CS 31: Intro to Systems Binary Representation. Kevin Webb Swarthmore College January 27, 2015

Transcription

1 CS 3: Intro to Systems Binary Representation Kevin Webb Swarthmore College January 27, 25

2 Reading Quiz

3 Abstraction User / Programmer Wants low complexity Applications Specific functionality Software library Reusable functionality Operating system Manage resources Complex devices Compute & I/O

4 Today Number systems and conversion Data types and storage: Sizes Representation Signedness

5 You can view binary file contents xxd (or hexdump C) to view binary file values: xxd a.out # a binary executable file Address: value of the next 6 bytes in memory : 7f45 4c46 2 : 2 3e : 4 84d (these weird numbers (f,c,e, ), are hexidecimal digits ) xxd myprog.c # binary ascii encoding of C source: : e63 6c c f 2e68 #i nc lu de <s td io.h : 3ea 696e 742 6d6 696e b a2 >\n in t ma in () { \n

6 Data Storage Lots of technologies out there: Magnetic (hard drive, floppy disk) Optical (CD / DVD / Blu-Ray) Electronic (RAM, registers, ) Focus on electronic for now We ll see (and build) digital circuits soon Relatively easy to differentiate two states Voltage present Voltage absent

7 Bits and Bytes Bit: a or value (binary) HW represents as two different voltages : the presence of voltage (high voltage) : the absence of voltage (low voltage) Byte: 8 bits, the smallest addressable unit Memory: (address) [] [] [2] Other names: 4 bits: Nibble Word : Depends on system, often 4 bytes

8 How many unique values can we represent with 9 bits? One bit: two values ( or ) Two bits: four values (,,, or ) Three bits: eight values (,,,, ) A. 8 B. 8 C. 256 D. 52 E. Some other number of values.

9 How many values? bit: 2 bits: 3 bits: 4 bits: 6 values N bits: 2 N values

10 C types and their (typical!) sizes byte: char, unsigned char 2 bytes: short, unsigned short 4 bytes: int, unsigned int, float 8 bytes: long long, unsigned long long, double 4 or 8 bytes: long, unsigned long unsigned long v; short s; long long ll; printf( %lu %lu %lu\n, sizeof(v), sizeof(s), sizeof(ll)); // prints out number of bytes How do we use this storage space (bits) to represent a value?

11 Let s start with what we know Decimal number system (Base ) Sequence of digits in range [, 9] 6425 Digit #4 Digit #

12 What is the significance of the N th digit number in this number system? What does it contribute to the overall value? 6425 A. d N * B. d N * C. d N * N D. d N * N E. d N * d N Digit #4 d 4 Digit # d Consider the meaning of d 3 (the value 4) above. What is it contributing to the total value?

13 Positional Notation The meaning of a digit depends on its position in a number. A number, written as the sequence of digits d n d n- d 2 d d in base b represents the value d n * b n + d n- * b n d 2 * b 2 + d * b + d * b

14 Decimal: Base Used by humans A number, written as the sequence of digits d n d n- d 2 d d where d is in {,,2,3,4,5,6,7,8,9}, represents the value d n * n + d n- * n d 2 * 2 + d * + d * 6425 = 6 * * 3 + * * + 5 *

15 Binary: Base 2 Used by computers A number, written as the sequence of digits d n d n- d 2 d d where d is in {,}, represents the value d n * 2 n + d n- * 2 n d 2 * d * 2 + d * 2

16 What is the value of in decimal? A number, written as the sequence of digits d n d n- d 2 d d where d is in {,}, represents the value d n * 2 n + d n- * 2 n d 2 * d * 2 + d * 2 A. 26 B. 53 C. 6 D. 6 E. 28

17 Other (common) number systems. Base : decimal Base 2: binary Base 6: hexadecimal (memory addresses) Base 8: octal Base 64 (Commonly used on the Internet, e.g. attachments).

18 Hexadecimal: Base 6 Indicated by prefacing number with x A number, written as the sequence of digits d n d n- d 2 d d where d is in {,,2,3,4,5,6,7,8,9,A,B,C,D,E,F}, represents the value d n * 6 n + d n- * 6 n d 2 * d * 6 + d * 6

19 What is the value of xb7 in decimal? A. 397 B. 49 C. 49 D. 437 E = 256

20 Hexadecimal: Base 6 Indicated by prefacing number with x Like binary, base is power of 2 Fewer digits to represent same value Each digit is a nibble, or half a byte A number, written as the sequence of digits d n d n- d 2 d d where d is in {,,2,3,4,5,6,7,8,9,A,B,C,D,E,F}, represents the value d n * 6 n + d n- * 6 n d 2 * d * 6 + d * 6

21 Each hex digit is a nibble Hex digit:6 values, 2 4 = 6 -> 4 bits / digit x B 7 Four-bit value: Four-bit value: B (decimal ) Four-bit value: 7 In binary: B 7

22 Converting Decimal -> Binary Two methods: division by two remainder powers of two and subtraction

23 Method : decimal value D, binary result b (b i is ith digit): i = while (D > ) if D is odd Example: Converting 5 set b i to if D is even set b i to i++ D = D/2 idea: D = b example: D = 5 a = D = b D = 52 a = D/2 = b/2 D = 26 a2 = D/2 = b/2 D = 3 a3 = D/2 = b/2 D = 6 a4 = D/2 = b/2 D = 3 a5 = = D = a6 = D = a7 = 5 =

24 Method : decimal value D, binary result b (b i is ith digit): i = while (D > ) if D is odd Example: Converting 5 set b i to if D is even set b i to i++ D = D/2 idea: D = b example: D = 5 a = D/2 = b/2 D = 52 a = D/2 = b/2 D = 26 a2 = D/2 = b/2 D = 3 a3 = D/2 = b/2 D = 6 a4 = D/2 = b/2 D = 3 a5 = = D = a6 = D = a7 = 5 =

25 Method : decimal value D, binary result b (b i is ith digit): i = while (D > ) if D is odd Example: Converting 5 set b i to if D is even set b i to i++ D = D/2 idea: D = b example: D = 5 a = D/2 = b/2 D = 52 a = D/2 = b/2 D = 26 a2 = D/2 = b/2 D = 3 a3 = D/2 = b/2 D = 6 a4 = D/2 = b/2 D = 3 a5 = = D = a6 = D = a7 = 5 =

26 Method 2 2 =, 2 = 2, 2 2 = 4, 2 3 = 8, 2 4 = 6, 2 5 = 32, 2 6 = 64, 2 7 = 28 To convert 5: Find largest power of two that s less than 5 (64) Subtract 64 (5 64 = 4), put a in d 6 Subtract 32 (4 32 = 9), put a in d 5 Skip 6, it s larger than 9, put a in d 4 Subtract 8 (9 8 = ), put a in d 3 Skip 4 and 2, put a in d 2 and d Subtract ( = ), put a in d (Done)

27 What is the value of 357 in binary? A. B. C. D. E. 2 =, 2 = 2, 2 2 = 4, 2 3 = 8, 2 4 = 6, 2 5 = 32, 2 6 = 64, 2 7 = 28, 2 8 = 256

28 So far: Unsigned Integers With N bits, can represent values: to 2 n - We can always add s to the front of a number without changing it: = = = byte: char, unsigned char 2 bytes: short, unsigned short 4 bytes: int, unsigned int, float 8 bytes: long long, unsigned long long, double 4 or 8 bytes: long, unsigned long

29 Representing Signed Values One option (used for floats, NOT integers) Let the first bit represent the sign means positive means negative For example: -> 5 -> -5 Problem with this scheme?

30 Floating Point Representation bit for sign sign exponent fraction 8 bits for exponent 23 bits for precision value = (-) sign *.fraction * 2 (exponent-27) let's just plug in some values and try it out x4ac49ba: sign = exp = 29 fraction = = * *2 2 = I don t expect you to memorize this

31 Up Next: Binary Arithmetic