Background/Review on Numbers and Computers (lecture)

Transcription

1 Bakground/Review on Numbers and Computers (leture) ICS312 Mahine-Level and Systems Programming Henri Casanova

2 Numbers and Computers Throughout this ourse we will use binary and hexadeimal representations of numbers need to be aware of the ways in whih the omputer stores numbers So let us go through a simple review before we start learning how to write assembly ode Numbers in different bases Number representation in omputers and basi arithmeti More to ome later on arithmeti

3 Numbers and bases We are used to thinking of numbers as written in deimal, that is, in base = 2* * = 1* * *10 0 Eah number is deomposed into a sum of terms Eah term is the produt of two fators A digit (from 0 to 9) The base (10) raised to a power orresponding to the digit s position in the number 136 = + 0* * * * *10 0 = We typially don t write (an infinite number of) leading 0 s

4 Numbers and Bases Any number an be written in base b, using b digits If b = 10 we have deimal with 10 digits [0-9] If b = 2 we have binary with 2 digits [0,1], whih are also alled bits If b = 8 we have otal with 8 digits [0-7] If b = 16 we have hexadeimal with 16 digits [0-9,A,B,C,D,E,F] Computers use binary internally It s easy to assoiate two states to a urrent Low voltage = 0, high voltage = 1 Assoiating 16 states to a urrent is more ompliated and error-prone However, binary is umbersome The lower the base the longer the numbers! It s really diffiult for a human to remember binary Therefore we, as humans, like to use higher bases Bases that are powers of 2 make for easy translation to binary, and thus are partiularly useful, and in partiular hexadeimal

5 Binary Numbers Counting in binary: A binary number with d bits orresponds to integer values between 0 and 2 d Example: An integer stored in 8 bits has values between 0 and = 255

6 Converting from Binary to Deimal We denote by XXXX 2 a binary representation of a number and by XXXX 10 a deimal representation Converting from binary to deimal is straightforward: = 1* *2 4 +1*2 2 +1*2 1 = 1* *16 + 1*4 + 1*2 = The rightmost bit of a binary number is alled the least signifiant bit The leftmost non-zero bit of a binary number is alled the most signifiant bit If the least signifiant bit is 0, then the number is even, otherwise it s odd

7 Converting from Deimal to Binary The onversion proeeds by a series of integer divisions by 2, and by reording the remainder of the division Integer division a/b: a = b* q + remainder, where all are integers Example: onverting into binary Divide 37 by 2: 37 = 2* Divide 18 by 2: 18 = 2*9 + 0 Divide 9 by 2: 9 = 2*4 + 1 Divide 4 by 2: 4 = 2*2 + 0 Divide 2 by 2: 2 = 2*1 + 0 Divide 1 by 2: 1 = 2*0 + 1 Result: The least signifiant bit is omputed first The most signifiant bit is omputed last Note that if we ontinue dividing, we get extraneous leading 0s

8 Binary Arithmeti Adding a 0 to the right of a binary number multiplies it by = = = = Adding two binary numbers is just like adding deimal numbers: using a arry With no previous arry = 0 = 1 = 1 = 0 With a previous arry = 1 = 0 = 0 = 1

9 Binary Addition = = = =

10 Counting in Hexadeimal 0 16 =0 10 A 16 = = E 16 = =1 10 B 16 = = F 16 = =2 10 C 16 = = = =3 10 D 16 = = = =4 10 E 16 = = = =5 10 F 16 = = = = = A 16 = = = = B 16 = = = = C 16 = = = = D 16 = =39 10

11 Converting from hex to deimal This is again straightforward A203DE 16 = 10* * * * *16 0 = 10,617,822 10

12 Converting from deimal to hex Use the same idea as for binary Example: onvert = 77* = 4* = 0* Result: 4D5 16

13 Hexadeimal addition A 2 3 F D = D F 5 2 = D 1 F F A 4 D F = D E =

14 Why is hexadeimal useful? We need to think in binary beause omputers operate on binary quantities But binary is umbersome However, hexadeimal makes it possible to represent binary quantities in a ompat form Conversions bak and forth from binary to hex are straightforward Just onvert hex digits into 4-bit numbers Just onvert 4-bit binary numbers into hex digits

15 Converting from hex to binary Consider A43FE2 16 We onvert eah hex digit into a 4-bit binary number: A 16 : : : F 16 : E 16 : : We glue them all together: A43FE2 16 = Note that: You must have the leading 0 s for the 4-bit numbers, whih is what a omputer would store anyway It all works beause F 16 = 15 10, and a 4-bit number has maximum value of = 15 10

16 Converting from binary to hex Let s onvert into hex We split it in 4-bit numbers, whih we onvert separately First we add leading 0 s to have a number of bits that s a multiple of 4: Then we onvert : : : A : F 16 And the result: = 12AF 16

17 Integer representation A omputer needs to store integers in memory/registers Stored using different numbers of bytes (1 byte = 8 bits): 1-byte: byte 2-byte: half word (or word ) 4-byte: word (or double word ) 8-byte: double word (or paragraph, or quadword ) Different omputers have used different word sizes, so it s always a bit onfusing to just talk about a word without any ontext Regardless of the number of bytes, integers are stored in binary Integers ome in two flavors: Unsigned: values from 0 to 2b -1 Signed: negatives values, with about the same number of negative values as the number of positive values You an atually delare variables as signed or unsigned in some high-level programming languages, like C

18 Sign-Magnitude Storing unsigned integers is easy: just store the bits of the integer s binary representation Storing signed integer raises a question: how to store the sign? One approah is alled sign-magnitude: reserve the leftmost bit to represent the sign denotes denotes It s very easy to negate a number: just flip the leftmost bit Unfortunately, sign-magnitude ompliates the logi of the CPU (i.e., ICS331-type stuff) There are two representations for zero: and Some operations are thus more ompliated to implement in hardware

19 One s omplement Another idea to store a negative number is to take the omplement (i.e., flip all bits) of its positive ounterpart Example: I want to store integer = = Simple, but still two representations for zero: and It turns out that omputer logi to deal with 1 s omplement arithmeti is ompliated Note: it s easy to ompute the 1 s omplement of a number represented in hexadeimal let s onsider: Subtrat eah hex digit from F: F-5=A, F-7=8 1 s omplement of is A8 16

20 Two s omplement While sign-magnitude and 1 s omplement were used in older omputers, nowadays all omputers use 2 s omplement Computing the 2 s omplement is in two steps: Compute the 1 s omplement of the positive number Add 1 to the result The gives the representation of the negative number Example: Let s represent = or s omplement: or A8 Add one: or A9 Let s invert again We start with A9 Invert: 56 Add one: 57, whih represents 87 10

21 Two s omplement Note that when adding 1 in the seond step a arry may be generated but is ignored! Differene between arithmeti and omputer arithmeti When adding two X-bit quantities in a omputer one always obtain another X-bit quantity (X=8, 16, 32, ) Example: Computing 2 s omplement of Take the invert: Add one: with a arry generated! Should be a 9-bit quantity: Therefore 0 has only one representation: a signed byte an store values from -128 to +127 (128 <0 values, and 128 >=0 values) It turns out that 2 s omplement makes for very simple arithmeti logi when building ALUs From now on we always assumed 2 s omplement representation Important: The leftmost bit still indiates the sign of the number (0: positive, 1: negative) In hex, if the left-most digit is 8, 9, A, B, C, D, E, or F, then the number is negative, otherwise it is positive

22 Ranges of Numbers For 1-byte values Unsigned Smallest value: 00 (010) Largest value: FF (25510) Signed Smallest value: 80 (-12810) Largest value: 7F (+12710) For 2-byte values et. Unsigned Smallest value: 0000 (010) Largest value: FFFF (65,53510) Signed Smallest value: 8000 (-32,76810) Largest value: 7FFF (+32,76710)

23 The Task of the (Assembly) Programmer The omputer simply stores data as bits The omputer internally has no idea what the data means It doesn t know whether numbers are signed or unsigned We, as programmers have preise interpretations of what bits mean I store a 4-byte signed integer, I store a 1-byte integer whih is an ASCII ode When using a high-level language like C, we say what data means I delare x as an int and y as an unsigned har But when writing assembly ode, we don t have a notion of data types The ISA provides many instrutions that operate on all types of data It s our role to use the instrutions that orrespond to the data e.g., if you used the signed multipliation instrution on unsigned numbers, you ll just get a wrong results but no warning/error This is one of the diffiulties of assembly programming And 2 s omplement appears magi...

24 The Magi of 2 s Complement Say I have two 1-byte values, A3 and 17, and I add them together: A = BA16 If my interpretation of the numbers is unsigned: A3 16 = = 2310 BA 16 = and indeed, = If my interpretation of the numbers is signed: A3 16 = = 2310 BA 16 = and indeed, = So, as long as I stik to my interpretation, the binary addition does the right thing assuming 2 s omplement notation!!! Same thing for the subtration

25 Conlusion We ll ome bak to numbers and arithmeti when we use arithmeti assembly instrutions