28th Apr, 2018
Gujarat Arts and Science College, Ahmedabad, Gujarat, INDIA. (Government College, Pincode - 380006.)
Question
Asked 4th Jul, 2013
Why do we need various number base conversions (number system conversions), e.g octal to hexadecimal?
What are the applications of various no. Systems?
Most recent answer
Dear Sandeep Pariyani,
We communicate with each other in a particular language made of letters or words. We normally type letters or words through keyboard of the computer, but computer does not understand the words and letters. Rather, those words and letters are translated into numbers. This means that computers understand only numbers. We know the decimal (base 10) system, and are very comfortable with performing operations using this system, it is also important for us to understand that the decimal system is not the only system in the world. By studying other number systems such as binary (base 2), quaternary (base 4), octal (base 8), hexadecimal (base 16) and so forth, we will gain a better understanding of how number systems work in general. Number systems are the technique to represent numbers in the computer system architecture, every value that you are saving or getting into/from computer memory has a defined number system. As Computer architecture supports following number systems so we need to study them and also need to know the conversion technique between them.
- Binary number system
- Octal number system
- Decimal number system
- Hexadecimal (Hex) number system
1) Binary Number System
A Binary number system has only two digits that are 0 and 1. Every number (value) represents with 0 and 1 in this number system. The base of binary number system is 2, because it has only two digits.
Applications:
Binary is found in computer technology. All computer language and programming is based on the 2-digit number system used in digital encoding (is the process of taking data and representing it with discreet bits of information).
The most common application for the binary number system can be found in computer technology. All computer language and programming is based on the 2-digit number system used in digital encoding. Digital encoding is the process of taking data and representing it with discreet bits of information. These discreet bits consist of the 0s and 1s of the binary system.
2) Octal number system
Octal number system has only eight (8) digits from 0 to 7. Every number (value) represents with 0,1,2,3,4,5,6 and 7 in this number system. The base of octal number system is 8, because it has only 8 digits.
Applications:
Octal used to be used widely within IT basically how hexadecimal is used today. It was used as an ideal abbreviation of binary because the bit size (amount of bits that could be handled at one go) of the CPUs that were used at the time that being 26 and 36 bit. After CPUs converted to bit sizes of 32 and 64 bit octal was no longer utilised because, though both are still dividable by 8 they can also be divided by 16 which is far more efficient. There are some places however where octal is still widely utilised including within file permissions within unix (the source code for linux, mac os and android along with other operating systems) to avoid having to add support for symbols for hexadecimal. It is also used within digital displays that also have no support for symbols.
Octal was used widely in the Electronics and Computer Industry, as although digital electronics is based on gates with only two states and is therefore fundamentally binary, binary numbers can quickly become long and hard to transcribe without errors. Their octal equivalents are much shorter and easier to remember, and have a straight-forward way of conversion to/from binary.
The PDP-11 computer made by the Digital Equipment Corporation used the octal numeric system exclusively for displaying memory addresses and content.
Unix file system permissions have three sets (user, group, others) of three bit permissions (read, write, execute), which is naturally represented in octal.
The use of octal numbers has declined as most modern computers no longer base their word length on multiples of three bits, (they are based on multiples of four bits, so hexadecimal is more widely used).
3) Decimal number system
Decimal number system has only ten (10) digits from 0 to 9. Every number (value) represents with 0,1,2,3,4,5,6, 7,8 and 9 in this number system. The base of decimal number system is 10, because it has only 10 digits.
Applications:
We use decimals every day, while dealing with money, weight, length etc. Decimal numbers are used in situations where more precision is required, than the whole numbers can provide. For example, when we calculate our weight on the weighing machine, we do not always find the weight equal to a whole number on the scale. In order to know our exact weight we must understand what the decimal value on the scale means. This section deals with the concept of decimals in three important fields of our daily life.
4) Hexadecimal number system
A Hexadecimal number system has sixteen (16) alphanumeric values from 0 to 9 and A to F. Every number (value) represents with 0,1,2,3,4,5,6, 7,8,9,A,B,C,D,E and F in this number system. The base of hexadecimal number system is 16, because it has 16 alphanumeric values. Here A is 10, B is 11, C is 12, D is 13, E is 14 and F is 15.
Applications:
A computer understands information composed of only Zeros and Ones. Therefore, when we type some letters or words, data is processed by the computer in the form of 0s and 1s. A computer can understand positional number system where there are only a few symbols called digits and these symbols represent different values depending on the position they occupy in the number. Computer is usually designed to process hexadecimal number.
For example, the images you see on your computer screen have been encoded with a binary line for each pixel. If a screen is using a 16-bit code, then each pixel has been told what color to display based on which bits are 0s and which bits are 1s. As a result, 2^16 represents 65,536 different colors! We also find the binary number system in a branch of mathematics known as Boolean algebra. This field of mathematics is concerned with logic and truth values. Here, statements that are either true or false are then assigned a 0 or 1.
Hexadecimal numbering system is often used by programmers to simplify the binary numbering system. Since 16 is equivalent to 24, there is a linear relationship between the numbers 2 and 16.
This means that one hexadecimal digit is equivalent to four binary digits. Computers use binary numbering system while humans use hexadecimal numbering system to shorten binary and make it easier to understand.
Hexadecimals are used in the following:
- To define locations in memory. Hexadecimals can characterise every byte as two hexadecimal digits only compared to eight digits when using binary.
- To define colours on web pages. Each primary colour – red, green and blue is characterised by two hexadecimal digits. The format being used is #RRGGBB. RR stands for red, GG stands for green and BB stands for blue.
- To represent Media Access Control (MAC) addresses. MAC addresses consist of 12-digit hexadecimal numbers. The format being used is either MM:MM:MM:SS:SS:SS or MMMM-MMSS-SSSS. The first 6 digits of the MAC address represent the ID of the adapter manufacturer while the last 6 digits represent the serial number of the adapter.
- To display error messages. Hexadecimals are used to define the memory location of the error. This is useful for programmers in finding and fixing errors.
You can also go through below mentioned book for better understanding of Number System:
BOOK:
8085 Microprocessor Assembly Language Programming by Samir G. Pandya, LAP Lambert Academic Publishing ( 2017-03-16 ), Germany.
Link of above mentioned BOOK:
I hope I have answered your question.
With Best Wishes,
Samir G. Pandya
2 Recommendations
Popular answers (1)
for digital hardware, the natural numbering system is binary (base 2). but expressing anything in binary causes way too many bits to be used, so , it's not practical. hexadecimal allows us to use 4x fewer bits, so, it is very useful. besides no processor uses anything less than 4bits, so, hexadecimal is the most common when expressing numbers in a way that is most suitable for digital hardware.
example 1101 1010 is the 8 bit numbef which can be expressed a lot more cleaner as the he equivalent DA.
on the other hand, humans are most comfortable with DECIMAL which is base 10, as opposed to HEXADECIMAL which base 16. In DECIMAL , 1101 1010 would be
218
4 Recommendations
All Answers (6)
The number based conversions are essential in digital electronics..mostly in all digital system,we have the input in decimal format..but it takes as binary number for the computation by decimal to binary conversion..and we use the hexadecimal number to make coding for microprocessor but it converts that to binary for computation after the computation the result will be in hexadecimal format by inverse conversion..
2 Recommendations
3rd Oct, 2013
Indian Institute of Technology (Banaras Hindu University) Varanasi
Normally we use base 10 or decimal number system in our day to day life. But when computers are made as decimal representation is difficult for them binary computation were made. But representing large numbers in binary is difficult to read so the hexadecimal representation of merging 4 binary bits came into pictures. Same case with octal representation
2 Recommendations
for digital hardware, the natural numbering system is binary (base 2). but expressing anything in binary causes way too many bits to be used, so , it's not practical. hexadecimal allows us to use 4x fewer bits, so, it is very useful. besides no processor uses anything less than 4bits, so, hexadecimal is the most common when expressing numbers in a way that is most suitable for digital hardware.
example 1101 1010 is the 8 bit numbef which can be expressed a lot more cleaner as the he equivalent DA.
on the other hand, humans are most comfortable with DECIMAL which is base 10, as opposed to HEXADECIMAL which base 16. In DECIMAL , 1101 1010 would be
218
4 Recommendations
See also the question below about the one of the possible number-system convertions:
2 Recommendations
Cyril Mechkov, Tolga Soyata, veera raghava Grandhi, Ravichandran Palanisamy - Thank you very much
2 Recommendations
28th Apr, 2018
Gujarat Arts and Science College, Ahmedabad, Gujarat, INDIA. (Government College, Pincode - 380006.)
Dear Sandeep Pariyani,
We communicate with each other in a particular language made of letters or words. We normally type letters or words through keyboard of the computer, but computer does not understand the words and letters. Rather, those words and letters are translated into numbers. This means that computers understand only numbers. We know the decimal (base 10) system, and are very comfortable with performing operations using this system, it is also important for us to understand that the decimal system is not the only system in the world. By studying other number systems such as binary (base 2), quaternary (base 4), octal (base 8), hexadecimal (base 16) and so forth, we will gain a better understanding of how number systems work in general. Number systems are the technique to represent numbers in the computer system architecture, every value that you are saving or getting into/from computer memory has a defined number system. As Computer architecture supports following number systems so we need to study them and also need to know the conversion technique between them.
- Binary number system
- Octal number system
- Decimal number system
- Hexadecimal (Hex) number system
1) Binary Number System
A Binary number system has only two digits that are 0 and 1. Every number (value) represents with 0 and 1 in this number system. The base of binary number system is 2, because it has only two digits.
Applications:
Binary is found in computer technology. All computer language and programming is based on the 2-digit number system used in digital encoding (is the process of taking data and representing it with discreet bits of information).
The most common application for the binary number system can be found in computer technology. All computer language and programming is based on the 2-digit number system used in digital encoding. Digital encoding is the process of taking data and representing it with discreet bits of information. These discreet bits consist of the 0s and 1s of the binary system.
2) Octal number system
Octal number system has only eight (8) digits from 0 to 7. Every number (value) represents with 0,1,2,3,4,5,6 and 7 in this number system. The base of octal number system is 8, because it has only 8 digits.
Applications:
Octal used to be used widely within IT basically how hexadecimal is used today. It was used as an ideal abbreviation of binary because the bit size (amount of bits that could be handled at one go) of the CPUs that were used at the time that being 26 and 36 bit. After CPUs converted to bit sizes of 32 and 64 bit octal was no longer utilised because, though both are still dividable by 8 they can also be divided by 16 which is far more efficient. There are some places however where octal is still widely utilised including within file permissions within unix (the source code for linux, mac os and android along with other operating systems) to avoid having to add support for symbols for hexadecimal. It is also used within digital displays that also have no support for symbols.
Octal was used widely in the Electronics and Computer Industry, as although digital electronics is based on gates with only two states and is therefore fundamentally binary, binary numbers can quickly become long and hard to transcribe without errors. Their octal equivalents are much shorter and easier to remember, and have a straight-forward way of conversion to/from binary.
The PDP-11 computer made by the Digital Equipment Corporation used the octal numeric system exclusively for displaying memory addresses and content.
Unix file system permissions have three sets (user, group, others) of three bit permissions (read, write, execute), which is naturally represented in octal.
The use of octal numbers has declined as most modern computers no longer base their word length on multiples of three bits, (they are based on multiples of four bits, so hexadecimal is more widely used).
3) Decimal number system
Decimal number system has only ten (10) digits from 0 to 9. Every number (value) represents with 0,1,2,3,4,5,6, 7,8 and 9 in this number system. The base of decimal number system is 10, because it has only 10 digits.
Applications:
We use decimals every day, while dealing with money, weight, length etc. Decimal numbers are used in situations where more precision is required, than the whole numbers can provide. For example, when we calculate our weight on the weighing machine, we do not always find the weight equal to a whole number on the scale. In order to know our exact weight we must understand what the decimal value on the scale means. This section deals with the concept of decimals in three important fields of our daily life.
4) Hexadecimal number system
A Hexadecimal number system has sixteen (16) alphanumeric values from 0 to 9 and A to F. Every number (value) represents with 0,1,2,3,4,5,6, 7,8,9,A,B,C,D,E and F in this number system. The base of hexadecimal number system is 16, because it has 16 alphanumeric values. Here A is 10, B is 11, C is 12, D is 13, E is 14 and F is 15.
Applications:
A computer understands information composed of only Zeros and Ones. Therefore, when we type some letters or words, data is processed by the computer in the form of 0s and 1s. A computer can understand positional number system where there are only a few symbols called digits and these symbols represent different values depending on the position they occupy in the number. Computer is usually designed to process hexadecimal number.
For example, the images you see on your computer screen have been encoded with a binary line for each pixel. If a screen is using a 16-bit code, then each pixel has been told what color to display based on which bits are 0s and which bits are 1s. As a result, 2^16 represents 65,536 different colors! We also find the binary number system in a branch of mathematics known as Boolean algebra. This field of mathematics is concerned with logic and truth values. Here, statements that are either true or false are then assigned a 0 or 1.
Hexadecimal numbering system is often used by programmers to simplify the binary numbering system. Since 16 is equivalent to 24, there is a linear relationship between the numbers 2 and 16.
This means that one hexadecimal digit is equivalent to four binary digits. Computers use binary numbering system while humans use hexadecimal numbering system to shorten binary and make it easier to understand.
Hexadecimals are used in the following:
- To define locations in memory. Hexadecimals can characterise every byte as two hexadecimal digits only compared to eight digits when using binary.
- To define colours on web pages. Each primary colour – red, green and blue is characterised by two hexadecimal digits. The format being used is #RRGGBB. RR stands for red, GG stands for green and BB stands for blue.
- To represent Media Access Control (MAC) addresses. MAC addresses consist of 12-digit hexadecimal numbers. The format being used is either MM:MM:MM:SS:SS:SS or MMMM-MMSS-SSSS. The first 6 digits of the MAC address represent the ID of the adapter manufacturer while the last 6 digits represent the serial number of the adapter.
- To display error messages. Hexadecimals are used to define the memory location of the error. This is useful for programmers in finding and fixing errors.
You can also go through below mentioned book for better understanding of Number System:
BOOK:
8085 Microprocessor Assembly Language Programming by Samir G. Pandya, LAP Lambert Academic Publishing ( 2017-03-16 ), Germany.
Link of above mentioned BOOK:
I hope I have answered your question.
With Best Wishes,
Samir G. Pandya
2 Recommendations
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