Number Systems II MA1S1. Tristan McLoughlin. November 30, 2013

Transcription

1 Number Systems II MA1S1 Tristan McLoughlin November 30, numeral system

2 Simple computer programs that use integers will be limited to the range of integers from 2 31 up to (which is the number that has 31 1 s in binary). However, it is possible to write programs that will deal with a larger range of integers. You can arrange your program to use more than 32 bits to store each integer, for example to use several rows of 32 bits. However, the program will then generally have to be able to implement its own carrying rules and so forth for addition and subtraction of these bigger integers. So you will not simply be able to use the ordinary plus and times that you can use with regular integers.

3 Why octal and Hex? We can now explain why computer people are fond of base 16 or hex. Octal looks easier to read (no need to worry about the new digits a for ten, etc) but in computers we are frequently considering 32 bits at a time. Using the 3 binary for one octal rule this allows us to write out the 32 bits quickly, but it takes us eleven octal digits. The messy part is that we really don t quite use the eleventh octal digit fully. It can be at most (11) 2 = 3. With hex, we have a 4 binary digits for one hex rule and 32 binary digits or bits exactly uses up 8 hex digits.

4 ASCII Computers use binary for everything, not just numbers. For example, text is encoded in binary by numbering all the letters and symbols. The most well used method for doing this is called ASCII (an acronym that stands for American Standard Code for Information Interchange). and it uses 7 binary digits or bits for each letter.

5 ASCII On a UNIX system, you can find out what the ASCII code is by typing the command: man ascii at the command line prompt. Another place to find this information is at Essentially what you will find is...

6 The first 32 characters in the ASCII-table are unprintable control codes and are used to control peripherals such as printers. DEC OCT HEX BIN Symbol HTML Number HTML Name Description NUL � Null char SOH  Start of Heading STX  Start of Text ETX  End of Text EOT  End of Transmission ENQ � Enquiry ACK  Acknowledgment BEL  Bell BS  Back Space HT Horizontal Tab A LF Line Feed B VT  Vertical Tab C FF  Form Feed D CR Carriage Return E SO  Shift Out / X-On F SI  Shift In / X-Off DLE  Data Line Escape DC1  Device Control 1 (oft. XON) DC2  Device Control DC3  Device Control 3 (oft. XOFF) DC4  Device Control NAK  Negative Acknowledgement SYN  Synchronous Idle ETB  End of Transmit Block CAN  Cancel EM  End of Medium Page 1 of 7

7 There first 32 items are codes for invisible or non-printible objects that are useful in organising messages of text etc.

8 Safari Power Saver Click to Start Flash Plug-in ASCII control characters (character code 0-31) The first 32 characters in the ASCII-table are unprintable control codes and are used to control peripherals such as printers. ASCII Code - The extended ASCII table DEC OCT HEX BIN Symbol HTML Number HTML Name Description NUL EM �  End Null of char Medium A SOH SUB   Substitute Start of Heading B STX ESC   Escape Start of Text C ETX FS   File End Separator of Text D EOT GS   Group End of Separator Transmission E ENQ RS �  Record EnquirySeparator F ACK US   Unit Acknowledgment Separator BEL  Bell Safari Power Saver BS  Back Space Click to Start Flash Plug-in HT Horizontal Tab A LF Line Feed B VT  Vertical Tab C FF  Form Feed ASCII printable characters (character code ) Codes are 0D common for all the different CR variations of the ASCII table, they are Carriage called Return printable characters, represent letters, 0E digits, punctuation marks, SO and a few  miscellaneous symbols. You Shift will Out find / almost X-On every character your keyboard. Character 127 represents the command DEL F SI  Shift In / X-Off DEC 16 OCT 020 HEX 10 BIN Symbol DLE HTML Number  HTML Name Description Data Line Escape DC1  Space Device Control 1 (oft. XON) 28/1

9 The remaining, no , are printable objects like punctuation...

10 printers. your keyboard. Character 127 represents the command DEL. DEC DEC OCT OCT HEX HEX BIN BIN Symbol Symbol HTML HTML Number HTML HTML Name Name Description NUL � Null Space char SOH! ! Start Exclamation of Heading mark STX "  " " Start Double of Text quotes (or speech marks) ETX #   End Number of Text EOT $  $ End Dollar of Transmission ENQ % � % Enquiry Procenttecken ACK &  & & Acknowledgment Ampersand BEL '  ' Bell Single quote BS (  ( Back Open Space parenthesis (or open bracket) HT ) ) Horizontal Close parenthesis Tab (or close bracket) A2A LF * * Line Asterisk Feed B2B VT +  + Vertical Plus Tab C2C FF, , Form Comma Feed D2D CR -  Carriage Hyphen Return E2E SO. . Shift Period, Out dot / X-On or full stop F2F SI /  / Shift Slash In or / X-Off divide DLE 0  0 Data ZeroLine Escape DC1 1  1 Device One Control 1 (oft. XON) DC2 2  � Device Two Control  Three DC3 4   Device Control 3 (oft. XOFF) Four DC4   Device Control 4 Five NAK 6   Negative Acknowledgement Six SYN 7  &#; Synchronous Idle Seven

11 ... numbers... and letters both upper case

12 The first 32 characters in the ASCII-table are unprintable control codes and are used to control periph ASCII control characters (character code 0-31) printers F ?? Question mark DEC 64 OCT 100 HEX 40 BIN HTML HTML Name Description At symbol A NUL  � Uppercase Null char A B SOH B  Uppercase Start of Heading B C STX C  Uppercase Start of Text C D ETX D  Uppercase End of Text D E EOT E  Uppercase End of Transmission E F ENQ F � Uppercase Enquiry F G ACK G  Uppercase Acknowledgment G H BEL H  Uppercase Bell H I BS I  Uppercase Back Space I A J HT J Uppercase Horizontal J Tab ASCII Code - The extended ASCII table B0A K LF  Uppercase Line FeedK C0B L VT L  Uppercase Vertical Tab L D0C M FF M  Uppercase Form Feed M E0D N CR N Uppercase Carriage NReturn F0E O SO O  Uppercase Shift Out O/ X-On F P SI P  Uppercase Shift In / PX-Off Q DLE Q  Uppercase Data Line QEscape R DC1 R  Uppercase Device Control R 1 (of

13 ... and lower case...

14 codes and are used to control peripherals such ASCII control characters (character code 0-31) The first 32 characters in the ASCII-table are unprintable control ASCII Code printers. - The extended ASCII table 110 DEC 16 OCT HEX 6E BIN Symbol n HTML n Number HTML Name Description Lowercase n F NUL o � o Null Lowercase char o SOH p  p Start Lowercase of Heading p STX q  q Start Lowercase of Text q ETX r  r End of Text Lowercase r EOT s   End of Transmission Lowercase s ENQ t � t Enquiry Lowercase t ACK u  u Acknowledgment Lowercase u BEL v  v Bell Lowercase v BS  Back Space w w Lowercase w HT Horizontal Tab A x LF x Lowercase x Line Feed B y VT y  Lowercase y Vertical Tab A 0C z FF z  Lowercase z Form Feed B 0D { CR { Opening brace Carriage Return C 0E SO  Vertical bar Shift Out / X-On D 0F SI }   Shift Closing In / X-Off brace E DLE ~  ~ Data Equivalency Line Escape sign - tilde F DC1   Device Delete Control 1 (oft. XON) DC2  Device Control 2 Safari Power Saver Click to Start Flash Plug-in DC3  Device Control 3 (oft. XOFF)

15 Obviously it s not likely useful to remember all this, but you can see that the symbol A (capital A) is given a code (101) 8 = (41) 16 = (6) 10 and that the rest of the capital letters follow A in the usual order. This means that A uses the 7 bits in ASCII but computers almost invariably allocate 8 bits to store each letter. If you look, you will see that there are no codes for accented letters like á or è (which you might need in Irish or French), no codes for the Greek or Russian letters, no codes for Arabic or Hindu. In fact 8 bits (or 26 total symbols) is nowhere near enough to cope with all the alphabets of the World. This is a reflection of the fact that ASCII goes back to the early days of computers when memory was relatively very scarce compared to now, and also when the computer industry was mostly American. The modern system (not yet universally used) is called UNICODE and it allocates 16 bits for each character. Even with 2 16 = 636 possible codes, there is a difficulty accommodating all the worlds writing systems (including Chinese, Japanese, mathematical symbols, etc).

16 Converting fractions to binary So far we have talked about integers both positive and negative. We now look at a way to convert fractions to binary. You see if we start with, say 34 we can say that is We know 6 = (110)2 and if we could work out how to represent 4 as 0. something in binary then we would have 34 = = (110. something)2. To work out what something should be, we work backwards from the answer.

17 Say the digits we want are b 1, b 2, b 3,... and so 4 = (0.b1b2b3b4 )2 We don t know any of b 1, b 2, b 3,... yet but we know they should be base 2 digits and so each one is either 0 or 1. We can write the above equation as a formula and we have If we multiply both sides by 2, we get 4 = b1 2 + b b b = b1 + b2 2 + b b In other words multiplying by 2 just moves the binary point and we have 8 = (b1.b2b3b4 )2

18 Now if we take the whole number part of both sides we get 1 on the left and b 1 on the right. So we must have b 1 = 1. But if we take the fractional parts of both sides we have 3 = (0.b2b3b4 )2 We are now in a similar situation to where we began (but not with the same fraction) and we can repeat the trick we just did. Double both sides again 6 = (b2.b3b4b )2 Take whole number parts of both sides: b 2 = 1. Take fractional parts of both sides. 1 = (0.b3b4b )2 We can repeat our trick as often as we want to uncover as many of the values b 1, b 2, b 3, etc as we have the patience to discover.

19 What we have is a method, in fact a repetitive method where we repeat similar instructions many times. We call a method like this an algorithm, and this kind of thing is quite easy to programme on a computer because one of the programming instructions in almost any computer language is REPEAT (meaning repeat a certain sequence of steps from where you left off the last time).

20 In this case we can go a few more times through the steps to see how we get on. Double both sides again. 2 = (b3.b4bb6 )2 Whole number parts: b 3 = 0. Fractional parts: Double both sides again. 2 = (0.b4bb6 )2 4 = (b4.bb6b7 )2 Whole number parts: b 4 = 0. Fractional parts: 4 = (0.bb6b7 )2 This is getting monotonous, but you see the idea. You can get as many of the b s as you like.

21 In fact, if you look carefully, you will see that it has now reached repetition and not just monotony. We are back to the same fraction as we began with 4. If we compare 4 = (0.bb6b7 )2 to the starting one 4 = (0.b1b2b3b4 )2 we realise that everything will unfold again exactly as before. We must find b = b 1 = 1, b 6 = b 2 = 1, b 7 = b 3 = 0, b 8 = b 4 = 0, b 9 = b = b 1 and so we have a repeating pattern of digits So we can write the binary expansion of 4 down fully as a repeating pattern and our original number as 4 = (0.1100)2 34 = ( )2

22 Floating point format storage We have seen that in order to cope with numbers that are allowed to have fractional parts, computers use a binary version of the usual decimal point. We called it a binary point as decimal refers to base 10. Recall that what we mean by digits after the decimal point has to do with multiples of 1/10, 1/100 = 1/10 2 = 10 2, etc. So the number means = We use the binary point in the same way with powers of 1/2. So ( ) 2 = As in the familiar decimal system, every number can be written in binary using a binary point and as for decimals, there can sometimes be infinitely many digits after the point.

23 Binary Scientific Notation What we do next is use a binary version of scientific notation. The usual decimal scientific notation is like this = We refer to the.4321 part (a number between 1 and 10 or between -1 and -10 for negative numbers) as the mantissa. The power (in this case the 4) is called the exponent. Another decimal example is = and here the mantissa is.678 while the exponent is 3.

24 This is all based on the fact that multiplying or dividing by powers of 10 simply moves the decimal point around. In binary, what happens is that multiplying or dividing by powers of 2 moves the binary point. (101) 2 = (10.1) 2 = = (101) ( ) 2 = ( ) This last is an example of a number in the binary version of scientific notation. The mantissa is ( ) 2 and we can always arrange (no matter what number we are dealing with) to have the mantissa between 1 and 2. In fact always less than 2, and so of the form 1.something. The exponent in this last example is 3 the power that goes on the 2. For negative numbers we would need a minus sign in front.

25 What can thus write every number in this binary version of scientific notation. That saves us from having to record where to put the binary point, because it is always in the same place. Or really, the exponent tells us how far to move the point from that standard place. Computers then normally allocate a fixed number of bits for storing such numbers. The usual default is to allocate 32 bits in total (though 64 is quite common also). Within the 32 bits they have to store the mantissa and the exponent. The mantissa is already in binary, but we also need the exponent in binary. So in ( ) the mantissa is +( ) 2 while the exponent is 3 = (11) 2. Computers usually allocate 24 bits for storing the mantissa (including its possible sign) and the remaining 8 bits for the exponent.

26 In our example, 24 bits is plenty for the mantissa and we would need to make it longer to fill up the 24 bits: ( ) 2 will be the same as ( ) 2. However, there are numbers that need more than 24 binary digits in the mantissa, and what we must then do is round off. In fact, we have to chop off the mantissa after 23 binary places (or more usually we will round up or down depending on whether the digit in the next place is 1 or 0). Filling out the example number ( ) into 32 bits using this system, we might get: We are keeping bit 1 for a possible sign on the mantissa and we also need to allow the possibility of negative exponents. For example ( ) 2 = (1.11) is negative and so has a negative mantissa (1.11) 2. Because it is less than 1 in absolute value, it also has a negative exponent 4 = (100) 2.

27 To be a bit more accurate about how computers really do things, they normally put the sign bit (of the mantissa) first (or in their idea of the most prominent place), then put the 8 bits of the exponent next and the remaining 23 bits of the mantissa at the end. So a better picture for ( ) is this: ± exponent mantissa less sign This is just explained for the sake of greater accuracy but is not our main concern.

28 The web site: goes into quite a bit of detail about how this is done. What you get on (under floating point) tells you the outcome in examples but there are many refinements used in practice that are not evident from that and that also we won t discuss. The method we have sketched is called single precision floating point storage. There are some details we have not gone into here(such as how to store 0). Another common method, called double precision, uses 64 bits to store each number, 3 for the mantissa (including one for the sign) and 11 for the exponent.