Number Systems and Data Representation CS221


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1 Number Systems and Data Representation CS221 Inside today s computers, data is represented as 1 s and 0 s. These 1 s and 0 s might be stored magnetically on a disk, or as a state in a transistor, core, or vacuum tube. To perform useful operations on these 1 s and 0 s we have to organize them together into patterns that make up codes. First we will discuss numeric values for the codes, and then ways that the codes represent textual values. Each 1 and 0 is called a bit. 4 bits is called a nibble. 8 bits is called a byte. 16/32/64/128 bits is called a word. The number depends upon the number of bits that can simultaneously be operated upon by the processor. Most processors today use 32 bits. Since we only have 1 s and 0 s available, the number system generally used inside the computer is the binary number system. We are used to the decimal number system. In decimal, we have ten symbols (09) that we can use. When all of the symbols are used up, we need to combine together two symbols to represent the next number (10). The most significant digit becomes our first digit, and the least significant digit cycles back to 0. The least significant digit increments until the symbols are used up, and which point the most significant digit is incremented to the next larger digit. This is just the familiar process of counting! The same process applies to the binary number system, but we only have two symbols available (01). When we run out of symbols, we need to combine them and increment the digits just as we would in a decimal, base 10, number system: Let s start counting to illustrate: Decimal Binary ? 17?
2 Binary to Decimal Conversion The binary number system is positional where each binary digit has a weight based upon its position relative to the least significant bit (LSB). To convert a binary number to decimal, sum together the weights. Each weight is calculated by multiplying the bit by 2 i, where i is the index relative to the LSB Weight = 1* * * * *2 4 = 1*1 + 1*2 + 0*4 + 0*8 + 1*16 = = 19 (base 10, decimal) What is 1111 in base 10? What is in base 10? Decimal to Binary Conversion To convert a decimal number to its equivalent binary number representation, the easiest method is to use repeated division by 2. Each time, save the remainder. The first remainder is the least significant bit. The last remainder is the most significant bit. What is 19 in binary? 19 / 2 = 9 remainder 1 LSB 9 / 2 = 4 remainder 1 4 / 2 = 2 remainder 0 2 / 2 = 1 remainder 0 1 / 2 = 0 remainder 1 MSB Our final number is If using a calculator, watch to see if the quotient has a fraction. If it is 0.5 then the remainder was 1, and repeat the process with the whole part minus the fraction. What is 83 in binary? What is 100 in binary? Octal Numbers The octal number system uses base 8 instead of base 10 or base 2. This is sometimes convenient since many computer operations are based on bytes (8 bits). In octal, we have 8 digits at our disposal, 07. Let s do the counting again:
3 Decimal Octal Octal to Decimal Conversion Converting octal to decimal is just like converting binary to decimal, except instead of powers of 2, we use powers of 8. That is, the LSB is 8 0, the next is 8 1, then 8 2, etc. To convert 172 in octal to decimal: Weight = 2* * *8 2 = 2*1 + 7*8 + 1*64 = 122 base 10 What is 104 in octal converted to decimal? What is 381 in octal converted to decimal? Decimal to Octal Conversion Converting decimal to octal is just like converting decimal to binary, except instead of dividing by 2, we divide by 8. To convert 122 to octal: 122 / 8 = 15 remainder 2 15 / 8 = 1 remainder 7 1 / 8 = 0 remainder 1 = 172 octal
4 If using a calculator to perform the divisions, the result will include a decimal fraction instead of a remainder. The remainder can be obtained by multiplying the decimal fraction by 8. For example, 122 / 8 = Then multiply 0.25 * 8 to get a remainder of 2. Octal to Binary, Binary to Octal Octal becomes very useful in converting to binary, because it is quite simple. The conversion can be done by looking at 3 bit combinations, and then concatenating them together. Here is the equivalent for each individual octal digit and binary representation: Octal Binary To convert back and forth between octal and binary, simply substitute the proper pattern for each octal digit with the corresponding three binary digits. For example, 372 in octal becomes or in binary. 777 in octal becomes or in binary. The same applies in the other direction: in binary becomes or 472 in octal. What is 441 octal in binary? What is binary in octal? Since it is so easy to convert back and forth between octal and binary, octal is sometimes used to represent binary codes. Octal is most useful if the binary code happens to be a multiple of 3 bits long. Sometimes it is quicker to convert decimal to binary by first converting decimal to octal, and then octal to binary. Hexadecimal Numbering The hexadecimal numbering system is the most common system seen today in representing raw computer data. This is because it is very convenient to represent groups of 4 bits. Consequently, one byte (8 bits) can be represented by two groups of four bits easily in hexadecimal.
5 Hexadecimal uses a base 16 numbering system. This means that we have 16 symbols to use for digits. Consequently, we must invent new digits beyond 9. The digits used in hex are the letters A,B,C,D,E, and F. If we start counting, we get the table below: Decimal Hexadecimal Binary A B C D E F Hex to Decimal Conversion Converting hex to decimal is just like converting binary to decimal, except instead of powers of 2, we use powers of 16. That is, the LSB is 16 0, the next is 16 1, then 16 2, etc. To convert 15E in hex to decimal: 1 5 E Weight = 14* * *16 2 = 14*1 + 5*16 + 1*256 = 350 base 10 What is F00 in hex converted to decimal? Decimal to Hex Conversion Converting decimal to hex is just like converting decimal to binary, except instead of dividing by 2, we divide by 16. To convert 350 to hex:
6 350 / 16 = 21 remainder 14 = E 21 / 16 = 1 remainder 5 1 / 16 = 0 remainder 1 So we get 15E for 350. Again, note that if a calculator is being used, you may multiple the fraction remainder by 16 to produce the remainder. 350/16 = Then to get the remainder, * 16 = 14. What is 33 in hex? Hex to Binary Conversion, Binary to Hex Going from hex to binary is similar to the process of converting from octal to binary. One must simply look up or compute the binary pattern of 4 bits for each hex code, and concatenate the codes together. To convert AE to binary: A = 1010 E = 1110 So AE in binary is The same process applies in reverse by grouping together 4 bits at a time and then look up the hex digit for each group. Binary broken up into groups of 4: (note the 0 added as padding on the MSB to get up to 4 bits) = 625 Hex How would you convert from Hex to Octal? E.g. convert B2F to octal. BCD Code When numbers, letters, words, or other pieces of data are represented by a special group of symbols, then we say that the data is encoded. The group of symbols is called a code. For example, Morse Code is a series of dashes and dots to represent letters. Decimal numbers may be represented using binary notation. When this is done, it is called a straight binary encoding.
7 Most systems use straight binary encoding to represent numbers. However, there are times when it is easier to use another means to represent numbers, particularly when we may want to retain some properties of decimal number systems. BCD, or binary coded decimal, is one such representation. BCD works by encoding each digit of a decimal number by its binary equivalent. Since a decimal digit can be as large as 9, we need 4 bits to code it (1001). Here is an example used to represent the decimal number 874: 8 = = = 0100 Note that 4 bits are always used for each digit. This results in the wasted space of bit patterns 9F. If one of these patterns ever shows up when using BCD, it could be flagged as an error condition. What is 53 decimal in BCD? What is encoded in BCD in decimal? Keep in mind that BCD is not a number system like binary, octal, decimal or hexadecimal. Instead, it is an encoding scheme. It is actually equivalent to the decimal system with an encoding to binary. A BCD number is NOT the same as its binary equivalent number. The number 10 (decimal) could be represented as 1010 in binary encoding, but would be represented as in BCD. The main advantage in using BCD is that it is easy to convert back and forth between decimal. Alphanumeric Codes So far we have only been discussing the representation of numbers. We would also like to represent textual characters. This includes letters, numbers, punctuation marks, and other special characters like % or / that appear on a typical keyboard. The most common code is ASCII, the American Standard Code for Information Interchange. ASCII is comprised of a 7 bit code. This means that we have 2 7 or 128 possible code groups. This is plenty for English as well as control functions such as the carriage return or linefeed functions. Another code is EBCDIC, extended binary coded decimal interchange code. EBCDIC was used on some mainframe systems but is now out of favor. The table at the back of your assembly book has a complete listing of ASCII codes. A small sample is shown below.
8 Character ASCII Hex Decimal A B C Z 5A 90 Since most computers store data in bytes, or 8 bits, there is 1 bit that may go unused (or used for parity, described below). Could we represent something like Chinese in ASCII, which has many more than 128 characters? One way to represent languages such as Chinese is to use a different code. UNICODE is an international standard used to represent multiple languages. To address the shortcomings of ASCII, UNICODE uses 16 bits per character. This allows 216 or characters. When this is insufficient, there are extensions to support over a million characters. Most of the work in UNICODE has gone in efforts to define the codes for the particular languages. If you would like more information, visit There are Unicode definitions for lots of languages, even some that are madeup. Some example Unicode characters for Tengwar (a script of Elvish from Tolkien lore) is shown below. Side note: A common problem with software development occurs when a product needs to support foreign languages. Many products in the US are initially developed assuming ASCII. However, if the product is successful and needs to support something like French or German, it is often a difficult task to move the program from ASCII to UNICODE. Software developers need to carefully take into account future needs and design the software appropriately. If it is known up front that other languages may need to be supported, it is a relatively simple matter to support UNICODE up front, or to remove dependencies on ASCII.
9 Identifying Code and Data Although we might know the binary contents of memory, the actual usage of the memory is not always clear. This is because in the von Neumann model, memory could store data or instructions. Consequently, either the contents of memory could be a computer instruction, or it could be data. If it is data, there is no way to tell if the data represents ASCII encoding, a binary number, or maybe something else. A program must keep track of its data and the type of representation used to avoid possible confusion. High level languages like C++ or Java impose restrictions on the way variables and instructions are manipulated (data type restrictions) but assembly language does not. In assembly language, there are few restrictions, but the programmer is burdened with the responsibility of taking care of many details.
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