Bits, Bytes, and Integers

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1 Bits, Bytes, and Integers with contributions from Dr. Bin Ren, College of William & Mary 1

2 Bits, Bytes, and Integers Representing information as bits Bit-level manipulations Integers Representation: unsigned and signed Conversion, casting Expanding, truncating Addition, negation, multiplication, shifting Summary Representations in memory, pointers, strings 2

3 Everything is bits Each bit is 0 or 1 By encoding/interpreting sets of bits in various ways Computers determine what to do (instructions) and represent and manipulate numbers, sets, strings, etc Why bits? Electronic Implementation Easy to store with bistable elements Reliably transmitted on noisy and inaccurate wires 1.1V 0.9V 0.2V 0.0V

4 The Decimal System and Bases base 10 (decimal): digits 0-9 e.g., = = in the decimal system, 10 is the base, or radix any integer > 1 can be a base base 2 has two digits: 0 and 1 bit = binary digit = =

5 Converting to Decimal Base eight (octal): digits = Base 16 (hexadecimal): digits 0-9 and A-F 13C 16 = Base 2 (binary): digits 0, = In general, radix r representations use the first r chars in {0 9, A...Z} and have the form d n-1 d n-2 d 1 d 0. Summing d n-1 r n-1 + d n-2 r n d 0 r 0 converts to base 10. 5

6 Converting from Decimal to Binary convert 1693 to binary use a divisor of 2 to obtain the following sequence of quotients and remainders dividend quotient remainder read remainders in reverse order =

7 More Base Conversion Practice convert to base 10 by multiplication of powers = ( ) 10 convert from base 10 by repeated division = ( ) 8 converting base x to base y: convert base x to base 10 then convert base 10 to base y 7

8 More Base Conversion Practice Convert from base = ( ) 3 and check = ( ) 16 and check Another way to convert from decimal to base n for n = 2 n 8 n 7 n 6 n 5 n 4 n 3 n 2 n 1 n From LEFT TO RIGHT, ask how many and subtract (219) 10 = ( ) 2 = ( ) 16 8

9 Converting Between Hex and Binary chart of values decimal hex binary decimal hex binary A B C D E F 1111 to convert from binary to hex start at right of binary number convert each group of 4 digits into a hex value e.g., convert to hex binary: hex: 6 E C 9

10 Converting Between Hex and Binary chart of values decimal hex binary decimal hex binary A B C D E F 1111 to convert from hex to binary replace each hex digit with its binary equivalent e.g., convert 8A5 16 to binary hex: 8 A 5 binary:

11 Octal 2 4 = 16 and 2 3 = 8 power = # of bits per hex/octal digit Binary to Hex every 4 bits = 1 hex digit Octal base 8 digits 0-7 Binary to Octal Every 3 bits = 1 octal digit DEC OCT HEX BIN Notes A B C D E F

12 Every Base is Base 10 In general, 10 X = X = = = = = = = = =

13 Other Binary Numbers Base 2 Number Representation Represent as Represent as [0011] 2 Represent X 10 4 as X

14 Encoding Byte Values Byte = 8 bits Binary to Decimal: 010 to Hexadecimal 0016 to FF16 useful for writing binary values concisely write FA1D37B16 in C as 0xFA1D37B 0xfa1d37b A B C D E F

15 Example Data Representations C Data Type Typical 32-bit Typical 64-bit x86-64 char short int long float double long double 10/16 pointer

16 Today: Bits, Bytes, and Integers Representing information as bits Bit-level manipulations Integers Representation: unsigned and signed Conversion, casting Expanding, truncating Addition, negation, multiplication, shifting Summary Representations in memory, pointers, strings 16

17 Boolean Algebra Developed by George Boole in 19th Century Algebraic representation of logic Encode True as 1 and False as 0 And Or A&B = 1 when both A=1 and B=1 A B = 1 when either A=1 or B=1 Not ~A = 1 when A=0 Exclusive-Or (Xor) A^B = 1 when either A=1 or B=1, but not both 17

18 General Boolean Algebras Operate on Bit Vectors Operations applied bitwise & ^ ~ All of the Properties of Boolean Algebra Apply 18

19 Example: Representing & Manipulating Sets Representation Width w bit vector represents subsets of {0,, w 1} a j = 1 if j A { 0, 3, 5, 6 } { 0, 2, 4, 6 } Operations & Intersection { 0, 6 } Union { 0, 2, 3, 4, 5, 6 } ^ Symmetric difference { 2, 3, 4, 5 } ~ Complement { 1, 3, 5, 7 } 19

20 Bit-Level Operations in C Operations &,, ~, ^ available in C Apply to any integral data type long, int, short, char, unsigned View arguments as bit vectors Arguments applied bit-wise Examples (char data type) ~0x41 0xBE ~ ~0x00 0xFF ~ x69 & 0x55 0x & x69 0x55 0x7D

21 Contrast: Logic Operations in C Contrast to Bit-level Operators &&,,! View 0 as False Anything nonzero as True Always return 0 or 1 Early termination Examples (char data type)!0x41 0x00!0x00 0x01!!0x41 0x01 0x69 && 0x55 0x01 0x69 0x55 0x01 p && *p (avoids null pointer access) 21

22 Contrast: Logic Operations in C Contrast to Bit-level Operators &&,,! View 0 as False Anything nonzero as True Always return 0 or 1 Early termination Examples (char data type)!0x41 0x00!0x00 0x01!!0x41 0x01 Watch out for && vs. & (and vs. ) one of the more common oopsies in C programming 0x69 && 0x55 0x01 0x69 0x55 0x01 p && *p (avoids null pointer access) 22

23 Shift Operations Left Shift: x << y Shift bit-vector x left y positions Throw away extra bits on left Fill with 0 s on right Right Shift: x >> y Shift bit-vector x right y positions Throw away extra bits on right Logical shift Fill with 0 s on left Arithmetic shift Replicate most significant bit on left Undefined Behavior Shift amount < 0 or word size Argument x << Log. >> Arith. >> Argument x << Log. >> Arith. >>

24 Bits, Bytes, and Integers Representing information as bits Bit-level manipulations Integers Representation: unsigned and signed Conversion, casting Expanding, truncating Addition, negation, multiplication, shifting Summary Representations in memory, pointers, strings Summary 24

25 Finite Precision the space to store integers in a computer is limited forced to deal with finite precision different machines use a varying number of bits for its word size, from 4 to 256 bits nominal size of integer and pointer data 32 and 64 bits are the current preferred sizes in general, we can store 2 n different values with n bits 1 bit: 2 values (0 and 1) 2 bits: 4 values (00, 01, 10, 11) 4 bits: 16 values we ve seen 0..15, but no negative values 25

26 Number of Values Address space depends on word size 2 word-size-in-#bits Is it big enough? 64-bit high-end machines becoming more prevalent Portability issues insensitive to sizes of different data types # bytes # bits # of values (2 #bits) low high E E E E E E E E E E E E+38 26

27 Negative Values so far, we ve seen the number of positive integers possible, but no negative values common sense tells us to split the number of bit patterns into two groups of roughly the same size: one for positive values and one for negative values don t forget 0 many ways to split these values have been developed over the years two s complement is the most popular unsigned only represents positive values 27

28 Encoding Integers Unsigned w 1 B2U(X) x i 2 i i 0 Two s Complement w 2 B2T(X) x w 1 2 w 1 x i 2 i i 0 short int x = 15213; short int y = ; C short 2 bytes long Decimal Hex Binary x B 6D y C Sign Bit For 2 s complement, most significant bit indicates sign 0 for nonnegative 1 for negative Sign Bit 28

29 Two-complement Encoding Example (Cont.) x = 15213: y = : Weight Sum

30 Two s Complement Simple Conversion conversion using a two-step process reverse the bits of the positive representation add 1 to the result e.g., reverse all bits add 1 = -9 only one representation for = one more negative number than positive number 30

31 Two s Complement Alternate Conversion alternate conversion using a two-step process reading from right to left, copy all values up to and including the first 1 reverse the remainder of the bits e.g.,

32 Numeric Ranges Unsigned Values UMin = UMax = 2 w Two s Complement Values TMin = 2 w TMax = 2 w Other Values Minus Values for W = 16 Decimal Hex Binary UMax FF FF TMax F FF TMin FF FF

33 Values for Different Word Sizes W UMax ,535 4,294,967,295 18,446,744,073,709,551,615 TMax ,767 2,147,483,647 9,223,372,036,854,775,807 TMin ,768-2,147,483,648-9,223,372,036,854,775,808 Observations TMin = TMax + 1 Asymmetric range UMax = 2 * TMax + 1 C Programming #include <limits.h> Declares constants, e.g., ULONG_MAX LONG_MAX LONG_MIN Values platform specific 33

34 Unsigned & Signed Numeric Values X B2U(X) B2T(X) Equivalence Same encodings for nonnegative values Uniqueness Every bit pattern represents unique integer value Each representable integer has unique bit encoding Can Invert Mappings U2B(x) = B2U -1 (x) Bit pattern for unsigned integer T2B(x) = B2T -1 (x) Bit pattern for two s comp integer 34

35 Bits, Bytes, and Integers Representing information as bits Bit-level manipulations Integers Representation: unsigned and signed Conversion, casting Expanding, truncating Addition, negation, multiplication, shifting Summary Representations in memory, pointers, strings 35

36 Mapping Between Signed & Unsigned Two s Complement x T2B T2U X B2U Unsigned ux Maintain Same Bit Pattern Unsigned ux U2B U2T X B2T Two s Complement x Maintain Same Bit Pattern Mappings between unsigned and two s complement numbers: Keep bit representations and reinterpret 36

37 Mapping Signed Unsigned Bits Signed Unsigned T2U U2T

38 Mapping Signed Unsigned Bits Signed Unsigned = /

39 Relation between Signed & Unsigned Two s Complement x T2B T2U X B2U Unsigned ux Maintain Same Bit Pattern ux x w Large negative weight becomes Large positive weight 39

40 Conversion Visualized 2 s Comp. Unsigned Ordering Inversion Negative Big Positive UMax UMax 1 TMax TMax + 1 TMax Unsigned Range 2 s Complement Range TMin 40

41 Signed vs. Unsigned in C Constants By default are considered to be signed integers Unsigned if have U as suffix 0U, U Casting Explicit casting between signed & unsigned same as U2T and T2U int tx, ty; unsigned ux, uy; tx = (int) ux; uy = (unsigned) ty; Implicit casting also occurs via assignments and procedure calls tx = ux; uy = ty; 41

42 Casting Surprises Expression Evaluation If there is a mix of unsigned and signed in single expression, signed values implicitly cast to unsigned Including comparison operations <, >, ==, <=, >= Examples for W = 32: TMIN = -2,147,483,648, TMAX = 2,147,483,647 Constant 1 Constant 2 Relation Evaluation 0 0 0U == unsigned < signed U > unsigned > signed U U < unsigned > signed (unsigned) > unsigned U < unsigned (int) U > signed 42

43 Summary Casting Signed Unsigned: Basic Rules Bit pattern is maintained But reinterpreted Can have unexpected effects: adding or subtracting 2 w Expression containing signed and unsigned int int is cast to unsigned!! 43

44 Bits, Bytes, and Integers Representing information as bits Bit-level manipulations Integers Representation: unsigned and signed Conversion, casting Expanding, truncating Addition, negation, multiplication, shifting Summary Representations in memory, pointers, strings 44

45 Sign Extension Task: Given w-bit signed integer x Convert it to w+k-bit integer with same value Rule: Make k copies of sign bit: X = x w 1,, x w 1, x w 1, x w 2,, x 0 k copies of MSB X w X k w 45

46 Sign Extension Example short int x = 15213; int ix = (int) x; short int y = ; int iy = (int) y; Decimal Hex Binary x B 6D ix B 6D y C iy FF FF C Converting from smaller to larger integer data type C automatically performs sign extension 46

47 Summary: Expanding, Truncating: Basic Rules Expanding (e.g., short int to int) Unsigned: zeros added Signed: sign extension Both yield expected result Truncating (e.g., unsigned to unsigned short) Unsigned/signed: bits are truncated Result reinterpreted Unsigned: mod operation Signed: similar to mod For small numbers yields expected behavior 47

48 Bits, Bytes, and Integers Representing information as bits Bit-level manipulations Integers Representation: unsigned and signed Conversion, casting Expanding, truncating Addition, negation, multiplication, shifting Representations in memory, pointers, strings Summary 48

49 Unsigned Addition Operands: w bits True Sum: w+1 bits u + v u + v Discard Carry: w bits UAdd w (u, v) Standard Addition Function Ignores carry output Implements Modular Arithmetic s = UAdd w (u, v) = u + v mod 2 w 49

50 Visualizing (Mathematical) Integer Addition Integer Addition 4-bit integers u, v Compute true sum Add 4 (u, v) Values increase linearly with u and v Forms planar surface Add 4 (u, v) Integer Addition u v

51 Visualizing Unsigned Addition Wraps Around If true sum 2 w At most once Overflow UAdd 4 (u, v) True Sum 2 w+1 Overflow w 0 Modular Sum u v 14 51

52 Two s Complement Addition Operands: w bits True Sum: w+1 bits u + v u + v Discard Carry: w bits TAdd w (u, v) TAdd and UAdd have Identical Bit-Level Behavior Signed vs. unsigned addition in C: int s, t, u, v; s = (int) ((unsigned) u + (unsigned) v); t = u + v Will give s == t 52

53 TAdd Overflow Functionality True sum requires w+1 bits Drop off MSB Treat remaining bits as 2 s comp. integer True Sum 2 w 1 2 w PosOver TAdd Result w w NegOver 53

54 Visualizing 2 s Complement Addition NegOver Values 4-bit two s comp. Range from -8 to +7 Wraps Around TAdd 4 (u, v) If sum 2 w 1 Becomes negative At most once If sum < 2 w 1 Becomes positive At most once u v PosOver 54

55 Multiplication Goal: Computing Product of w-bit numbers x, y Either signed or unsigned But, exact results can be bigger than w bits Unsigned: up to 2w bits Result range: 0 x * y (2 w 1) 2 = 2 2w 2 w Two s complement min (negative): Up to 2w-1 bits Result range: x * y ( 2 w 1 )*(2 w 1 1) = 2 2w w 1 Two s complement max (positive): Up to 2w bits, but only for (TMin w ) 2 Result range: x * y ( 2 w 1 ) 2 = 2 2w 2 So, maintaining exact results would need to keep expanding word size with each product computed is done in software, if needed e.g., by arbitrary precision arithmetic packages 55

56 Unsigned Multiplication in C Operands: w bits True Product: 2*w bits Discard w bits: w bits u v u * v UMult w (u, v) Standard Multiplication Function Ignores high order w bits Implements Modular Arithmetic UMult w (u, v) = u v mod 2 w 56

57 Signed Multiplication in C Operands: w bits True Product: 2*w bits Discard w bits: w bits u v u * v TMult w (u, v) Standard Multiplication Function Ignores high order w bits Some of which are different for signed vs. unsigned multiplication Lower bits are the same 57

58 Power-of-2 Multiply with Shift Operation u << k gives u * 2 k Both signed and unsigned Operands: w bits True Product: w+k bits u 2 k Examples u << 3 == u * 8 (u << 5) (u << 3) == u * 24 Most machines shift and add faster than multiply Compiler generates this code automatically * u 2 k Discard k bits: w bits UMult w (u, 2 k ) TMult w (u, 2 k ) k

59 Unsigned Power-of-2 Divide with Shift Quotient of Unsigned by Power of 2 u >> k gives u / 2 k Uses logical shift k u Operands: / 2 k Division: Result: u / 2 k u / 2 k Binary Point Division Computed Hex Binary x B 6D x >> D B x >> B x >> B

60 Bits, Bytes, and Integers Representing information as bits Bit-level manipulations Integers Representation: unsigned and signed Conversion, casting Expanding, truncating Addition, negation, multiplication, shifting Summary Representations in memory, pointers, strings 60

61 Arithmetic: Basic Rules Addition: Unsigned/signed: Normal addition followed by truncate, same operation on bit level Unsigned: addition mod 2 w Mathematical addition + possible subtraction of 2 w Signed: modified addition mod 2 w (result in proper range) Mathematical addition + possible addition or subtraction of 2 w Multiplication: Unsigned/signed: Normal multiplication followed by truncate, same operation on bit level Unsigned: multiplication mod 2 w Signed: modified multiplication mod 2 w (result in proper range) 61

62 Why Should I Use Unsigned? Don t use without understanding implications Easy to make mistakes unsigned i; for (i = cnt-2; i >= 0; i--) a[i] += a[i+1]; Can be very subtle #define DELTA sizeof(int) int i; for (i = CNT; i-delta >= 0; i-= DELTA)... 62

63 Counting Down with Unsigned Proper way to use unsigned as loop index unsigned i; for (i = cnt-2; i < cnt; i--) a[i] += a[i+1]; See Robert Seacord, Secure Coding in C and C++ C Standard guarantees that unsigned addition will behave like modular arithmetic 0 1 UMax Even better size_t i; for (i = cnt-2; i < cnt; i--) a[i] += a[i+1]; Data type size_t defined as unsigned value with length = word size Code will work even if cnt = UMax What if cnt is signed and < 0? 63

64 Why Should I Use Unsigned? (cont.) Do Use When Performing Modular Arithmetic Multiprecision arithmetic Do Use When Using Bits to Represent Sets Logical right shift, no sign extension 64

65 Bits, Bytes, and Integers Representing information as bits Bit-level manipulations Integers Representation: unsigned and signed Conversion, casting Expanding, truncating Addition, negation, multiplication, shifting Summary Representations in memory, pointers, strings 65

66 Byte-Oriented Memory Organization Programs refer to data by address Conceptually, envision it as a very large array of bytes In reality, it s not, but can think of it that way An address is like an index into that array and, a pointer variable stores an address Note: system provides private address spaces to each process Think of a process as a program being executed So, a program can clobber its own data, but not that of others 66

67 Machine Words Any given computer has a Word Size Nominal size of integer-valued data and of addresses Until recently, most machines used 32 bits (4 bytes) as word size Limits addresses to 4GB (2 32 bytes) Increasingly, machines have 64-bit word size Potentially, could have 18 EB (exabytes) of addressable memory That s 18.4 X Machines still support multiple data formats Fractions or multiples of word size Always integral number of bytes 67

68 Word-Oriented Memory Organization Addresses Specify Byte Locations Address of first byte in word Addresses of successive words differ by 4 (32-bit) or 8 (64-bit) 32-bit Words Addr = 0000?? Addr = 0004?? Addr = 0008?? Addr = 0012?? 64-bit Words Addr = 0000?? Addr = 0008?? Bytes Addr

69 Example Data Representations C Data Type Typical 32-bit Typical 64-bit x86-64 char short int long float double long double 10/16 pointer

70 Byte Ordering So, how are the bytes within a multi-byte word ordered in memory? Conventions Big Endian: Sun, PPC Mac, Internet Least significant byte has highest address Little Endian: x86, ARM processors running Android, ios, and Windows Least significant byte has lowest address 70

71 Byte Ordering Example Example Variable x has 4-byte value of 0x Address given by &x is 0x100 Big Endian Little Endian 0x100 0x101 0x102 0x x100 0x101 0x102 0x

72 Representing Integers Decimal: Binary: Hex: 3 B 6 D int A = 15213; IA32, x D 3B IA32, x C4 FF FF Sun B 6D int B = ; Sun FF FF C4 93 long int C = 15213; IA32 6D 3B x D 3B Two s complement representation Sun B 6D 72

73 Examining Data Representations Code to Print Byte Representation of Data Casting pointer to unsigned char * allows treatment as a byte array typedef unsigned char *pointer; void show_bytes(pointer start, size_t len){ size_t i; for (i = 0; i < len; i++) printf( %p\t0x%.2x\n",start+i, start[i]); printf("\n"); } Printf directives: %p: Print pointer %x: Print Hexadecimal 73

74 show_bytes Execution Example int a = 15213; printf("int a = 15213;\n"); show_bytes((pointer) &a, sizeof(int)); Result (Linux x86-64): int a = 15213; 0x7fffb7f71dbc 6d 0x7fffb7f71dbd 3b 0x7fffb7f71dbe 00 0x7fffb7f71dbf 00 74

75 Representing Pointers int B = ; int *P = &B; Sun EF FF FB 2C IA32 AC 28 F5 FF x C 1B FE 82 FD 7F Different compilers & machines assign different locations to objects Even get different results each time run program 75

76 Representing Strings Strings in C Represented by array of characters Each character encoded in ASCII format Standard 7-bit encoding of character set Character 0 has code 0x30 Digit i has code 0x30+i String should be null-terminated Final character = 0 Compatibility Byte ordering not an issue char S[6] = "18213"; IA Sun

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