Sessions 1, 2 and 3 Number Systems

Transcription

1 COMP 1113 Sessions 1, 2 and 3 Number Systems The goal of these three class sessions is to examine ways in which numerical and text information can be both stored in computer memory and how numerical information might be manipulated to simulate common arithmetic operations. In order to learn how it works, it is necessary to introduce some formalism. First, some definitions. A number is a symbol that represents or corresponds to a quantity of things. A number system (or, as some prefer, a numeral system) is a set of symbols and rules for representing a useful range of numbers. A wide variety of number systems have been used by people throughout history. In Canada, we use the decimal system, which we will look at in much more detail below. There is a lot of information available on the internet on historically important number systems. If you re interested, you might start with the site at Many of the number systems used by early civilizations had a distinct symbol for every quantity they might wish to express. As the size of their herds increased, such number systems either became increasingly awkward or less precise (for instance, by using a word meaning many for any quantity larger than some specific quantity). Positional Number Systems In a positional number system, the numerical significance of a symbol depends on where it appears in the representation of the number. Numbers are created by writing down a sequence of symbols (called digits) selected from a fixed set of symbols. The base or radix of the positional number system is the number of symbols in this fixed set. The radix point is a special symbol used to establish the actual position of each digit in the number. Sounds pretty complicated and weird? Read on. Example 1: The Decimal Number System. Our ordinary decimal number system is a positional number system. The base is 10, because we build up all numerical values using 10 basic symbols: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. In North America, it is common to use the period as the radix point (we call it the decimal point), but in some European countries, it is more common to use the comma as the radix point. Of course, you know how to work with decimal numbers, but you may not have thought about how their properties arise. Think about the counting process. Suppose we have a vast herd of goats to which we would like to attach a number. Before we start counting any of our goats, we would say that as far as we can say, we have 0 goats. Then one-by-one we start counting as our goats walk past in an orderly single-file line. a goat goes by we now have 1 goat another goat goes by we now have 2 goats another goat goes by we now have 3 goats this continues for a bit until we have counted 8 goats then another goat goes by, and so we have 9 goats now another goat comes, but we ve run out of symbols. Since we are working with a positional number system, what is done is we say we have 10 goats we ve gone once through our entire set of symbols (of which there are ten) and we about to start to go David W. Sabo (2014) Number Systems Page 1 of 50

2 through the set again. The number 10 is formed as a sequence of two of our basic symbols. now another goat comes we increment the rightmost digit of our count by 1, so we can say we have 11 goats. The means we have counted 1 ten of goats and 1 more. The 1 on the right is in position 2 and represents numbers of tens of goats. The 1 on the right is in the position of ones, and represents just numbers of individual goats. when we ve counted 19 goats (1 ten of goats and 9 individual goats), the next goat causes us to change our tally to 20 two tens of goats. This process could go on for a long time, but you can see how the basic rules of counting lead to the properties of the positional number system. When we need to increment position 1 but we are at the highest value symbol, what we do is increment the position to its left by 1, and write a 0. Then further counting can increment this digit by one at a time until we run out of symbols, at which point the digit to the left is again incremented by 1, and we write a 0. You know that in the end, decimal numbers look something like: position: digits in position 0 count as individuals or ones or 10 0 = 1 s. digits in position 1 count as tens or 10 1 = 10 s. digits in position 2 count as hundreds or 10 2 = 100 s. digits in position 3 count as thousands or 10 3 = 1000 s. Similarly, the pattern of positional value can extend to the right of the radix point (here, it is the decimal point of course) so that digits in position -1 count as tenths or as 10-1 s. digits in position -2 count as hundredths or as 10-2 s. digits in position -3 count as thousandths or as 10-3 s. Thus, a decimal number such as really represents the sum: 4 x x x x x x x In this way, a very small set of symbols (only ten) can be used to construct representations of an essentially unlimited range of quantities. The quantity represented by each symbol is not known until we know where in the number that symbol appears. So the leftmost 4 in the example represents a quantity of four thousand things. The rightmost 4 represents a quantity of four thousandths of a thing. Same symbol, but different quantities. The properties of decimal numbers are very familiar to you. This means that you know the basic structure of a positional number system. It also means that you have a mental image of the rather abstract concepts introduced in our definition of a positional number system at the beginning of this document. Much of the remainder of this document deals with applying the same principles when a base different from 10 is being used, and in converting the representations of numbers from one base to another. So, from now on, we will write numbers in the following form: d4d3d2d1d0.d-1d-2d-3d-4 followed by a subscript, b, indicating the base. Here the dk are the digits of the number. Page 2 of 50 Number Systems David W. Sabo (2014)

3 Then, the actual quantity represented by the number is Thus Similarly, + d4 x b 4 + d3 x b 3 + d2 x b 2 + d1 x b 1 + d0 + d-1 x b -1 + d2 x b -2 + d3 x b -3 + d4b = 4 x x x x x = 4 x x x x x 7-2 (You can use your calculator to determine that this last number is numerically equivalent to , rounded to twelve significant figures.) Converting to Base 10 (Decimal) Numbers The last example just above illustrates the procedure for converting numbers expressed in some base, b, into the equivalent number in base 10 (the equivalent decimal form). You just apply the basic definition of a positional number system, and then use your calculator (which gives answers in the decimal number system) to get the decimal value you require. Here s one more example: = 3 x x x x x x x x 5-3 = In this case, the conversion from the base 5 number to its base 10 equivalent gives an answer which can be stated exactly in seven significant digits. Often, conversions from some base to base 10 gives results which do not have terminating decimal parts, and illustrated by the base 7 example in the previous section above. Converting Numbers from Base 10 to Other Base Number Systems The process of converting decimal numbers into their equivalent in some other number base is more complicated than the reverse operation illustrated just above. If we restrict consideration to number bases which are whole numbers and greater than 1, then the following principles apply: the whole number part (to the left of the decimal point) converts to the whole number part (to the left of the radix point) of the result the fractional part (the part to the right of the decimal point) converts to the fractional part (the part to the right of the radix point) of the result The conversion of these two parts must be done separately, because they involve different procedures. i) conversion of the whole number part Suppose wish to convert the number to its equivalent in base 5. The procedure is as follows. Carry out successive divisions by 5, recording both the result and the remainder, until the result of 0 is obtained. Here the sequence of operations is: = 446 with remainder = 89 with remainder 1 David W. Sabo (2014) Number Systems Page 3 of 50

4 89 5 = 17 with remainder = 3 with remainder = 0 with remainder 3 Since we ve reached the result of 0, we stop. Now, the digits of this quantity expressed in base 5 is just the remainders, listed in order from bottom to top; that is = (You can use the methods of the previous section to confirm that is indeed equivalent to ). People refer to this procedure of dividing and recording remainders as the modular arithmetic method. It s fairly easy to see how it works. The base 5 equivalent of will have the general form d4d3d2d1d0, where the digits have values to ensure that d0 + d1(5) + d2(5 2 ) + d3(5 3 ) + d4(5 4 ) + = Now, divide both sides by 5: d0 d1 5 d2 5 d3 5 d Every term except the first in the numerator on the left-hand side of this equation is a multiple of 5 because of the way positional numbers are constructed. Thus, we can rewrite this equation as d d1 d25d35 d It is easy to see here that when 2233 is divided by 5, the result will be 446 with a remainder of 3 (since the fraction on the extreme right is the remainder divided by 5). The fraction on the right must equal the fraction on the left, since the rest of the expression on the left is a whole number. Thus, dividing by 5 and determining the remainder has isolated the value of the rightmost digit, d0, of the base 5 form of the number. If we repeat the process with the whole number parts of each side, we get 2 3 d1 d2 5 d3 5 d4 5 d d2 d3 5d When 446 is divided by 5, the remainder is 1, and this must be the value of d1, the second digit of the base 5 version of the number. Thus, each time we divide by 5, the remainder of the result is the next digit of the base 5 version of the number, working from right to left. Example 2: Convert to its equivalent in base 7. Apply the modular arithmetic method, with the divisor being 7 in this case. We get = with remainder = 8893 with remainder = 1270 with remainder = 181 with remainder = 25 with remainder = 3 with remainder 4 Page 4 of 50 Number Systems David W. Sabo (2014)

5 3 7 = 0 with remainder 3 Now, listing the remainders in reverse order (bottom to top in the list above) gives the desired answer: = We can verify this answer by checking that it s positional number system form does give the original decimal value: x x x x x x 7 6 = If you had a bit of trouble determining the remainders in these divisions by 7, note the following trick. If you use your calculator to do the first division in the list above, you get = The fractional part of this number (to the right of the decimal point) must be the remainder divided by 7 (see the patterns of the calculations on the previous pages). Thus remainder so remainder 7 x = That is, the remainder in this step of division is 4. Since remainders are always whole numbers, you need to use only a few digits of the fractional part to get a result from which the value of the remainder is obvious. Example 3: Convert to its equivalent in base 3. If we had a calculator that could do arithmetic with base 5 numbers, we could simply apply the modular arithmetic method straight away to get our answer. However, most calculators only do arithmetic in base 10. So, the strategy here will be to first convert into its decimal equivalent, and then we can use the modular arithmetic method to convert that number into its base 3 form. So = x x x 5 3 = Then = 189 with remainder = 63 with remainder = 21 with remainder = 7 with remainder = 2 with remainder = 0 with remainder 2 Thus, listing these remainders in order from top to bottom, we get = = David W. Sabo (2014) Number Systems Page 5 of 50

6 Example 4: Convert to its equivalent in base 9. This seems like more of the same. If you just plow ahead and do the arithmetic, you may overlook the fact that there is something very seriously wrong with the purported number If this is a base 5 number, then its digits must be selected from the set of just five basic symbols, conventionally {0, 1, 2, 3, and 4}. Hence a base 5 number cannot contain the symbols 7, 6, or even 5. The conversion request cannot be done, because is not a valid base 5 number, and so cannot be converted into a valid number in any other base. ii) conversion of the fractional part Conversion of the fractional part involves successive multiplication by the new base, retaining the whole number parts of the results. For instance, to convert to its equivalent in base 5, we would do the following x 5 = keep the 3, use the in the next stage x 5 = 4.44 keep the 4, use the 0.44 in the next stage 0.44 x 5 = 2.2 keep the 2, use the 0.2 in the next stage 0.2 x 5 = 1.0 keep the 1, there is nothing left to use in the next stage. The process is complete when the successive multiplication by the new base results in a whole number, since there is no fractional part of the result to take to the next stage. Then, the digits of the number in the new base is just the retained digits (listed following the words keep the above) in the order in which they appeared. Thus, = How can we tell that this is correct? The easiest confirmation is by applying the properties of a positional number system: = 3 x x x x 5-4 = = Example 5: Convert to base 7. We just follow the method explained above x 7 = keep the 0, use the in the next stage x 7 = keep the 6, use the in the next stage x 7 = keep the 4, use the in the next stage x 7 = keep the 4, use the in the next stage x 7 = keep the 3, use the in the next stage x 7 = keep the 5, use the in the next stage x 7 = keep the 1, use the in the next stage x 7 = keep the 5, use the in the next stage The fractional part at each step does not seem to be getting any closer to zero over the long run, and this process is showing no signs of terminating. In fact, in all likelihood, it will not terminate. This would mean that the decimal fraction, , does not have a terminating representation as a fraction in base 7. The best we could do in this case is just quote a certain number of digits as an approximation to the equivalent result: Page 6 of 50 Number Systems David W. Sabo (2014)

7 This example demonstrates that fractions which have a finite length representation in one base may not have a finite length representation in another base. In fact, the same thing would happen if you tried to convert the number 0.17 into its base 10 equivalent: 0.17 = 1 x 7-1 = endlessly repeating the six digit sequence So, when converting numbers from one base to another, a finite length whole number part always converts into a finite length whole number part, but the fractional parts may be finitely long in one system but not in another. This turns out to be the source of difficulties when floating point numbers are manipulated by computers (which work with numbers coded in a base 2 system, rather than a base 10 system). iii) conversion of decimal number which has both whole number and fractional parts In this case, as mentioned earlier, we do the whole number part using the modular arithmetic method, and the fractional part using the successive multiplication method, and put the two results together. Example 6: Convert into its equivalent in base 8. Round the fractional part to six digits if necessary. We need to do the whole number part and the fractional part separately. For the whole number part, use the modular arithmetic method: = 7684 with remainder = 960 with remainder = 120 with remainder = 15 with remainder = 1 with remainder = 0 with remainder 1 Therefore = For the fractional part, we use the successive multiplication method, multiplying by 8 each time. To round the fractional part to six digits if necessary, we must calculate until the process stops or until we actually have seven digits to the right of the radix point x 8 = keep the 4, use the in the next stage x 8 = keep the 1, use the in the next stage x 8 = keep the 4, use the in the next stage x 8 = keep the 2, use the in the next stage x 8 = keep the 2, use the in the next stage x 8 = keep the 3, use the in the next stage x 8 = keep the 3, use the in the next stage This is now seven steps with no sign of termination. So far, we have = Since the seventh digit is a 3, which is less than half of 8, we just drop the seventh digit and leave the sixth digit unchanged in order to round off to six places to the right of the radix point. Thus, our final answer here is David W. Sabo (2014) Number Systems Page 7 of 50

8 As a check, we note that: = x x x 8 5 = = verifying our result for the whole number part. For the fractional part, we get = You need to use a computer to get this many decimal places. Although this number is obviously not equal to , you do get , if you round it off to three decimal places, and so our result for the fractional part of the number is verified. The difference between this number and the desired value , is the result of rounding the fractional part of the octal representation to six digits. Names of Number Systems Positional number systems are often given names that reflect the value of their base. The most commonly used names are: base = 2 base = 3 base = 4 base = 5 base = 8 base = 10 base = 16 binary number system ternary number system quaternary number system quinary or quintal number system octal number system decimal number system hexadecimal number system The list above reflects common usage. (These names do not follow a precise grammatical pattern, and you can find internet sites promoting the use of names such as octonary instead of octal or banal instead of binary. ) The most important numbers systems involved in computer systems technology are the binary, octal, decimal and hexadecimal systems. Most of the remainder of this document will deal with just these four systems. Positional Number Systems With Bases Larger than 10 To write numbers in a positional system with base n, we need n symbols to use as digits. When n is 10 or less, it is conventional to simply use the first n symbols from the set that we use for decimal numbers. Thus, base 3 numbers use the symbol set {0, 1, 2}, and base 7 numbers would use the symbol set {0, 1, 2, 3, 4, 5, 6}, and so on. For bases, n, which are bigger than 10, obviously we need additional symbols. The convention is to adopt alphabetic characters, in alphabetic order. Thus, for example, to write numbers in hexadecimal form, we need 16 symbols. The set of symbols conventionally used is Page 8 of 50 Number Systems David W. Sabo (2014)

9 {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F} Thus, the symbol following 9 is A, the one following A is B, and so on. The ten usual numerical symbols along with the first six letters of the alphabet give us the sixteen symbols we need for expressing numbers in a base 16 number system. This means that A16 = 1010 B16 = 1110 C16 = 1210 D16 = 1310 E16 = 1410 F16 = 1510 If you needed to construct numbers with even larger bases, you would just continue to use further alphabetic characters: G, H, I, J, K, etc. Binary and Related Number Systems Having established some basic properties of positional number systems and methods for converting the representation of a number in one system into the equivalent representation in another number system, we can now turn our attention to numbers systems which are of greatest importance in computer technologies. Since digit electronic devices are capable of existing in one of two states, the most fundamental representations of data by computers makes use of the two symbols: {0, 1} a binary or base 2 system. Ultimately, all data representations stored electronically will have to be binary. However, binary representations are not particularly convenient for human beings, nor are humans particularly effective in working with binary numbers. This has led to the use of hexadecimal numbers as a convenient intermediate between the binary form used for digital data storage, and the decimal form we use in everyday communication. Historically, computer technologies have used the octal or base 8 system, and so we will look at that one briefly as well. Binary Numbers In a positional number system with base 2, the two symbols used for digits are conventionally 0 and 1. The positional values of the digits are powers of 2. The first few positions on either side of the radix point have the values: Although this pattern is what we would call a base 2 positional number system using the characteristics of a positional number system as they have been described in previous pages of this document, the application of binary numbers in digital computer design will require some modifications to this basic method. In what follows, we will refer to the above pattern as ordinary unsigned binary. The unsigned part just means we are not making any allowance for the possibility of negative numbers at this stage. The word ordinary distinguishes this type of binary number from other patterns that have been developed to allow for arithmetic operations involving negative numbers. These other ways of rewriting decimal values in binary form will have their own special names. David W. Sabo (2014) Number Systems Page 9 of 50

10 Conversions from binary to decimal are particularly easy in the sense that a digit can only be 0 (in which case there is no contribution to the value from that position in the number) or 1 (in which case, the contribution is just the position value). Thus = = Of course, you can also use the modular arithmetic and/or successive multiplication method for conversion of decimal numbers to their binary equivalent. Example 7: Convert to ordinary unsigned binary form. We proceed here much like we did in the previous example 6, except that now the divisor and multiplier we use is 2 instead of 8. So, using modular arithmetic to convert the whole number part, we get So, we conclude that = 83 with remainder = 41 with remainder = 20 with remainder = 10 with remainder = 5 with remainder = 2 with remainder = 1 with remainder = 0 with remainder = Convert the fractional part by successive multiplications by 2: x 2 = keep the 1, use the in the next stage x 2 = 1.25 keep the 1, use the 0.25 in the next stage x 2 = 0.5 keep the 0, use the 0.5 in the next stage 0.5 x 2 = 1.0 keep the 1 The residual fractional part is now zero, and so we re done with the conversion. The computations mean that = , listing the kept digits in the order in which they arose. Thus, the final answer in this case is: = For decimal numbers with more that two or three digits, there is a more efficient approach with far less chance of error in hand calculations that makes use of hexadecimal numbers. Details are given ahead. Page 10 of 50 Number Systems David W. Sabo (2014)

11 Hexadecimal Numbers Hexadecimal numbers use a positional number system with a base of 16. The sixteen basic symbols forming the digits of hexadecimal numbers are: hexadecimal digit: A B C D E F decimal value: Although people have been known to use the lower case characters {a, b, c, d, e, f} for the last six symbols in this set, it is strongly recommended that you always write these in upper case form as shown in the list above. Conversion from hexadecimal to decimal form is done using the positional values of digits of the number system. Each position has a value different by a factor of 16 in this case compared to the adjacent digits. Example 8: Convert E57AB.C916 to its equivalent decimal number. The problem here is solved by simply applying the positional values of each hexadecimal digit. We ll do this in two steps: writing out the positional value expression as a first step, and then translating the hexadecimal digits as necessary in the second step: 1 1 E57 ABC. 9 B A E16 C Although you can t tell for sure using a calculator that displays just eight digits to the right of the decimal point, the conversion above is exact to eight decimal places. There has been no rounding. (In fact, you can demonstrate that a binary number with n digits to the right of the radix point will convert exactly to a decimal number with no more than n digits to the right of the decimal point. As you ll see below, this means that a hexadecimal number with n digits to the right of the radix point will convert exactly to a decimal number with no more than 4n digits to the right of the decimal point. The unavoidable need for truncation in conversion of fractional parts between decimal, binary, octal, and hexadecimal occurs in conversion of decimal numbers to one of these other three forms, and not when the conversion is the other way around.) Conversion of decimal numbers to hexadecimal form is done using the modular arithmetic/successive multiplication method as usual. Example 9: Convert to its equivalent hexadecimal number. For the whole number part we have = 2035 with remainder 13 = D = 127 with remainder = 7 with remainder 15 = F16 David W. Sabo (2014) Number Systems Page 11 of 50

12 7 16 = 0 with remainder 7 So, = 7F3D16. For the fractional part, we have x 16 = keep the 11 = B16, use the in the next stage x 16 = 6.00 keep the 6 Thus = 0.B616 Thus, finally, = 7F3D.B616. as the required answer. Octal Numbers Octal numbers use a positional number system with a base of 8. The eight basic symbols forming the digits of octal numbers are {0, 1, 2, 3, 4, 5, 6, 7}, each having the same numerical value as the identical decimal digit. Octal numbers are not used as extensively in computer applications any more as they once were, but you will encounter them occasionally in certain areas of application. To convert from octal to decimal forms of a number, just use the positional values of each digit in the usual way. To convert from decimal to octal, use the modular arithmetic and successive multiplication methods in the usual way. Example 10: Convert into its decimal equivalent = Example 11: Convert into its octal equivalent. First, the whole number part: So, = 3441 with remainder = 430 with remainder = 53 with remainder = 6 with remainder = 0 with remainder = Page 12 of 50 Number Systems David W. Sabo (2014)

13 For the fractional part, So, x 8= keep the 3, use the in the next stage x 8= keep the 2, use the in the next stage x 8 = 7.00 keep the = So, the final answer is = Useful Relationships Between Binary, Octal, and Hexadecimal Numbers A binary digit can exist in one of two forms: 0 or 1. This means that two binary digits can exhibit four different patterns or states: either of two possible values in the second digit can be paired up with either of two possible values in the first digit, so that 2 x 2 = 4 different patterns of digits are possible. Extending this argument, three binary digits are capable of exhibiting a total of 8 different patterns each of the two possible values of the third digit can be paired up with any of the four possible patterns that the first two digits can have. In fact, if you think about it, the number of possible patterns of three zeros and ones is exactly equal to the number of rows in a truth table for an expression involving three logical variables. If we tabulate these 8 patterns, and then interpret them as binary numbers, we get the following decimal equivalents: 3-digit binary number decimal equivalent With a three-digit binary number, we can represent the range of values that a single digit of an octal number can display. This means that each digit in an octal number is equivalent to three digits in the corresponding binary form of that number, and each triplet of digits in a binary number (counting from the radix point, either left or right this is important!) is equivalent to one digit in the corresponding octal form of that number. This gives us a quick way to convert from octal to binary forms of numbers (or the reverse). Before exploiting this property of octal and binary numbers, we take a look at an even more useful similar connection between binary and hexadecimal numbers. There are sixteen distinct symbols in the hexadecimal number system, which is exactly the number of distinct patterns of zeros and ones in a group of four binary digits. (Again, think of the number of rows in a truth table for an expression involving four logical variables.) It is easy to confirm the following equivalences: David W. Sabo (2014) Number Systems Page 13 of 50

14 4-digit binary number hexadecimal equivalent A B C D E F The small numbers along the left are just decimal numbers labeling the rows (as we did earlier in the course in labeling rows of 4-variable truth tables). This table demonstrates that each digit in the hexadecimal representation of a number corresponds to four digits in the binary representation of that number, and each group of four digits in the binary representation (counting fours starting at the radix point, either right or left) corresponds to one digit in the hexadecimal representation of that number. This provides a fast way to convert between binary and hexadecimal representations of a number. Example 12: Determine the octal and hexadecimal representation of the binary number Before this section, the strategy we would have to use to accomplish these tasks would be to first convert the binary number to decimal form (a fairly difficult task because of the number of digits in this binary number) and then we could use the modular arithmetic and successive multiplication methods to convert that decimal value to octal and to hexadecimal. In view of the special relationships just described between binary, octal and hexadecimal representations, this task can be done by grouping the binary digits appropriately and translating the groups directly into either octal or hexadecimal digits. To convert to the octal representation, we need to recopy the binary number very carefully and insert separators between each group of 3 digits, counting left and right from the radix point. Then below each group of three, we jot the octal digit equivalent: 1/011/00 0/110/101/0 10/010/100110/100/11 0/101/ You can get these octal digits from the table just above, or just use the fact that in the three digit groupings, the place values are 1, 2, and 4 reading from right to left (eg = = 610 = 68). Notice that the values of triplets at the extreme right and extreme left are worked out as if zeros are added to the outside of them to complete the triplet. Thus, the single 1 on the left is evaluated as if it were a triplet 001, and the single 1 on the extreme right is evaluated as if it were the triplet 100. You know that appending zeros to either the right or left of a decimal number does not change its value, and the same is true for numbers in any base. Page 14 of 50 Number Systems David W. Sabo (2014)

15 So, we can now write: = From the number of digits involved in both numbers, you can easily see that this method is superior to first converting the binary number to decimal, and then the decimal number to octal. A similar procedure is used to obtain the hexadecimal equivalent, only now the original binary number is chopped up into groups of four digits counting from the radix point: 10/1100/ 0110/1010/ 1001/0100/.1101/0011/ 0101/ C 6 A 9 4 D In the first row of translations, we ve written down the decimal equivalent of each group of four digits. Then in the second row, we ve written down the corresponding hexadecimal digits. In this case, the partial groups of four digits on either end are completed by adding zeros to the outside as required. Thus the group 10 on the left end is expanded to 0010, equivalent to 216, and the group 10 on the right is expanded to 1000, equivalent to 816. We always add the extra zeros to the outside of the group. So, we can state = 2C6A94.D35816 There are two important implications of this special relationship that exists between binary, octal and hexadecimal numbers. (i) Digital computers work with binary numbers only, but human beings are very poorly equipped to work reliably with binary numbers because we have trouble parsing long lists of only two different symbols reliably. On the other hand, humans can deal with shorter lists of a larger number of possible symbols very well. (After all, even the dullest of us can handle words of one syllable strings of three or four symbols selected from a set of 26 different symbols.) So, in practice, because there is this simple equivalence between groups of binary digits and either octal or hexadecimal digits, it is common for low-level computer data to be displayed to humans in hexadecimal (and sometimes octal) form. You will see this when we talk about the ASCII code for storing text in computer memory. (ii) We can also use this relationship between binary and hexadecimal numbers to streamline conversions between decimal and binary numbers. Particularly when the decimal number involved has more than two or three digits, or the binary number has more than eight or ten digits, we strongly recommend that conversions between decimal and binary be done using a hexadecimal intermediate. If you do these conversions directly, the number of steps in the calculation will be so great that it is almost impossible to avoid making an error. Example 13: Convert to ordinary unsigned binary form. As usual, the whole number and fractional parts have to be converted using separate methods. If we do the modular arithmetic for the whole number part and the successive multiplication for the fractional part using the base 2, the steps look like: = with remainder x 2 = = with remainder x 2 = David W. Sabo (2014) Number Systems Page 15 of 50

16 = with remainder x 2 = = 5317 with remainder x 2 = = 2658 with remainder x 2 = = 1329 with remainder x 2 = = 664 with remainder = 332 with remainder = 166 with remainder = 83 with remainder = 41 with remainder = 20 with remainder = 10 with remainder = 5 with remainder = 2 with remainder = 1 with remainder = 0 with remainder 1 So, writing the remainders from the modular arithmetic calculation in reverse order, and the whole number parts from the successive multiplication calculation in the order in which they appeared, we get that = If we had used base 16 as an intermediate in this conversion, the calculations would have gone as follows: = 5317 with remainder x 16 = = 332 with remainder x 16 = = 20 with remainder = 1 with remainder = 0 with remainder 1 This means that = 14C59.2C16 using the facts that = = 00012, 416 = 01002, C16 = 11002, 516 = 01012, 916 = 10012, and 216 = We could even have done this conversion through an octal intermediate: = with remainder x 8 = = 1329 with remainder x 8 = = 166 with remainder = 20 with remainder = 2 with remainder = 0 with remainder 2 Then, using the fact that 18 = 0012, 28 = 0102, 38 = 0112, 48 = 1002, and 68 = 1102, Page 16 of 50 Number Systems David W. Sabo (2014)

17 we get = = which is identical (except for leading and trailing zeros) to the previous two results. The point of this example is that the large amount of easily messed up detail present in the eventual binary number is packaged into more humanly manageable chunks by using either a hexadecimal or octal intermediate in the conversion from decimal to binary forms. Addition of Unsigned Binary Numbers In order to understand some of the issues we will address shortly in looking at methods for representing numbers in digital computers, you will need to be able carry out the addition of two unsigned binary numbers by hand. The process is a simple adaptation of the method we use routinely for adding decimal numbers. We would display the work involved in adding to something like: 11 carries We work from right to left here: first, add 4 to 2, write the result 6 in the rightmost position then, add 7 to 8. The result is 15, which is too big to write in one space. The 5 is written in the second column, and the 1 is carried over to the third column, written just above the 2 there. now, add the digits in the third column from the right: = 10. The digit 0 is written down and the 1 is carried to the next column. now add the digits in the fourth column from the right: = 7. This single digit is written down, there is nothing to carry to the next column. finally, add the digits in the last column on the left: = 14. Both are written down since there are no more columns to carry into. The end result is the sum Whenever the sum of digits in a column exceeds 9, the excess (which is just the left-most digit of that column sum) must be carried to the next column leftwards. The same pattern works with unsigned binary numbers. In this case, there are really only five distinct situations we need to take into account: Whenever the result of the addition is two digits, the leftmost digit is carried to the next column leftwards (and should be written down explicitly as carries in this course). Example 14: Add the following two unsigned binary numbers, showing all carries. David W. Sabo (2014) Number Systems Page 17 of 50

18 Confirm your end result by converting all numbers to decimal form and performing the addition. We ll display the addition in the traditional form as a whole, and explain each column in notes to follow: carries result column number The bottom row of italicized digits number the columns from right to left. The work done here is as follows: column 1 has = 10 (binary). Write down the 0 and carry the 1 to the next column. column 2 now has = 112. Write down the 1 on the right, and carry to 1 on the left to the next column. column 3 now has = 102. Write down the 0 and carry the 1 to the next column. column 4 now has = 12. Write down the 1. There is nothing to carry to the next column. column 5 has = 12. Write down the 1. There is nothing to carry to the next column. column 6 has = 102. Write down the 0 and carry the 1 to the next column. column 7 now has = 102. Write down the 0 and carry the 1 to the next column. column 8 now has = 102. Write down the 0 and carry the 1 to the next column. column 9 now has = 102. Write down the 0 and carry the 1 to the next column. column 10 now has = 102. Write down the 0 and carry the 1 to the next column. column 11 now has = 102. Write down the 0 and carry the 1 to the next column. column 12 now has = 112. Since this is the leftmost position of the numbers being added, write down both digits, completing the result. We need to confirm the result using decimal arithmetic. Converting the two original numbers to decimal via a hexadecimal intermediate gives: = B6316 = 11 x x = = CB716 = 12 x x = and for the result of the binary addition = 181A16 = 1 x x x = But, using a calculator, = , and so the binary result is confirmed. Page 18 of 50 Number Systems David W. Sabo (2014)

19 Methods of Representing Numbers in Digital Computing With this theory of number systems behind us, it is now time to address some practical issues as far as devising binary number representations for use in digital computing. Although it is easy to generalize conventions that we use for decimal numbers into written ordinary binary forms, these do not lend themselves to straightforward computer implementation. There are two issues we need to take account of: (i) the distinction between whole numbers (integers) and floating point numbers You have already seen one bothersome distinction between the results of converting whole numbers into binary form and of converting numbers with fractional parts into binary form namely that when a fractional part is present, it may not be possible (in fact, usually is not possible) to express the binary equivalent of the fractional part in a finite number of digits. This means that techniques for coding floating point numbers in a binary form will have to allow for truncation and hence the usual occurrence of inexact conversions. A second problem is that while whole numbers that arise in calculations are often the result of enumeration of actual objects and thus, do not usually have extremely large values, it is not uncommon to have floating point values which range over many, many decimal digit positions. For instance, typical time intervals for a microprocessor operation are of the order of nanoseconds, or seconds. On the other hand, the number of such operations a computer performs in one second is of the order of, say, Since there are approximately seconds in one year, then a typical pc performs about cycles per year. Any capability of manipulating floating point arithmetic by computer will have to be able to cope with values having such widely different orders of magnitude. (ii) distinguishing between positive and negative numbers In ordinary written decimal arithmetic, we distinguish between positive and negative values by appending a minus sign in front of numbers measuring negative values. However, in computer application, we only have two symbols available, 0 and 1, both of which are used for coding the actual numerical value. If we introduced a minus sign, as a distinct symbol, we would suddenly be working with a three-symbol system, and conventional computing equipment cannot handle that. So, we must come up with other ways of distinguishing between positive and negative values. The way we address both of these issues has implications for what will be required in developing algorithms for carrying out arithmetic operations on the binary numbers that result. When we work with decimal numbers on paper, whether whole numbers or numbers with fractional parts, it number of digits in the number is not usually an issue we have to think much about. If we encounter a larger number, we just use up more space on the paper. However, when numbers are stored in a binary form in computer memory, the number of binary digits, or bits, available for each number has to be fixed in advance. The computer hardware is not designed to be able to store different numbers with different numbers of binary digits, because the binary digits are being electronically coded into an array of actual physical memory devices. The principles we will describe below do not require a specific number of bits to be available for each number, but to illustrate them, we will have to assume a specific number of bits. For most of the examples to follow, we will work with 8-bit binary representations for whole numbers. In computer technology, 8 bits of memory is called 1 byte, which is the smallest directly accessible amount of memory on most computers. At the end we ll look very briefly at how easily the patterns established show up in larger numbers, such as 16-bit or 32-bit numbers. (You could also illustrate these binary representations using, say, 5 bit binary numbers or 11-bit binary numbers, or whatever. We ve chosen to use 8-bit binary representations as illustrations here because they are simple enough to prevent hand calculations from becoming really tedious; they are extensive enough to give some variety in the results you see; and there are quite a number of practical situations in which 8-bit binary numbers are used.) David W. Sabo (2014) Number Systems Page 19 of 50

20 We start by describing five commonly-used approaches to coding whole numbers, all but one of which takes sign into account. Computer Representation of Integers (i) (modulo 2 n ) unsigned binary numbers This format is what we ve called ordinary unsigned binary form so far. Numbers are coded in binary form using n bits. When n = 8, the range of values that can be represented is up to = = = a total of 2 8 = 256 different values. (Obviously, in general, the range of decimal numbers represented in n-bit unsigned binary form is 010 to 2 n 1). Conversion from decimal to unsigned binary form is done using the modular arithmetic method, and conversion from unsigned binary to decimal using the positional values of the binary digits, as has been demonstrated earlier in this document. Example 15: Write the following decimal numbers in 8-bit unsigned binary form: 46, 182, -92, and 465. For 4610, we ll use the modular arithmetic method directly, since 46 is quite a small number: 46 2 = 23 with remainder = 11 with remainder = 5 with remainder = 2 with remainder = 1 with remainder = 0 with remainder 1 Listing the remainders in reverse order, we get 4610 = This result is just 6 digits long, so to get the 8-bit form of 4610, we need to pad on the left with two zero digits. Thus, 4610 = in 8-bit unsigned binary form. Since is a somewhat larger value, we ll carry out the conversion to binary form through a hexadecimal intermediate: Thus, = 11 with remainder = 0 with remainder 11 = B = B616 = (where we have used the fact that B16 = 1110 = and 616 = 610 = 01102). Page 20 of 50 Number Systems David W. Sabo (2014)

21 For the remaining two values, we are out of luck is a negative number, but the unsigned binary format is incapable of storing sign information. Hence, cannot be coded in unsigned binary form. Since the largest value expressible in an 8-bit unsigned binary number is = 255, we are unable to code in 8-bit unsigned binary form. (ii) signed magnitude form A very obvious way to include sign information in the binary representation of a number is to use one of the available digits to distinguish between positive an negative values. In the signed magnitude form, the left-most digit is used for this purpose: 0 _ indicates a positive value 1 _ indicates a negative value. The remaining n 1 digits are then used to store the numerical value without sign (the magnitude) in unsigned binary form. Thus, for 8-bit signed magnitude numbers, the largest magnitude we can store is the unsigned binary value , a string of seven 1 s, which is equivalent to So, in 8-bit signed magnitude form, we can store any integer between (= ) and (= ). An oddity of the signed magnitude form is that there are two distinct representations of the number 010: and = = -010 The signed magnitude form is easy to set up, but it turns out to be quite awkward to formulate basic arithmetic operations using the signed magnitude form. The basic idea of using a single bit to distinguish between positive and negative numbers is used in the standard methods of coding floating point numbers. Example 16: Express the decimal values 87, -63, 149, and -738 in 8-bit signed magnitude form. The technique is to start by converting the numerical part to binary form ignoring the sign. Then assemble the numerical part (padded with zeros on the left to give exactly seven digits) with the single bit at the left which indicates the sign. So, using the methods already illustrated several times, we can determine that 8710 = which is exactly 7 digits. So, since is positive, the result we need is 8-bit signed magnitude form For the second number, we have 6310 = , which is just six digits long. We need a seven digit numerical part, so we must append a zero to the left of these six digits. Finally, since is negative, the sign bit at the extreme left of the 8-bit representation is set to 1: David W. Sabo (2014) Number Systems Page 21 of 50