zone and Numeric Fields of 4 bits each.
Some of its character codes are given below:
iii. Numerical Character Codes:
The numeric character codes shown earlier are used to represent number, which often need to be modified by being given positive or negative signs, as the case may be.
For numeric character codes, either in EBCDIC or ASCII-8, the zone code of the last number is replaced by a sign code of 4 bits, which are:
1111 – For unsigned number, that is, positive
1100 – Signed positive number
1101 – Signed negative number
For example, the decimal number 746 as represented by numeric character codes under the two systems, with their respective signs, are as shown below:
Now these numeric character representations are not suitable for arithmetic computations and so before mathematical processing, these codes are converted to equivalent BCD numbers of 8-4-2-1 type and the last 4 bit code for signs are put at the end — the set being called Packed Decimal Number. After the necessary arithmetic computations, the Packed Decimal Numbers are reconverted back to either EBCDIC or ASCII-8 codes for displaying output, as the case may be.
The procedure for converting to Packed Decimal Number is to delete the zone bits of EBCDIC or ASCII-8 codes of numbers and pack them together to get the equivalent BCD number and then put the sign bits from the numeric codes at the end. For example, by converting the decimal number 746 from the ASCII-8 or EBCDIC to the BCD code, we will get 0111 0100 0110 to represent 746.
Now the sign zone would be used after this code as:
iv. PC-ASCII:
The ASCII code developed for the Personal Computer is a 7 bit code for representing 128 different characters, as detailed below. Among these, the first 32 codes are used for various control purposes like, Line Feed, Form Feed, Return Key press, Bell, etc., which are mainly for printer and communication services, and these codes cannot be printed.
Later on, this 7 bit code has been extended to 8 bits to define up to 256 different characters, the later ones being mostly graphical characters. Incidentally, there are two sets of 8 bit ASCII set, called Extended ASCII characters — one for IBM characters and the other for EPSON characters, the latter includes characters for Italic prints.
The standard ASCII codes are:
v. Floating Point Numbers:
The accuracy or precision of the number manipulated by the computers is limited by the storage space available, which depends on the size of its registers. If a machine operates at 8 bit level, then the maximum value of an integer that can be stored is 28 = 256 only. With a 16 bit machine, the maximum value of an integer would be 216= 65,536, provided all the 16 bits are used to represent the number, which is called unsigned number.
In case of signed numbers, the Most Significant Bit [MSB], that is the bit 15 is used for denoting the sign — it is 0 if it is a positive number and 1, if it is a negative number. But this system reduces the maximum value of the number to be represented, it becoming 27 = 128. Again, as you have seen, the negative numbers are represented in 1’s and 2’s complement forms.
Under these systems the range of numbers that can be stored are:
In some softwares like BASIC, you can use integers defined as “double precision”, where the maximum value of the number that can be stored is doubled, but, this is done by the software.
In view of the above limitations in the maximum value, positive or negative, that can be stored, the higher values of numbers are stored in exponential forms, which in the decimal system is given by a number raised to the power of 10. For example, 18907426 could be represented as 18.9×106.
Naturally in this process, some amount of accuracy is lost, as the digits which cannot be stored, called overflow, is truncated or rounded off. In case of truncating the excess digits are simply discarded, where as in case of rounding off, the last stored digit is increased by 1 if the discarded digit after it is 5 or more.
The real numbers containing fractions, called floating point numbers are also stored in the exponential form, where the number is divided into two parts — the whole number as the characteristic and the fractional part as the mantissa — the same system which is followed for using log tables in schools.
Generally, 32 bits are used to represent floating point numbers, the 1st bit is reserved for the sign of the number, the next 7 bits to accommodate the exponent, and the last 24 bits to store the mantissa. The sign of the exponent is also stored in the 7 bits reserved for it by modifying the actual exponent. Before storing, the floating point number is converted to standard exponential form, which in case of binary numbers, requires placing the binary point before the first occurrence of 1, as shown below:
When 7 bits are used to store the exponent, its negative value is taken care of by adding 26 = 64 to it and then stored. The empty spaces are filled up with 0s to form 7 digits in storage. The whole operation of storing numbers in such form is done by the software and that is why it takes longer time to operate with floating point numbers, which is called number crunching.
vi. Truth Tables:
The computer’s superb ability to process data in different form for solving our problems, to a large extent, depend on its ability to carry out logical operations. The general logical operators are OR, AND, XOR, NOT, etc. which are used in the hardware in the form of logic gates. In software, they indicate whether the relations evaluated are true or not. A truth table gives the outcome of using the logical operators, which is used in logic circuits.
Generally, simple logic gates have two inputs and one output. The bits passing through the gate is modified, depending on how the gate has been formed. For example, in case of OR operator, if either of the expressions connected by the OR operator is true represented by 1, then the output is 1, indicating the whole expression to be True.
The X and Y denote the two inputs and O denotes the output, with 0 indicating False and 1 indicating True. The NOT logic gate has one input and one output and it reverses the input, if it is 0 it becomes 1 and vice versa.