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1 APPENDIX C The Hexadecimal Number System and Memory Addressing U nderstanding the number system and the coding system that computers use to store data and communicate with each other is fundamental to understanding how computers work. Early attempts to invent an electronic computing device met with disappointing results as long as inventors tried to use the decimal number system, with the digits 0 9. Then John Atanasoff proposed using a coding system that expressed everything in terms of different sequences of only two numerals: one represented by the presence of a charge and one represented by the absence of a charge. The numbering system that can be supported by the expression of only two numerals is called base 2, or binary; it was invented by Ada Lovelace many years before, using the numerals 0 and 1. Under Atanasoff s design, all numbers and other characters would be converted to this binary number system, and all storage, comparisons, and arithmetic would be done using it. Even today, this is one of the basic principles of computers. Every character or number entered into a computer is first converted into a series of 0s and 1s. Many coding schemes and techniques have been invented to manipulate these 0s and 1s, called bits for binary digits. The most widespread binary coding scheme for microcomputers, which is recognized as the microcomputer standard, is called ASCII (American Standard Code for Information Interchange). (Appendix B lists the binary code for the basic 127- character set.) In ASCII, each character is assigned an 8-bit code called a byte. The byte has become the universal single unit of data storage in computers everywhere. Table C-1 lists common terms used in the discussion of how numbers are stored in computers. Term Definition Bit A numeral in the binary number system: a 0 or a 1. Byte Kilobyte Megabyte 8 bits bytes, which is 2 to the 10th power, often rounded to 1000 bytes. Either 1024 kilobytes or 1000 kilobytes, depending on what has come to be standard practice in different situations. For example, when calculating floppy disk capacities, 1 megabyte = 1000 kilobytes; when calculating hard drive capacity, traditionally, 1 megabyte = 1024 kilobytes. Table C-1 (continued) 1107

4 1110 APPENDIX C The Hexadecimal Number System and Memory Addressing Figure C-2 2 trucks 1 crate 0 cartons 2 boxes 2 singles Joe s shipment of 197 widgets: 2 truckloads, 1 crate, 0 cartons, 2 boxes, and a group of 2 singles You can easily apply the widget-packing analogy to another base. If Joe used 10 instead of three, he would be using base 10 (decimal) rules. So, numbering systems differ by the different numbers of units they group together. In the hex number system, we group by 16. So, if Joe were shipping in groups of 16, as in Figure C-3, single widgets could be shipped out up to 15, but 16 widgets would make one box. Sixteen boxes would make one carton, which would contain 16 16, or 256, widgets. Sixteen cartons would make one crate, which would contain , or 4096, widgets. Suppose Joe receives an order for 197 widgets to be packed in groups of 16. He will not be able to fill a carton (256 widgets), so he ships out 12 boxes (16 widgets each) and five single widgets: = 192, and = 197 Figure C-3 1 truckload holds 16 crates (65,536 widgets) 1 crate holds 16 cartons (4,096 widgets) 1 carton holds 16 boxes (256 widgets) 1 box holds 16 widgets Widgets displayed in truckloads, crates, cartons, boxes, and singles grouped in 16s single widgets

5 Appendix C 1111 You approach an obstacle if you attempt to write the number in hex. How are you going to express 12 boxes and 5 singles? In hex, you need single numerals to represent the numbers 10, 11, 12, 13, 14, and 15 in decimal. Hex uses the letters A through F for the numbers 10 through 15. Table C-2 shows values expressed in the decimal, hex, and binary numbering systems. In the second column in Table C-2, you are counting in the hex number system. For example, 12 is represented with a C. So you say that Joe packs C boxes and 5 singles. The hex number for decimal 197 is C5 (see Figure C-4). C Figure C-4 Hex C5 represented as C boxes and 5 singles = 197 decimal Decimal Hex Binary Decimal Hex Binary Decimal Hex Binary E C F D E F A B Table C-2 (continued)

6 1112 APPENDIX C The Hexadecimal Number System and Memory Addressing Decimal Hex Binary Decimal Hex Binary Decimal Hex Binary 12 C A D B Table C-2 Decimal, hex, and binary values For a little practice, calculate the hex values of the decimal values 14, 259, 75, and 1024 and the decimal values of FFh and A11h. How Exponents Are Used to Express Place Value If you are comfortable with using exponents, you know that writing numbers raised to a power is the same as multiplying that number by itself the power number of times. For example, 3 4 = = 81. Using exponents in expressing numbers can also help us easily see place value, because the place value for each place is really the base number multiplied by itself a number of times, based on the place value position. For instance, look back at Figure C-1. A truckload is really , or 81, units, which can be written as 3 4. A crate is really 3 3 3, or 27, units. The numbers in Figure C-1 can therefore be written like this: Truckload = 3 4 Crate = 3 3 Carton = 3 2 Box = 3 1 Single = 3 0 (Any number raised to the 0 power equals 1.) Therefore, we can express the numbers in Figure C-2 as multiples of truckloads, crates, cartons, boxes, and singles like this: Truckloads Crates Cartons Boxes Singles (base 3) Decimal equivalent When we sum up the numbers in the last row above, we get 197. We just converted a base 3 number (21022) to a base 10 number (197). Binary Number System It was stated earlier that it is easier for computers to convert from binary to hex or from hex to binary than to convert between binary and decimal. Let s see just how easy. Recall that the binary number system only has two numerals, or bits: 0 and 1. If our friend Joe in shipping operated a binary shipping system, he would pack like this: 2 widgets in a box, 2 boxes in one carton (4 widgets), two cartons in one crate (8 widgets), and two crates in one truckload (16 widgets). In Figure C-5, Joe is asked

7 Appendix C 1113 to pack 13 widgets. He packs 1 crate (8 widgets), 1 carton (4 widgets), no boxes, and 1 single. The number 13 in binary is 1101: (1 2 3 ) + (1 2 2 ) + (0 2 1 ) + (1 2 0 ) = = 13 C Figure C-5 1 crate 1 carton 0 boxes 1 single Binary 1101 = 13 displayed as crates, cartons, boxes, and singles Now let s see how to convert binary to hex and back again. The largest 4-bit number in binary is This number in decimal and hex is: binary 1111 = 1 group of 8 = 8 1 group of 4 = 4 1 group of 2 = 2 1 single = 1 TOTAL = 15 (decimal) Therefore, 1111 (binary) = 15 (decimal) = F (hex) This last calculation is very important when working with computers: F is the largest numeral in the hex number system and it only takes 4 bits to write this largest hex numeral: F (hex) = 1111 (binary). So, every hex numeral (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F) can be converted into a 4-bit binary number. Look back at the first 16 entries in Table C-2 for these binary values. Add leading zeroes to the binary numbers as necessary. When converting from hex to binary, take each hex numeral and convert it to a 4-bit binary number, and string all the 4-bit groups together. Fortunately, when working with computers, you will almost never work with more than two hex numerals at a time. Here are some examples: 1. To convert hex F8 to binary, do the following: F = 1111, and 8 = Therefore, F8 = (usually written ). 2. To convert hex 9A to binary, do the following: 9 = 1001, and A = Therefore, 9A =

9 Appendix C 1115 FFFFF converted to decimal: = 15 1 = = = = = 3, = = 61, = 15 65,536 = 983,040 TOTAL = 1,048,575 C Remember that FFFFF is the last memory address in upper memory. The very next memory address is the first address of extended memory, which is defined as memory above 1 MB. If you add 1 to the number above, you get 1,048,576, which is equal to , which is the definition of 1 megabyte. Displaying Memory with DOS DEBUG In Figure C-6 you see the results of the beginning of upper memory displayed. The DOS DEBUG command displays the contents of memory. Memory addresses are displayed in hex segment-and-offset values. To enter DEBUG, type the following command at the C prompt and press Enter: C:\> DEBUG First address to dump Dump next group of addresses Figure C-6 Memory dump: -d A000:0 You create a memory dump when you tell the system to record the contents of memory to a file or display it on screen; a dump can be useful when troubleshooting. Type the following dump command to display the beginning of upper memory (the hyphen in the command is the DEBUG command prompt) and press Enter: -d A000:0

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