Hexadecimal Number System

11 Key excerpts on "Hexadecimal Number System"

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  Computer Systems Architecture

  • Aharon Yadin(Author)
  • 2016(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  It is based on the 10 decimal digits with the addition of A, B, C, D, E, and F. It should be noted that while the letters A–F are text characters, when used as part of the hexadecimal system they represent numerals. A is the equivalent of 10, B is 11, C is 12, and so on up to F, which is the equivalent of 15. 52 ◾ Computer Systems Architecture The migration of binary numbers to hexadecimal numbers is done by splitting the binary number into groups of four bits and using Table 2.3 for the appropriate value. For example: 3 212 580 11 0001 0000 0101 0010 0100 310524 10 2 16 , , = = Both bases (8 and 16) are powers of two, and this provides a simple and straightforward migration to and from binary numbers. Since 8 = 2 3 , then every three bits represents one octal digit, and since 16 = 2 4 , every 4 bits represents one hexadecimal digit. For example, the number 0101 5 2 = ( ) in octal, decimal and hexadecimal This can be easily figured out using the migration formulas described above. On the other hand, the number 1011 11 2 1 0 16 = = B The underlining meaning of these migrations is that any binary number can be replaced by an octal or hexadecimal number without any complex calculations, just by grouping the bits and figuring out the value of each such group. As with the delimiters that separate different groups of thousands in the decimal system, the split used in binary numbers always starts from the right side. For a simple migration, one may use the migration table ( Table 2.3 ). TABLE 2.3 Migration Table Binary (2) Decimal (10) Base 4 (4) Octal (8) Hexadecimal (16) 0 0 0 0 0 1 1 1 1 1 10 2 2 2 2 11 3 3 3 3 100 4 10 4 4 101 5 11 5 5 110 6 12 6 6 111 7 13 7 7 1000 8 20 10 8 1001 9 21 11 9 1010 10 22 12 A 1011 11 23 13 B 1100 12 30 14 C 1101 13 31 15 D 1110 14 32 16 E 1111 15 33 17 F Data Representation ◾ 53 Using the numbers in the table, it is easy to represent the binary numbers in other bases that are powers of two.

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  Principles Of Computer Programming NQF3 SB

  TVET FIRST

  • S Sasti D Sasti(Author)
  • 2019(Publication Date)
  • Macmillan

    (Publisher)

  30 Topic 1 Table 2.1: Properties of decimal, binary, octal, and Hexadecimal Number Systems Number system Base Number of digits Range of digits Decimal Base 10 Represented with 10 digits. 0–9 Binary Base 2 Represented with two digits. 0 and 1 Octal Base 8 Represented with eight digits. 0–7 Hexadecimal Base 16 Represented with 16 digits and characters. To achieve this, the first 10 digits are numeric and the next six are letters of the alphabet or characters: 0–9 and A–F. As you can see, the Hexadecimal Number System is not easily represented because there are 16 possible digits. Letters of the alphabet are used to represent the 11 th to 16 th digits. See Table 2.2 for a representation of the 16 digits compared with the decimal number system. Table 2.2: The 16 digits in the Hexadecimal Number System Decimal 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Hexadecimal 0 1 2 3 4 5 6 7 8 9 A B C D E F Since there are different ways of representing numbers, we have to be able to convert between the different number systems. We will now look at how to convert between the different number systems. 2.1.3 Converting between number systems How digits are represented in a number system Let’s first look at how digits are represented in a number system. Example 2.1: Representing decimal and binary numbers The digits of a number are represented by the base of the number raised to a particular power. To understand this concept, we will look at the decimal number system because you are familiar with it. Decimal numbers: Base 10 Below is an example of a decimal number. Each digit represents a value of 10 n , where: l 10 represents the base. l n represents the power. The power of the last digit starts from 0 and increases by 1 for each preceding digit. 68413 10 Value of number Base of number Figure 2.1: Parts of a decimal number

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  Principles of Modern Digital Design

  • Parag K. Lala(Author)
  • 2007(Publication Date)
  • Wiley-Interscience

    (Publisher)

  1 Number Systems and Binary Codes 1.1 INTRODUCTION In conventional arithmetic, a number system based on ten units (0 to 9) is used. However, arithmetic and logic circuits used in computers and other digital systems operate with only 0 ’ s and 1 ’ s because it is very dif fi cult to design circuits that require ten distinct states. The number system with the basic symbols 0 and 1 is called binary. Although digital systems use binary numbers for their internal operations, communication with the external world has to be done in decimal systems. In order to simplify the communication, every decimal number may be represented by a unique sequence of binary digits; this is known as binary encoding . In this chapter we discuss number systems in general and the binary system in particular. In addition, we consider the octal and Hexadecimal Number Systems and fi xed-and fl oating-point representation of numbers. The chapter ends with a discussion on weighted and nonweighted binary encoding of decimal digits. 1.2 DECIMAL NUMBERS The invention of decimal number systems has been the most important factor in the devel-opment of science and technology. The term decimal comes from the Latin word for “ ten. ” The decimal number system uses positional number representation, which means that the value of each digit is determined by its position in a number. The base (also called radix) of a number system is the number of symbols that the system contains. The decimal system has ten symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9; in other words it has a base of 10. Each position in the decimal system is 10 times more signi fi cant than the pre-vious position. For example, consider the four-digit number 2725: Notice that the 2 in the 10 3 position has a different value than the 2 in the 10 1 position. The value of a decimal number is determined by multiplying each digit of the number by the 1 Principles of Modern Digital Design , by Parag K. Lala Copyright © 2007 John Wiley & Sons, Inc.

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  Information Technology

  An Introduction for Today's Digital World

  • Richard Fox(Author)
  • 2020(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  Section 4.3 .

  SECTION 4.1 ACTIVITY

  1.  As noted, Roman numerals can be used to represent decimal values but in a non-positional way. Why do Roman numerals not qualify for this? 2.  Research the Babylonian numeral system and write down the following values: 1, 4, 5, 10, 15, 20, 50.

  4.2      THE BINARY NUMBERING SYSTEM

  Why is all this important? It wouldn’t be if our computers used decimal, but computers store, transmit, and process all information in binary. As an IT specialist, you may never have to worry about looking at or expressing data in binary, but the use of non-decimal numbering systems does come up from time to time. Perhaps the most common occurrence of non-decimal numbering that you will see will occur with network communication. Internet protocol (IP) addresses are sometimes denoted in binary, and IPv6 addresses are denoted in hexadecimal. Octal may also be used in some computer applications. In this section, we look at binary to understand storage sizes and how to convert between binary and decimal. In Section 4.3 , we look at octal and hexadecimal. In Section 4.4 , we look at how to represent negative numbers and fractions in binary.

  4.2.1      BITS AND STORAGE SIZES

  Binary is probably the most important numbering system outside of decimal. Because of the digital nature of computers, everything has to be represented as high and no (low) voltage or ons and offs. We assign these the values of 1 and 0, respectively, and use binary numbers for convenience.

  We call a single 1 or 0 value a bit (for binary digit). We cannot do much with a single bit, so we combine bits together for more significant values. We group 8 bits together into a unit called a byte. Most computers do not operate on individual bytes, instead moving and computing data in larger sizes called words

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  Elementary Technical Mathematics, 12th

  • Dale Ewen(Author)
  • 2018(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  This system of numeration is based on the historical and natural way humans have used their fingers to count objects and maintain records of their possessions by groups of tens. Computers use a binary number system , which has only two symbols or numerals: 0 and 1. These can represent the two positions in a transistor, “off” and “on.” Off is assigned 0, and on is assigned 1. Table 16.1 should help you to understand the relationship between these two systems by comparing place values in the decimal system with place values in the binary system. Table 16.1 Decimal System Millions Hundred thousands Ten thousands Thousands Hundreds Tens Ones 10 6 10 3 10 3 10 3 10 3 10 3 10 1,000,000 10 5 10 3 10 3 10 3 10 3 10 100,000 10 4 10 3 10 3 10 3 10 10,000 10 3 10 3 10 3 10 1000 10 2 10 3 10 100 10 1 10 10 10 0 0 0 Binary System Sixty-fours Thirty-twos Sixteens Eights Fours Twos Ones 2 6 2 3 2 3 2 3 2 3 2 3 2 64 1000000 2 2 5 2 3 2 3 2 3 2 3 2 32 100000 2 2 4 2 3 2 3 2 3 2 16 10000 2 2 3 2 3 2 3 2 8 1000 2 2 2 2 3 2 4 100 2 2 1 2 2 10 2 2 0 1 1 1 2 Note: In base ten, there are powers of ten and ten numerals. In base two, there are powers of two and two numerals. Copyright 2019 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 514 CHAPTER 16 ◆ Binary and Hexadecimal Numbers Some notation is needed to distinguish between decimal and binary numbers. The nota-tion generally used is a subscript after the number to indicate the base of the system in which the number is written. For instance, 1101 10 is a decimal number, and 1101 2 is a binary number.

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  Microelectronic Systems N2 Checkbook

  The Checkbook Series

  • R E Vears(Author)
  • 2013(Publication Date)
  • Butterworth-Heinemann

    (Publisher)

  5 6 2 Thus, 101110010 2 =562 8 (see Problem 5) 8 In order to convert an octal number into binary, the process in paragraph 7 is reversed. For example, consider the octal number 473 g . This number may be con-verted into binary form by: (i) spacing out the octal digits, and 4 7 3 (ii) converting each octal digit into binary form. 100 111 011 Thus, 473 g = 100111011 2 . (See Problem 6) 9 The characteristics of a hexadecimal numbering system are: (i) sixteen distinct counting digits (0,1, 2,3,4,5,6,7,8,9, A, B,C,D,E and F), and (ii) a place value (or weight) for each digit, organised in ascending powers of sixteen starting from the right (see column 4 of Table 1). Since everyday situations make use of a decimal numbering system, only ten distinct counting digits have been devised. A hexadecimal system requires six more counting digits, and for this purpose, letters A to F are used. These letters correspond to decimal values 10 to 15. Thus, for example, the hexadecimal number 1A4E 16 may be considered as follows: 1A4E 16 (hex) = 1 X 16 3 + A X 16 2 + 4 X 16 1 + E X 16° = 1 X 4096 + A X 256 + 4 X 16 + E X 1 = 1 X 4096 + 10 X 256 + 4 X 16 + 14X 1 = 4096 + 2560 + 64 + 1 4 = 6734 10 . (See Problem 7) 21 In order to express a binary number in hexadecimal, its bits are arranged in groups of four, starting from the right, and a hexadecimal symbol is assigned to each group. For example, consider the binary number 1110011110101001 2 . This number may be expressed in its hexadecimal form by: (i) grouping bits in fours from the right, and UJjJ (jjUA ^01_g ljjgj, (ii) assigning hexadecimal symbols to each group. E 7 A 9 Thus, 1110011110101001 2 = E7A9 16 . (See Problem 9) In order to convert a hexadecimal number into binary, the process in paragraph 10 is reversed. For example, consider the hexadecimal number A3FB 16 .

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  Foundations of Computing

  Essential for Computing Studies, Profession And Entrance Examinations - 5th Edition

  • Pradeep K. Sinha, Priti Sinha(Authors)
  • 2022(Publication Date)
  • BPB Publications

    (Publisher)

  memory dump . Working with memory dumps in binary form would be difficult and error prone for computer professionals. Hence, computer professionals often use octal and Hexadecimal Number Systems, as shortcut notations for binary. They are described here.

  Octal Number System

  In octal number system, the base is 8. Hence, there are only eight symbols or digits: 0, 1, 2, 3, 4, 5, 6, and 7 (8 and 9 do not exist in this system). The largest single digit is 7 (one less than the base 8). Each position in an octal number represents a power of the base (8). Therefore, decimal equivalent of octal number 2057 (written as 20578) is:

  (2 x 83 ) + (0 x 82 ) + (5 x 81 ) + (7 x 80 ) = 1024 + 0 + 40 + 7 = 1071

  Hence, 20578 = 107110

  Since there are only 8 digits in octal number system, 3 bits (2 3 = 8) are sufficient to represent any octal number in binary (see Figure 2.1 ).

  Hexadecimal Number System

  In Hexadecimal Number System, the base is 16. Hence, there are 16 symbols or digits. The first 10 digits are the same digits of decimal number system - 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The remaining six digits are the symbols A, B, C, D, E, and F, representing decimal values 10, 11, 12, 13, 14, and 15, respectively. Hence, the largest single digit is F or 15 (one less than the base 16). Each position in Hexadecimal Number System represents a power of the base (16). Therefore, decimal equivalent of hexadecimal number 1AF (written as 1AF16) is:

  (1 x 162 ) + (A x 161 ) + (F x 160 ) = (1 x 256) + (10 x 16) + (15 x 1) = 256 +160 + 15 = 431

  Hence, 1AF16 = 43110

  Since there are only 16 digits in Hexadecimal Number System, 4 bits (2 4 = 16) are sufficient to represent any hexadecimal number in binary.

  CONVERTING FROM ONE NUMBER SYSTEM TO ANOTHER

  Input data fed to computers and final output values are to be in decimal because decimal numbers are more meaningful to us than are numbers expressed in any other number system. Hence, computer professionals are often required to convert numbers in other number systems to decimal and vice-versa. They use various methods to convert numbers from one base to another. A method of converting from another base to decimal, and a method of converting from decimal to another base are described here.

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  Digital Electronic Circuits

  Principles and Practices

  • Shuqin Lou, Chunling Yang(Authors)
  • 2019(Publication Date)
  • De Gruyter

    (Publisher)

  2 Number systems and codes 2.1 Introduction Digital signals are generally represented by a series of binary digits. Binary number system and digital codes are fundamental to digital circuits and digital systems. First, this chapter introduces number systems including binary, decimal, octal, and hexa-decimal, and conversion between different number systems. Then binary arithmetic operations with binary numbers are introduced. Finally, digital codes such as binary-coded decimal (BCD), excess-3 code, the Gray code, and the American Standard Code for Information Interchange (ASCII) code are covered. The objectives of this chapter are to – Describe binary, decimal, hexadecimal, and octal number system – Convert number representation between different number systems – Apply arithmetic operations to binary numbers – Express signed binary numbers in sign-magnitude, one ’ s complement, two ’ s complement – Describe BCD, excess-3 code, and Gray code. 2.2 Number systems Number system s are ways to represent the quantities. You are familiar with the decimal number system since you use decimal number in your daily lives. However, a digital circuit and a digital system belong to a two-valued system. A binary number system is the most efficient way to represent quantities. Hexadecimal and octal number systems are used primarily as a compact way of writing binary number since long binary numbers are difficult to read and write. In this section, we will begin with the structure of decimal number system that you are familiar with. This will help you more easily understand the structure of a binary number system, a hexadecimal system, and an octal system. Then, the structure of a binary number system, a hexadecimal system, and an octal system are introduced, because the same quantity can be represented by a decimal number, or an equivalent binary number, hexadecimal number, and octal number. Conversion between different number systems is introduced at the end of this section.

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  Mathematics for Information Technology

  • Alfred Basta, Stephan DeLong, Nadine Basta, , Alfred Basta, Stephan DeLong, Nadine Basta(Authors)
  • 2013(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Find the decimal numbers with the shown representations. 12. 0001 1011 1110 0011 14. 1111.1110 1011 0010 13. 0101 0001 1111.1100 1101 15. 0000.0011 0100 1110 Write an explanation for the the solution of the following. 16. We have introduced one method for generating a Gray code and sug-gested an alternative approach to represent the integers from 0 through 15, inclusive. Begin with a string of four 0s (we use four digits because we know that there are 16 possible strings of four binary digits) and then successively flip the rightmost bit that produces a completely new string of digits. When we’re through, you will have 16 distinct sets of four binary digits to which we can assign the numbers 0 through 15. Demonstrate this procedure generates a Gray code. Summary In this chapter, you learned about: The decimal, binary, and Hexadecimal Number Systems. The arithmetic operations and applications of the binary number system. The relationship between binary and hex, decimal and hex, and binary and octal. The binary and 8421 codes. Glossary ASCII (American Standard Code for Information Interchange) An encryption device for the English al-phabet using binary numbers. base The number of distinct characters employed for numeration within a particular number system. Also called the radix. base-16 number system A system of numeration using 16 distinct symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F. Also called the Hexadecimal Number System. base-10 number system The decimal number system. Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

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  Digital Computer Structure and Design

  • R. Townsend(Author)
  • 2014(Publication Date)
  • Butterworth-Heinemann

    (Publisher)

  Chapter 4 Number Representation in Computers Number Systems The number systems, with which we are most familiar, all use positional notation to represent a number. Although various number bases have been used by different races and nations throughout history, the base of 10 has been commonly accepted and is now universally established in the decimal system. It appears to have its origin in counting on the fingers, and the word 'digit' is derived from the Latin word for finger. The procedure of counting numbers greater than the base is to increase the digit which is one place to the left by one, each time the digit is increased by ten steps. One can therefore represent a decimal number: a n . . . α 3 α 2 αχα 0 M ^ e f orrn: N = a 0 10° +0X1O 1 + Λ ι 10 2 +0 3 1Ο 3 ...+0„1Ο The ordering of the coefficients is reversed in the polynomial in contrast to the number, which is a relic of the fact that our number system came to us from Arabia, in which writing is normally from right to left. The system can be generalised for a uniform system of any base as the polynomial N = a 0 b° +a x b l +a 2 b 2 +a 3 b 3 .. .+a n b n where b is the base, and the coefficients are a 0 . . .a n which are normally written down to represent the number. A step further in complication are the nonuniform based number systems which are now happily being outmoded, but some of which we still have with us for a time, such as tons, hundredweights, and quarters; seconds, minutes, hours and days; and these can be represented as the polynomial. N = a 0 b 0 +a 1 (b 0 )b 1 +a 2 (b 0 b 1 )b 2 +a 3 (b 0 b x b 2 )b 3 ... where b 0 = 1 Table 4.1 shows the numbers up to 16 represented in bases up to 10, 67

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  The Architecture of Computer Hardware, Systems Software, and Networking

  An Information Technology Approach

  • Irv Englander, Wilson Wong(Authors)
  • 2021(Publication Date)
  • Wiley

    (Publisher)

  It may be necessary to mentally add 0s to the left end of the number to convert the most significant digit. This is most easily illustrated with an example: EXAMPLE Convert 13754 8 to base 10: 1 8 8 3 11 8 88 7 95 8 760 5 765 8 6120 4 6124 10 68 NUMBER SYSTEMS EXAMPLE Let us convert 11010111011000 2 to hexadecimal. Grouping the binary number by fours from the right, we have 0011 0101 1101 1000 2 or 35D8 16 Note that we added two zeros at the left end of the binary number to create groups of four. The conversion in the other direction works identically. Thus, 275331 8 becomes 010 111 101 011 011 001 2 For practice, now convert this value to hexadecimal. You should get 17AD9 16 as an answer. Most computer manufacturers today prefer to use hexadecimal, since a 16-bit or 32-bit number can be represented exactly by a four- or eight-digit hexadecimal number. (How many octal digits would be required?) A few manufacturers still use octal representation for some applications. You might ask why it is necessary to represent data in binary form at all. After all, the binary form is used within the computer, where it is usually invisible to the user. There are many occasions, however, where the ability to read the binary data is very useful. Remember that the computer stores both instructions and data in binary form. When debugging a program, it may be desirable to be able to read the program’s instructions and to determine intermediate data steps that the computer is using. Older computers used to provide binary dumps for this purpose. Binary dumps were complete octal listings of everything stored in memory at the time the dump was requested. Even today it is sometimes important, for example, to be able to read the binary data from a disk to recover a lost or damaged file. Modern computer operating systems and networks present a variety of troubleshooting data in hexadecimal form. Conversions between binary and hexadecimal notation are used frequently.

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