The document discusses number systems, including decimal, binary, octal, and hexadecimal systems. It provides details on converting between different number systems, such as binary to decimal, binary to hexadecimal, and hexadecimal to decimal. Methods discussed include using place values, doubling, and grouping bits. Examples are provided for addition, subtraction, multiplication, and division in binary systems as well as converting between binary, decimal, and hexadecimal representations.
A digital system can understand positional number system only where there are only a few symbols called digits and these symbols represent different values depending on the position they occupy in the number.
Chapter 2.1 introduction to number systemISMT College
Binary Number System, Decimal Number System, Octal Number System, Hexadecimal Number System, Conversion, Binary Arithmetic, Signed Binary Number Representation, 1's complement, 2's complement, 9's complement, 10's complement
Inductive programming incorporates all approaches which are concerned with learning programs or algorithms from incomplete (formal) specifications. Possible inputs in an IP system are a set of training inputs and corresponding outputs or an output evaluation function, describing the desired behavior of the intended program, traces or action sequences which describe the process of calculating specific outputs, constraints for the program to be induced concerning its time efficiency or its complexity, various kinds of background knowledge such as standard data types, predefined functions to be used, program schemes or templates describing the data flow of the intended program, heuristics for guiding the search for a solution or other biases.
Output of an IP system is a program in some arbitrary programming language containing conditionals and loop or recursive control structures, or any other kind of Turing-complete representation language.
In many applications the output program must be correct with respect to the examples and partial specification, and this leads to the consideration of inductive programming as a special area inside automatic programming or program synthesis, usually opposed to 'deductive' program synthesis, where the specification is usually complete.
In other cases, inductive programming is seen as a more general area where any declarative programming or representation language can be used and we may even have some degree of error in the examples, as in general machine learning, the more specific area of structure mining or the area of symbolic artificial intelligence. A distinctive feature is the number of examples or partial specification needed. Typically, inductive programming techniques can learn from just a few examples.
The diversity of inductive programming usually comes from the applications and the languages that are used: apart from logic programming and functional programming, other programming paradigms and representation languages have been used or suggested in inductive programming, such as functional logic programming, constraint
programming, probabilistic programming
Research on the inductive synthesis of recursive functional programs started in the early 1970s and was brought onto firm theoretical foundations with the seminal THESIS system of Summers[6] and work of Biermann.[7] These approaches were split into two phases: first, input-output examples are transformed into non-recursive programs (traces) using a small set of basic operators; second, regularities in the traces are searched for and used to fold them into a recursive program. The main results until the mid 1980s are surveyed by Smith.[8] Due to
Binary addition, Binary subtraction, Negative number representation, Subtraction using 1’s complement and 2’s complement, Binary multiplication and division, Arithmetic in octal, hexadecimal number system, BCD and Excess – 3 arithmetic
A digital system can understand positional number system only where there are only a few symbols called digits and these symbols represent different values depending on the position they occupy in the number.
Chapter 2.1 introduction to number systemISMT College
Binary Number System, Decimal Number System, Octal Number System, Hexadecimal Number System, Conversion, Binary Arithmetic, Signed Binary Number Representation, 1's complement, 2's complement, 9's complement, 10's complement
Inductive programming incorporates all approaches which are concerned with learning programs or algorithms from incomplete (formal) specifications. Possible inputs in an IP system are a set of training inputs and corresponding outputs or an output evaluation function, describing the desired behavior of the intended program, traces or action sequences which describe the process of calculating specific outputs, constraints for the program to be induced concerning its time efficiency or its complexity, various kinds of background knowledge such as standard data types, predefined functions to be used, program schemes or templates describing the data flow of the intended program, heuristics for guiding the search for a solution or other biases.
Output of an IP system is a program in some arbitrary programming language containing conditionals and loop or recursive control structures, or any other kind of Turing-complete representation language.
In many applications the output program must be correct with respect to the examples and partial specification, and this leads to the consideration of inductive programming as a special area inside automatic programming or program synthesis, usually opposed to 'deductive' program synthesis, where the specification is usually complete.
In other cases, inductive programming is seen as a more general area where any declarative programming or representation language can be used and we may even have some degree of error in the examples, as in general machine learning, the more specific area of structure mining or the area of symbolic artificial intelligence. A distinctive feature is the number of examples or partial specification needed. Typically, inductive programming techniques can learn from just a few examples.
The diversity of inductive programming usually comes from the applications and the languages that are used: apart from logic programming and functional programming, other programming paradigms and representation languages have been used or suggested in inductive programming, such as functional logic programming, constraint
programming, probabilistic programming
Research on the inductive synthesis of recursive functional programs started in the early 1970s and was brought onto firm theoretical foundations with the seminal THESIS system of Summers[6] and work of Biermann.[7] These approaches were split into two phases: first, input-output examples are transformed into non-recursive programs (traces) using a small set of basic operators; second, regularities in the traces are searched for and used to fold them into a recursive program. The main results until the mid 1980s are surveyed by Smith.[8] Due to
Binary addition, Binary subtraction, Negative number representation, Subtraction using 1’s complement and 2’s complement, Binary multiplication and division, Arithmetic in octal, hexadecimal number system, BCD and Excess – 3 arithmetic
Chapter 2 Data Representation on CPU (part 1)Frankie Jones
This topic introduces the numbering systems: decimal, binary, octal and hexadecimal. The topic covers the conversion between numbering systems, binary arithmetic, one's complement, two's complement, signed number and coding system. This topic also covers the digital logic components.
Contents:
1.What is number system?
2.Conversions of number from one radix to another
3.Complements (1's, 2's, 9's, 10's)
4.Binary Arithmetic ( Addition, subtraction, multiplication, division)
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
How to Create Map Views in the Odoo 17 ERPCeline George
The map views are useful for providing a geographical representation of data. They allow users to visualize and analyze the data in a more intuitive manner.
We all have good and bad thoughts from time to time and situation to situation. We are bombarded daily with spiraling thoughts(both negative and positive) creating all-consuming feel , making us difficult to manage with associated suffering. Good thoughts are like our Mob Signal (Positive thought) amidst noise(negative thought) in the atmosphere. Negative thoughts like noise outweigh positive thoughts. These thoughts often create unwanted confusion, trouble, stress and frustration in our mind as well as chaos in our physical world. Negative thoughts are also known as “distorted thinking”.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
The Indian economy is classified into different sectors to simplify the analysis and understanding of economic activities. For Class 10, it's essential to grasp the sectors of the Indian economy, understand their characteristics, and recognize their importance. This guide will provide detailed notes on the Sectors of the Indian Economy Class 10, using specific long-tail keywords to enhance comprehension.
For more information, visit-www.vavaclasses.com
Sectors of the Indian Economy - Class 10 Study Notes pdf
B sc3 unit 2 number system
1. KUD BSc 3rd SEM : ELECTRONICS : OPTO AND DIGITAL ELECTRONICS
BY : MAHIBOOB ALI K MULLA MSc , Mphil.
Asst.Prof.in electronics , SSGFG COLLEGE NARAGUND
2. Opto and Digital Electronics: Unit 2: Number System
Number System
The mastery over depiction & symbolization of the numbers through performing certain
conversion tasks on them to make then equivalent is called Number System. This forms the
fundamentals of Computers Number system. Computers recognize and converts common man’s
stereotyping of words and letters into numbers.
There are mainly four different groups of Number Systems:
•Decimal System
•Binary System
•Octal System
•Hexadecimal System
3. Decimal Number System
Decimal number system is used in our everyday business. Normally in this genre of rendition non-
identical positions are given to each numeral connoting the number depending upon their setting
points.The units, tens, hundreds, thousands and so on of the decimal number system take
consecutive places to the left of the decimal point to exemplify it. The decimal number system
appoints 10 as its plinth/ base and it houses the 10 diverging numerals, the digits are
0,1,2,3,4,5,6,7,8,9.
29. Subtraction by 1’s Complement
The steps to be followed in subtraction by 1’s
complement are:
i) To write down 1’s complement of the subtrahend.
ii) To add this with the minuend.
iii) If the result of addition has a carry over then it is
dropped and an 1 is added in the last bit.
iv) If there is no carry over, then 1’s complement of
the result of addition is obtained to get the final
result and it is negative.
30. Evaluate:
(i) 110101 – 100101
Solution:
1’s complement of 10011 is 011010. Hence
Minued - 1 1 0 1 0 1
1’s complement of subtrahend - 0 1 1 0 1 0
Carry over - 1 0 0 1 1 1 1
1
0 1 0 0 0 0
The required difference is 1 0 0 0 0
31. (ii) 101011 – 111001
Solution:
1’s complement of 111001 is 000110. Hence
Minued - 1 0 1 0 1 1
1’s complement - 0 0 0 1 1 0
1 1 0 0 0 1
Hence the difference is – 1 1 1 0
(iii) 1011.001 – 110.10
Solution:
1’s complement of 0110.100 is 1001.011 Hence
Minued - 1 0 1 1 . 0 0 1
1’s complement of subtrahend - 1 0 0 1 . 0 1 1
Carry over - 1 0 1 0 0 . 1 0 0
1
0 1 0 0 . 1 0 1
Hence the required difference is 100.101
32. (iv) 10110.01 – 11010.10
Solution:
1’s complement of 11010.10 is 00101.01
1 0 1 1 0 . 0 1
0 0 1 0 1 . 0 1
1 1 0 1 1 . 1 0
Hence the required difference is – 00100.01 i.e. – 100.01
33. Subtraction by 2’s Complement
With the help of subtraction by 2’s complement method we can
easily subtract two binary numbers.
The operation is carried out by means of the following steps:
(i) At first, 2’s complement of the subtrahend is found.
(ii) Then it is added to the minuend.
(iii) If the final carry over of the sum is 1, it is dropped and the
result is positive.
(iv) If there is no carry over, the two’s complement of the sum
will be the result and it is negative.
The following examples on subtraction by 2’s complement will
make the procedure clear:
34. Evaluate:
(i) 110110 - 10110
Solution:
The numbers of bits in the subtrahend is 5 while that of minuend is 6. We
make the number of bits in the subtrahend equal to that of minuend by taking
a `0’ in the sixth place of the subtrahend.
Now, 2’s complement of 010110 is (101101 + 1) i.e.101010. Adding this with
the minuend.
1 1 0 1 1 0 Minuend
1 0 1 0 1 0 2’s complement of subtrahend
Carry over 1 1 0 0 0 0 0 Result of addition
After dropping the carry over we get the result of subtraction to be 100000.
35. (ii) 10110 – 11010
Solution:
2’s complement of 11010 is (00101 + 1) i.e. 00110. Hence
Minued - 1 0 1 1 0
2’s complement of subtrahend - 0 0 1 1 0
Result of addition - 1 1 1 0 0
As there is no carry over, the result of subtraction is negative and is
obtained by writing the 2’s complement of 11100 i.e.(00011 + 1) or 00100.
Hence the difference is – 100.
36. (iii) 1010.11 – 1001.01
Solution:
2’s complement of 1001.01 is 0110.11. Hence
Minued - 1 0 1 0 . 1 1
2’s complement of subtrahend - 0 1 1 0 . 1 1
Carry over 1 0 0 0 1 . 1 0
After dropping the carry over we get the result of subtraction as 1.10.
37. (iv) 10100.01 – 11011.10
Solution:
2’s complement of 11011.10 is 00100.10. Hence
Minued - 1 0 1 0 0 . 0 1
2’s complement of subtrahend - 0 1 1 0 0 . 1 0
Result of addition - 1 1 0 0 0 . 1 1
As there is no carry over the result of subtraction is negative
and is obtained by writing the 2’s complement of 11000.11.
Hence the required result is – 00111.01.
40. Conversion from Binary to Hexadecimal number system
Since number numbers are type of positional number system. That
means weight of the positions from right to left are as 160, 161, 162,
163and so on. for the integer part and weight of the positions from left
to right are as 16-1, 16-2, 16-3and so on. for the fractional part.
Most
Significant
Bit (MSB)
Hexa Point Least Significant Bit (LSB)
16
2
16
1
16
0
16
-1
16
-2
16
-3
256 16 1 1/16 1/256 1/4096
41. Example − Convert binary number 1101010 into hexadecimal number.
First convert this into decimal number:
= (1101010)2
= 1x2 ↑ 6+1x2 ↑ 5+0x2 ↑ 4+1x2 ↑ 3+0x2 ↑ 2+1x2 ↑ 1+0x2 ↑ 0
= 64+32+0+8+0+2+0
= (106)10
Then, convert it into hexadecimal number
= (106)10
= 6x16 ↑ 1+10x16 ↑ 0
= (6A)16 which is answer.
However, there is also a direct method to convert a binary number into
hexadecimal number − grouping which is explained as following below.
42. Using Grouping :
Since, there are only 16 digits (from 0 to 7 and A to F) in hexadecimal number system, so we
can represent any digit of hexadecimal number system using only 4 bit as following below.
So, if you make each group of 4 bit of binary input number, then replace each group of
binary number from its equivalent hexadecial digits. That will be hexadecimal number of
given number. Note that you can add any number of 0’s in leftmost bit (or in most significant
bit) for integer part and add any number of 0’s in rightmost bit (or in least significant bit) for
fraction part for completing the group of 4 bit, this does not change value of input binary
number.
So, these are following steps to convert a binary number into hexadecimal number.
•Take binary number
•Divide the binary digits into groups of four (starting from right) for integer part and start
from left for fraction part.
•Convert each group of four binary digits to one hexadecimal digit.
This is simple algorithm where you have to grouped binary number and replace their
equivalent hexadecimal digit.
43. Example-1 − Convert binary number 1010101101001 into hexadecimal number. Since
there is no binary point here and no fractional part. So,
Therefore, Binary to hexadecimal is,
= (1010101101001)2
= (1 0101 0110 1001)2
= (0001 0101 0110 1001)2
= (1 5 6 9)16
= (1569)16
44. Example-2 − Convert binary number 001100101.110111 into hexadecimal number. Since there
is binary point here and fractional part. So,
Therefore, Binary to hexadecimal is,
= (001100101.110111)2
= (0 0110 0101 . 1101 1100)2
= (0110 0101 . 1101 1100)2
= (6 5 . D C)16
= (65.DC)16
These are above simple conversions binary number to hexadecimal number.
45. Conversion of Hexadecimal to Binary:
Hexadecimal to Binary conversion: Each hexadecimal digit is replaced by its equivalent decimal
digit and each decimal digit is replaced by equivalent 4- bit binary number.
Examples 1:convert hexa decimal (16AF)16 to binary.
Ans: 1H=0001B , 6H=0110B , AH=1010B , FH=1111B .Hence ,
Hexa decimal (16AF)H=(0001 0110 1010 1111) Binary.
Example 2: Convert Hexa decimal (E5C9)16 to Binary.
Ans: (E5C9) Hexa decimal = (1110 0101 1100 1001)Binary.
Example 3: Find binary equivalent of Hexa decimal (D7B8.C4E6) .
Ans: (D7B8.C4E6)H = (1101 0111 1011 1000.1100 0100 1110 0110)B.
Hexa 0 1 2 3 4 5 6 7 8 9 A B C D E F
Binary 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111
46. Conversion from Binary to Decimal number system
There are mainly two methods to convert a binary number into decimal
number − using positional notation, and using doubling. These methods are
explained are as following below.
Using Positional Notation
Since number numbers are type of positional number system. That means
weight of the positions from right to left are as 20, 21, 22, 23... and so on for
the integer part and weight of the positions from left to right are as 2-1, 2-2, 2-3,
2-4... and so on for the fractional part.
Most
Significant Bit
(MSB)
Binary Point Least Significant Bit (LSB)
2
2
2
1
2
0
2
-1
2
-2
2
-3
4 2 1 0.5 0.25 0.125
47. Assume any unsigned binary number is bnb(n-1) ... b1b0.b-1b-2 ... b(m-1)bm.
Then the decimal number is equal to the sum of binary digits (bn) times
their power of 2 (2n), i.e., bnb(n-1) ... b1b0.b-1b-2 ... b(m-1)bm = bnx2n+b(n-
1)x2(n-1)+ ... +b1x21+bx020+b-1x2-1+b-22-2+ ...
This is simple algorithm where you have to multiply positional value of
binary with their digit and get the sum of these steps.
Example-1 − Convert binary number 11001010 into decimal number.
Since there is no binary point here and no fractional part. So,
Binary to decimal is,
48. = (1010.1011)2
= 1x2 ↑ 3+0x2 ↑ 2+1x2 ↑ 1+0x2 ↑ 0+1x2 ↑ -1+0x2 ↑ -2+1x2 ↑ -3+1x2 ↑ -4
= 8+0+2+0+0.5+0+0.125+0.0625
= (10.6875)10
Using Doubling:
This is simple method to convert a binary number into decimal number, you need to start
from leftmost digit (or MSB) from the input. Take the most significant bit (MSB), right
down, then multiply by 2 with it and add second leftmost bit, store it as current result,
then again multiple by 2 with current result and add third leftmost bit, update this value as
current result and follow this till addition of least significant bit (LSB or rightmost bit).
Since you are doubling (multiplying by 2) each time, so this method is known as
Doubling.
These are simple algorithm is explained below in steps: − Write down the binary number.
•Starting from the left, double your previous total and add the current digit..Double your
current total and add the next leftmost digit..Repeat the previous step.S
49. For example, Convert binary number 11101110 into decimal number.
According to above algorithm, Binary to decimal is,
= (11101110)2
= 1
= 1*2+1 =3
= 3*2+1=7
= 7*2+0=14
= 14*2+1=29
= 29*2+1=59
= 59*2+1=119
= 119*2+0=238
= (238)10
These are above two simple methods to convert a binary number into decimal number.
50. Conversion from :Hexadecimal to Decimal number system
There are various indirect or direct methods to convert a hexadecimal number
into decimal number. In an indirect method, you need to convert a hexadecimal
number into binary or octal number, then you can convert it into decimal number.
Example − Convert hexadecimal number F1 into decimal number.
First convert it into binary number,
= (F1)16
= (1111 0001)2
Because in binary, value of F and 1 are 1111 and 0001 respectively. Then convert it
into decimal number multiplying power of its position of base.
= (1x2↑7+1x2↑6+1x2↑5+1x2↑4+0x2↑3+0x2↑2+0x2↑1+1x2↑0)10
= (241)10
51. However, there is a simple direct method to convert a hexadecimal number
to decimal number. Since, there are only 16 digits (from 0 to 9 and A to F)
in hexadecimal number system, so we can represent any digit of
hexadecimal number system using only 4 bit as following below.
Hexa 0 1 2 3 4 5 6 7
Binary 0000 0001 0010 0011 0100 0101 0110 0111
Hexa 8 9 A=10 B=11 C=12 D=13 E=14 F=15
Binary 1000 1001 1010 1011 1100 1101 1110 1111
Hexadecimal number system provides convenient way of converting large binary
numbers into more compact and smaller groups. These are weights of hexadecimal
of respective position of hexadecimal (value of base is 16).
Most Significant Bit (MSB) Hexa Point Least Significant Bit (LSB)
16 ↑ 2 16 ↑ 1 16 ↑ 0 16 ↑ -1 16 ↑ -2 16 ↑ -3
256 16 1 1/16 1/256 1/4096
52. Example-1 − Convert hexadecimal number ABCDEF into decimal number.Since
value of Symbols − A, B, C, D, E, F are 10, 11, 12, 13, 14, 15 respectively.
Therefore equivalent decimal number is,
= (ABCDEF)16
= (10x165+11x164+12x163+13x162+14x161+15x160)10
= (10485760+720896+49152+3328+224+15)10
= (11259375)10 which is answer.
Example-2 − Convert hexadecimal number 1F.01B into decimal number.
Since value of Symbols: B and F are 11 and 15 respectively. Therefore
equivalent decimal number is,
= (1F.01B)16
= (1x161+15x160 +0x16-1+1x16-2+11x16-3)10
= (31.0065918)10 which is answer.
53. CONVERTING DECIMAL TO HEXADECIMAL
Steps: Divide the decimal number by 16. Treat the division as an integer
division. Write down the remainder (in hexadecimal).
Divide the result again by 16. Treat the division as an integer division.
Repeat step 2 and 3 until result is 0.
The hex value is the digit sequence of the remainders from the last to first.
Note: a remainder in this topic refers to the left over value after
performing an integer division.
HEXADECIMAL 0 1 2 3 4 5 6 7 8 9
A B C D E F
DECIMAL 0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15
54. Example 2
Convert the number 256 DECIMAL to HEXADECIMAL
DIVISION RESULTREMAINDER (in HEX)
256 / 16 16 0
16 / 161 0
1 / 16 0 1
ANSWER 100
Example 3
Convert the number 921 DECIMAL to HEXADECIMAL
DIVISION RESULTREMAINDER (in HEX)
921 / 16 57 9
57 / 163 9
3 / 16 0 3
ANSWER 399
55. Example 4:
Convert the number 188 DECIMAL to HEXADECIMAL
DIVISION RESULTREMAINDER
(in HEX)
188 / 16 11 C (12 decimal)
11 / 160 B (11 decimal)
ANSWER BC
Note that here, the answer would not be 1112, but BC. Remember to write down the
remainder in hex, not decimal.
Example 5:
Convert the number 100 DECIMAL to HEXADECIMAL
DIVISION RESULTREMAINDER
(HEX)
100 / 16 6 4
6 / 16 0 6
ANSWER 64
56. Example 6
Convert the number 590 DECIMAL to HEXADECIMAL
DIVISION RESULT REMAINDER
(HEX)
590 / 16 36 E (14 decimal)
36 / 162 4 (4 decimal)
2 / 16 0 2 (2 decimal)
ANSWER 24E
Example 7:
Convert the number 1128 DECIMAL to HEXADECIMAL
DIVISION RESULT REMAINDER
1128/16 70 8
70/16 4 6
4/16 0 4
ANSWER:[ 468]H
Note: For fractional numbers successive multiplication is used.
57. Example : Convert Hexadecimal of 0.16 to Decimal.
Step 1
We multiply 0.16 by 16 and take the integer part
0.16 x 16 = 2.56
Integer part = 2
Fractional part = 0.56
As, fractional part is not equal to 0 so again multiply fractional part by 16.
Step 2
We multiply 0.56 by 16 and take the integer part
0.56 x 16 = 8.96
Integer part = 8
Fractional part = 0.96
As, fractional part is not equal to 0 so again multiply fractional part by 16.If zero stop else
continue till fractional part becomes zero or number of digits after decimal point.
Answer: (0.16)10 = (0.28F5C...)16
58. Octal number system:
The octal number system uses only eight digits (0 through 7) , but the
conversion from decimal to octal and binary to octal follows the same
pattern as for hexadecimal. It is a 3-bit binary code from (000 to 111).In octal
Digits 8 & 9 are not used.
64. BCD codes:
A binary code will have some unassigned bit combinations if the number of
element in the set is not a multiple power of two. The decimal digit from
such a set is the binary code distinguished among 10 elements must
contain at least 4 bits and this code is called BCD. So in BCD, 6 out of 16
possible combinations remain unassigned and each decimal digit is treated
differently and assigned the corresponding 4 bit code for every digit. A
number with K decimal digit will require 4K bits in BCD. For e.g. BCD for
132 is 0001 0011 0010.
Advantages
Even though THE BCD representations requires more numbers of bit then in
binary, it is easier for data manipulations and the observations for the user can
understand decimal codes rather than binary.
Decimal 0 1 2 3 4 5 6 7 8 9
8421 BCD 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001
65. BCD addition:
IN BCD addition the normal addition process is followed if the sum
is less than 10 (1010)2. But if the um is greater than or equal to 10
(1010)2 we have to add 6(0110) to the sum. The addition of 6
(0110)2 to the binary sum converts it to the correct digit and
produces a carry as required. This is because the carry is the most
significant bit position of the binary sum and decimal carry differ by
16-10=6
Example
4 (0100) + 5 (0101) = 9 (1001)
4 (0100) + 8 (1000) = 12 (1100) which is greater than 10 so adding
6 (0110) to it we get 10010. (0010)2 is 2
66. Gray code:
It is the non-weighted code and it is not arithmetic codes. That means there are no
specific weights assigned to the bit position. It has a very special feature that, only
one bit will change each time the decimal number is incremented as shown in fig.
As only one bit changes at a time, the gray code is called as a unit distance code.
The gray code is a cyclic code. Gray code cannot be used for arithmetic operation.
67. Application of Gray code:
Gray code is popularly used in the shaft position encoders.
A shaft position encoder produces a code word which represents the
angular position of the shaft.
Binary to Gray conversion: Example: Let assume the Binary code digits be
bo, b1, b2, b3 whereas the particular Gray Code can be attained based on
the following concept.
68. Ex : Convert binary value b3, b2, b1, b0 = 1101 to gray code g3, g2, g1, g0
70. Gray to Binary Code Converter:
This gray to binary conversion method also uses the working concept of the EX-OR logic
gate among the bits of gray as well as binary bits.
we can get the binary values like b3 = g3, b2 = b3 XOR g2, b1= b2 XOR g1, b0 = b1 XOR g0.
71. Example of Gray to Binary Code Converter:
Let assume the Gray Code digits g3, g2, g1, g0 whereas the particular Binary code digits
are bo, b1, b2, b3 can be attained based on the following concept.
we can get the binary values like b3 = g3, b2 = b3 XOR g2, b1= b2 XOR g1, b0 = b1 XOR g0.
72. For example take the gray value g3, g2, g1, g0 = 0011 and find the
binary code b3, b2, b1, b0
b3=g3=0
b2 = b3 XOR g2 = 0 XOR 0 =0
b1= b2 XOR g1= 0 XOR 1 = 1
b0= b1 XOR g0= 1 XOR 1 = 0
The final binary code for the value of gray 0011 is 0010
74. ASCII code:
ASCII (American Standerd Code For Information Interchange ) is a 7-bit
character set containing 128 characters. It contains the numbers from 0-9,
the upper and lower case English letters from A to Z, and some special
characters. The character sets used in modern computers (Key Boards) , in
HTML, and on the Internet, are all based on ASCII. Ex:Key Board