Number systems Fall, 2007


 Augustine Ramsey
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1 Number Systems What does 1234 mean? Before taking one step into computing, you need to understand the simplest mathematical idea, number systems. You will need to be comfortable with both binary (base 2) and hexadecimal (base 16) numbers. The former are essential to the internal working of the computer; the latter are encountered in much web work and graphics, and turn out to be a convenient way to package large binary numbers. Think of numbers in two ways. A number has an innate value, its ordinality, and it also has an expression such as The ordinal value of a number is something real, in the sense that 1234 is 1 more than 1233 and 1 less that 1235 and it can be mapped to a pile of pebbles or a herd of sheep of the same size. Think of the pile of pebbles. No matter how we choose to write the number 1234, or what kinds of things we are counting, it is equivalent to a pile of 1234 pebbles. The expression 1234, however, is not a fundamental property of a number since it depends upon which base we are using. Number systems are usually built on using the idea of "place", i.e. in decimal, for the number 6305: Place Digits Column Values In Decimal Sum Digits x Values Each digit is in a column so the digit tells you how many units of that column you have. In this case, we have 6 thousands, 3 hundreds, 0 tens, and 5 ones. The third row shows the values of the columns written as powers of the base, 10. If we had the same digits but in base 8, the implicit columns would be: Place Digits Column Values In Decimal Sum Digits x Values (in decimal) Notice that we are being a bit sneaky here. The 8, 64, 512 are written in decimal just so we have a real sense of them. Those values in base 8 would be 10, 100, The allowed digits in base 1
2 8 are 0, 1, 2, 3, 4, 5, 6, 7, that is, when you count in base 8, the count rolls over to 10 after 7. This shows that = If the base were sixteen, the number would be: Place Digits Column Values In Decimal Sum Digits x Values (in decimal) Again, the columns in hexadecimal would be the 1, 10, 100, 1000 columns if they were in base 16. The value of 6305 in base 16 would equal in base 10, or, more compactly as = The number 6305 would not be allowed in bases 2, 3, 4, 5, or 6 because no "6" digit is allowed in any of those bases. Moving between number systems Let's consider changing a number from base 10, decimal, to some other base, say 5. There are two ways to go from base 10 to base 5. For example, to convert 613 to base 5: Examine the base 10 number and starting from the highest power of 5, see what powers of 5 are in the number 613. The powers of 5 are 1, 5, 25, 125, 625 (5^0, 5^1, 5^2, 5^3, 5^4). 625 is too big, so check 125 and see that it will go 4 times, leaving a remainder of = 113. Next, 25 divides 113 just 4 times leaving 13. Finally, 5 goes into 13 twice and leaves 3, so the digits are, from the left: (125) (25) (5) (1) = = 613 So, in base 5, the decimal number 613 would be expressed as Another approach is to work from the least significant digit to the most significant (that is, working from right to left). To use this method, divide the decimal number by the base (5) to repeatedly find a quotient and remainder: number operation quotient remainder
3 As you see, the digits are extracted in reverse order, from right to left. If you think about this method, it suggests how number systems were invented in the first place. Suppose that in the distant past, you were lucky enough to own a large herd of sheep, say 613 (base 10) for our example. You might decide to count them by sorting them into groups based on the number of digits on one hand (5). So, you divide the sheep into groups of 5 and find you have 3 left over. There are a lot of 5groups (122, to be precise), so you gather them into groups of five 5groups and resulting in groups with 2 5groups left over. Well, that is still a lot of groups of 25 sheep, so you group them once more into larger groups with 5 groups of 25 in each. Finally you see that there are just 4 of these large groups ( 125groups ) with 4 groups of 25 left over, so in base 5 you have 4423 sheep: = (125) (25) (5) (1) = If you have a sweet tooth, consider this alternative: Suppose you have a small business in which you produce and package mint candies to sell at the local supermarket. Your mints are extremely popular in town, so each week you package and deliver several large cartons of mints. This week as it happens, you have made a total of 613 mints to package  that's base 10. You package them into 5packs, shrink wrap those into packages containing 5 of these 5packs, and then you make up boxes each containing 5 shrink wraps. Since you are starting with 613 candies, you obviously will have some "leftovers." We show this in Figure
4 Figure 11 Counting candies in base 5 So what does our final count actually yield? The totality of the mints that you have made this afternoon are pictured at the bottom (the 4 boxes) and the right side of the diagram, the leftovers. This total number is: 4 (125mint boxes) + 4 (25mint leftover shrinkwraps) + 2 (5mint leftover packs) + 3 leftover mints. Allinall there are mint candies; that is, candies in all. The real temptation that must be avoided is to just eat up all those leftovers!! 4
5 To be certain the idea is clear, convert to base 3. Figure 12 shows how we can think about this using 32 toothpicks grouped along the top of the figure. Group them in 3 and notice 2 left over Combine the sets of 3 in groups of 3 (9) with 1 left over Combine the sets of 9 into groups of 3 (27) with 0 left over And finally collect groups of 3 units of 27. There are 0 groups of 81, just that 1 group of 27 left over Left over So, = The operations were quotient remainder 32 \ 3 = mod 3 = 2 10 \ 3 = 3 10 mod 3 = 1 3 \ 3 = 1 3 mod 3 = 0 1 \ 3 = 0 1 mod 3 = 1 Figure Conversion from base 10 to base 5 using 32 toothpicks 5
6 Bits, Bytes and Nibbles Binary The binary number system, base 2, is just the most extreme case with only two digits, 0 and 1. so that [128] [64] [32] [16] [8] [4] [2] [1] = = = = 255 Check out the Visual Basic program available on line an try counting in binary to watch the digits roll over. Adding binary numbers Here is a simple little exercise, to do the following additions in decimal: Now, write the answers in binary: Try these additions in decimal: Now, write the answers in binary: At this point, you know everything you need to know to add two long binary numbers like:
7 The process is quite simple. If the two digits in a column are 0 and 0, the sum is 0 in the column; If the digits are 0 and 1, the answer is 1 in the column. What happens if two digits are the column are 1? At this point, we place a 0 in the answer but we also have a 1 to carry over to the next column! That means that there are actually three numbers to add in the next column, the two digits already there and the carry digit. Here is an example: Carry 1 Carry 1 Carry 0 In decimal, the first number is 11, the second is 9, the sum is 20 or in binary Systematically do the addition for the two longer numbers above. Check your answer by converting the two numbers to decimal, adding them, and converting the answer back to binary. Bytes and character codes Binary numbers are typically used by computers and not by humans. The size of the binary numbers that get used are determined by hardware considerations, i.e. how many digits are there in each word, or memory element, in a computer. That s a design question. In early computers, word size would be determined by cost and the market the machine was designed for. Some computers designed primarily for business use might have 32bit words; others aimed at the scientific and engineering market might contain 40 or 60bit words. IBM made a decision in the mid1960s that reshaped the design of memory. The company decided to build all of its machines around an 8bit standard with this 8bit element called a byte. For business calculations, 4 bytes (32 bits) could be treated as a unit or for scientific purposes one might use 8 bytes in a unit (64 bits). The byte had the further advantage that it was big enough to hold a single character code. A character code is simply this: if we have an electronic device that can store bits, 0 or 1, the only way to store a character is to assign a code to each character and store the code. For example the ASCII (or American Standard Code for Information Interchange) code for the letter A is 65 or in 8bit binary. The character code for a is 97 or These codes were not invented for computers but for teletype machines that replaced the telegraph for message transmission. Now here s a problem. With 8bits in a byte, there are only 256 possible codes, from to This works if the only language you care about is English, with maybe some European letters with accents, but what about the rest of the world? The answer was to move to a 16bit, or 2byte, code called Unicode to store the world s alphabets in. 16bits allows up to 65,536 characters. Another advantage of moving to the byte notation is just that it makes it easier to look at binary numbers if they are divided into groups of 8, like: rather than
8 Unary PS: there is actual one simpler number system, unary, that goes like this: 1 = 1 2 = 11 3 = = 1111 This gets old fast and we will not pursue it. Hexadecimal In a sense, hexadecimal is the other extreme we will find useful. In hexadecimal, the base is 16, so the digits are oops, we have run out of characters Well, hexadecimal first came into popular use among computer programmers in the days when the choice of characters that a computer could generate was extremely limited, i.e. the decimal digits, a few punctuation marks, and uppercase roman letters. So the characters chosen to represent [10] [11] [12] [13] [14] [15] were the letters A B C D E F, respectively, and our digits now are A B C D E F Please note that the letters A, B, C, D, E, F here are digits, not letters of the alphabet. Some calculations that we'd be interested in: [ ] [ ] [ ] [65536] [4096] [256] [16] [1] Who would want to use numbers in base 16? Anyone, like a programmer, who needs to examine a computer s memory to find out, for example, why a computer program crashed. To examine the memory (in the old days) meant to print out the contents of the computer s memory. As you might imagine, printing the memory in binary would entail a vast amount of paper and be almost impossible for anyone to grasp. On the other hand, if the memory were printed in hexadecimal, it might just be possible to understand what one is seeing. Why base 16 and not base 10. The answer is that there is an easy correspondence between binary and hexadecimal numbers. Look at what happens as one counts in binary from 0 to 15 (table below) and then goes one number further. The sequence of binary numbers 0000 to 1111 equals the hexadecimal digits 0 to F. When the binary number 1111 rolls over to 10000, the hexadecimal number F rolls over to 10. 8
9 binary hexadecimal decimal A B C D E F Figure comparison of small binary, hexadecimal, and decimal numbers. Shading at the bottom of the table shows where additional characters are required. To go from binary to hexadecimal, you can use the fact that 4 binary digits, 4 bits, can be converted immediately to hexadecimal, e.g., B D F 7 3 (Note that we begin the grouping into 4bits from the right side of the binary number) and viceversa 4 A 3 F (note: no initial "0" needed here) 9
10 The utility of being able to read base 16 is that many computer parameters are given in hexadecimal and you need to be comfortable with it. It is often used to describe color intensities in computer graphics and that is directly related to how computer memory is arranged. Early computers differed greatly in how memory was partitioned. Memory was described in terms of words of a certain number of bits with each word having its own address in memory. The most powerful computers for scientific calculation might use 60bit words, a more general purpose computer could have 36bit words, and smaller minicomputers could have words of 8 or 12 bits. IBM, with its introduction of the big System 360 computers in the late 1960 s moved to a different model where memory was described in bytes, or 8bit units, that could be used in 32 or 64bit words. That turned out to be a useful unit, as early microprocessors were typically based on 8bit words also. Gradually, everyone moved to describing computer storage in units of bytes and even the most inexperienced PC user feels at ease describing their computer as having 256 Megabytes (MB) of memory and 40 Gigabytes (GB) of disk storage (one KB (kilobyte) is 1024 bytes, one MB is 1024 kilobytes, one GB is 1024 MB). So, to review, let s look at the byte or 8bit number. 2^7 2^6 2^5 2^4 2^3 2^2 2^1 2^0 [128] [64] [32] [16] [8] [4] [2] [1] A one byte number can have values from to _ _ In decimal and hexadecimal those values are = 0 10 = to = = FF 16. Now, remember: there are 10 kinds of people in this world. Those who know binary, and those who don t. The Keyboard One last look at the funny little dichotomy involved in dealing with a computer. By now you know that everything is stored in a computer as numbers. Interestingly enough all of our communication with the computer through the keyboard is by way of characters. Well, sort of. Here is what happens. If you type the number 73 in responses to a question from the computer, what you are typing are the two characters 7 and 3, possibly followed by a third character, a return (Enter) character. When you press a key, say for the character 7, the keyboard sends a binary code, called a scan code, to the computer and signals an interrupt. The computer accepts the scan code and uses it to look in a table to find the American Standard Code for Information Interchange (ASCII) code for the character 7. This turns out to be a binary number (or or ). To convert 10
11 this into the number 7, a computer program that is trying to read the number 7 has to subtract (= ) from the binary code (subtract from to get = 7 10 ). After the 7 has been read and stored as a binary number ( ), the second character (3) is read and converted to a number, the 7 is multiplied by 10 to shift it to the 10s column and the 3 is added to produce, finally, the number 73. Easy, no? Fortunately, the code necessary to read in a multidigit number is built into most operating systems and programming languages and doesn t have to be reconstructed by every programmer. It should make clear to you, however, why computers are often very fussy about whether you have typed in a string of digits or a string of alphabetic characters. 11
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