– 1 – Number Systems Number of symbols = base of the system Most intuitive -- base 10 (decimal system) counting on fingertips symbols Computers use base 2 (binary system) only two symbols - 0 / 1 Many physical reasons for choosing Other useful bases 8 - octal 16 - hexadecimal

– 2 – Representing a number in base B General expression : A number U 3 U 2 U 1 U 0 in base B is effectively U 3 x B 3 + U 2 x B 2 + U 1 x B 1 + U 0 x B 0 Examples : 234 in base 10 is 2 x x x in base 2 is 1 x x x x 2 0 = 1 x x 10 0 in base 10

– 3 – Why Don’t Computers Use Base 10? Implementing Electronically Hard to store Hard to transmit Messy to implement digital logic functions Binary Electronic Implementation Easy to store with bistable elements Reliably transmitted on noisy and inaccurate wires Straightforward implementation of arithmetic functions 0.0V 0.5V 2.8V 3.3V 010

– 4 – Octal and hexadecimal Octal uses 0-7 for representing numbers Hexadecimal uses 0-9, a, b, c, d, e and f as digits a = decimal 10, b=decimal 11 and so on They are useful because translation from binary is easy Consider rewrite as and you get 5465 in octal rewrite as and you get b35 in hex Much more readable in octal and hex than in binary Also much easier to calculate equivalent decimal value It will really pay to learn to interpret hex numbers in this course

– 5 – Encoding Byte Values Byte = 8 bits Binary to Decimal: 0 10 to Hexadecimal to FF 16 Base 16 number representation Use characters ‘0’ to ‘9’ and ‘A’ to ‘F’ Write FA1D37B 16 in C as 0xFA1D37B o Or 0xfa1d37b A B C D E F Hex Decimal Binary

– 6 – Machine Words Machine Has “Word Size” Nominal size of integer-valued data Including addresses Most current machines are 32 bits (4 bytes) or 64 bits 32 bits Limits addresses to 4GB Becoming too small for memory-intensive applications High-end systems are 64 bits (8 bytes) Potentially address  1.8 X bytes Machines support multiple data formats Fractions or multiples of word size Always integral number of bytes

– 7 – Data Representations Sizes of C Objects (in Bytes) C Data TypeCompaq AlphaTypical 32-bitIntel IA32 int444 long int844 char111 short222 float444 double888 long double8810/12 char *844 o Or any other pointer

– 8 – Byte Ordering How should bytes within multi-byte word be ordered in memory? Conventions Sun’s, Mac’s are “Big Endian” machines Least significant byte has highest address Alphas, PC’s are “Little Endian” machines Least significant byte has lowest address

– 9 – Byte Ordering Example Big Endian Least significant byte has highest address Little Endian Least significant byte has lowest address Example Variable x has 4-byte representation 0x Address given by &x is 0x100 0x1000x1010x1020x x1000x1010x1020x Big Endian Little Endian

– 10 – Reading Byte-Reversed Listings Disassembly Text representation of binary machine code Generated by program that reads the machine code Example Fragment AddressInstruction CodeAssembly Rendition :5b pop %ebx :81 c3 ab add $0x12ab,%ebx c:83 bb cmpl $0x0,0x28(%ebx) Deciphering Numbers Value: 0x12ab Pad to 4 bytes: 0x000012ab Split into bytes: ab Reverse: ab

– 11 – Representing Integers int A = 15213; int B = ; long int C = 15213; Decimal: Binary: Hex: 3 B 6 D 6D 3B 00 Linux/Alpha A 3B 6D 00 Sun A 93 C4 FF Linux/Alpha B C4 93 FF Sun B Two’s complement representation (Covered later) 00 6D 3B 00 Alpha C 3B 6D 00 Sun C 6D 3B 00 Linux C

– 12 – Representing Pointers int B = ; int *P = &B; Alpha Address Hex: 1 F F F F F C A 0 Binary: A0 FC FF Alpha P Sun Address Hex: E F F F F B 2 C Binary: Different compilers & machines assign different locations to objects FB 2C EF FF Sun P FF BF D4 F8 Linux P Linux Address Hex: B F F F F 8 D 4 Binary:

– 13 – Representing Floats Float F = ; IEEE Single Precision Floating Point Representation Hex: D B Binary: : Not same as integer representation, but consistent across machines 00 B4 6D 46 Linux/Alpha F B D Sun F Can see some relation to integer representation, but not obvious IEEE Single Precision Floating Point Representation Hex: D B Binary: : IEEE Single Precision Floating Point Representation Hex: D B Binary: :

– 14 – char S[6] = "15213"; Representing Strings Strings in C Represented by array of characters Each character encoded in ASCII format Standard 7-bit encoding of character set Other encodings exist, but uncommon Character “0” has code 0x30 o Digit i has code 0x30 +i String should be null-terminated Final character = 0 Compatibility Byte ordering not an issue Data are single byte quantities Text files generally platform independent Except for different conventions of line termination character(s)! Linux/Alpha S Sun S

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

– 16 – A ~A ~B B Connection when A&~B | ~A&B Application of Boolean Algebra Applied to Digital Systems by Claude Shannon 1937 MIT Master’s Thesis Reason about networks of relay switches Encode closed switch as 1, open switch as 0 A&~B ~A&B = A^B

– 17 – Algebra Integer Arithmetic Addition is “sum” operation Multiplication is “product” operation – is additive inverse 0 is identity for sum 1 is identity for product Boolean Algebra  {0,1}, |, &, ~, 0, 1  forms a “Boolean algebra” Or is “sum” operation And is “product” operation ~ is “complement” operation (not additive inverse) 0 is identity for sum 1 is identity for product

– 18 – Commutativity A | B = B | AA + B = B + A A & B = B & AA * B = B * A Associativity (A | B) | C = A | (B | C)(A + B) + C = A + (B + C) (A & B) & C = A & (B & C)(A * B) * C = A * (B * C) Product distributes over sum A & (B | C) = (A & B) | (A & C)A * (B + C) = A * B + B * C Sum and product identities A | 0 = AA + 0 = A A & 1 = AA * 1 = A Zero is product annihilator A & 0 = 0A * 0 = 0 Cancellation of negation ~ (~ A) = A– (– A) = A Boolean Algebra  Integer Ring

– 19 – Boolean: Sum distributes over product A | (B & C) = (A | B) & (A | C)A + (B * C)  (A + B) * (B + C) Boolean: Idempotency A | A = AA + A  A “A is true” or “A is true” = “A is true” A & A = AA * A  A Boolean: Absorption A | (A & B) = AA + (A * B)  A “A is true” or “A is true and B is true” = “A is true” A & (A | B) = AA * (A + B)  A Boolean: Laws of Complements A | ~A = 1A + –A  1 “A is true” or “A is false” Ring: Every element has additive inverse A | ~A  0A + –A = 0 Boolean Algebra  Integer Ring

– 20 – Properties of & and ^ Boolean Ring  {0,1}, ^, &, , 0, 1  Identical to integers mod 2  is identity operation:  (A) = A A ^ A = 0 PropertyBoolean Ring Commutative sumA ^ B = B ^ A Commutative productA & B = B & A Associative sum(A ^ B) ^ C = A ^ (B ^ C) Associative product(A & B) & C = A & (B & C) Prod. over sumA & (B ^ C) = (A & B) ^ (B & C) 0 is sum identityA ^ 0 = A 1 is prod. identityA & 1 = A 0 is product annihilatorA & 0 = 0 Additive inverseA ^ A = 0

– 21 – Relations Between Operations DeMorgan’s Laws Express & in terms of |, and vice-versa A & B = ~(~A | ~B) o A and B are true if and only if neither A nor B is false A | B = ~(~A & ~B) o A or B are true if and only if A and B are not both false Exclusive-Or using Inclusive Or A ^ B = (~A & B) | (A & ~B) o Exactly one of A and B is true A ^ B = (A | B) & ~(A & B) o Either A is true, or B is true, but not both

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

– 23 – Bit-Level Operations in C Operations &, |, ~, ^ Available in C Apply to any “integral” data type long, int, short, char View arguments as bit vectors Arguments applied bit-wise Examples (Char data type) ~0x41 --> 0xBE ~ > ~0x00 --> 0xFF ~ > x69 & 0x55 --> 0x & > x69 | 0x55 --> 0x7D | >

– 24 – Contrast: Logic Operations in C Contrast to Logical 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)

– 25 – 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 right Useful with two’s complement integer representation Argument x << Log. >> Arith. >> Argument x << Log. >> Arith. >>

– 26 – Cool Stuff with Xor void funny(int *x, int *y) { *x = *x ^ *y; /* #1 */ *x = *x ^ *y; /* #1 */ *y = *x ^ *y; /* #2 */ *y = *x ^ *y; /* #2 */ *x = *x ^ *y; /* #3 */ *x = *x ^ *y; /* #3 */} Bitwise Xor is form of addition With extra property that every value is its own additive inverse A ^ A = 0 BA Begin BA^B 1 (A^B)^B = AA^B 2 A(A^B)^A = B 3 AB End *y*x

– 27 – Storing negative integers sign-magnitude of n bits in the word, the most significant bit is sign 1 indicates negative number, 0 a positive number remaining n-1 bits determine the magnitude 0 has two representations The range of valid values is -(2 (n-1) -1) to (2 (n-1) -1) Examples o 8 bit words can represent -127 to +127 o 16 bit words can represent to Advantages Very straightforward Simple to understand Disadvantage different machinery for addition and subtraction multiple representations of zero.

– 28 – One’s complement The most significant bit is effectively sign bit The range of numbers the same as signed magnitude Positive numbers stored as such Negative numbers are bit-wise inverted Example 4 bit storage – range -7 to +7 o -7 = 1000, +7 = 0111, +3 = 0011, -4 = bit storage – range = -127 t0 128 Zero has two representations : o o Hardly ever used for actual storage Useful for understanding 2’s compliment and some other operations

– 29 – Two’s Complement short int x = 15213; short int y = ; C short 2 bytes long Sign Bit For 2’s complement, most significant bit indicates sign 0 for nonnegative 1 for negative Unsigned Two’s Complement Sign Bit

– 30 – Two’s compliment is effectively one’s compliment with a “1” added to it. It is a number’s additive inverse Number added to its own two’s compliment results in zero Same circuitry can do arithmetic operations for positive and negative numbers The range is (-2 n-1) to (2 n-1 -1) Example : 13 + (-6) 13 in 4 bit binary is is 0110 It’s one’s complement is 1001 Two’s complement is 1010 Add 1101 and 1010 in 4 bits, ignoring the carry The answer is 0111, with carry of 1, which in decimal is 7

– 31 – Power-of-2 Multiply with Shift Operation u << k gives u * 2 k Both signed and unsigned Examples u << 3==u * 8 u << 5 - u << 3==u * 24 Most machines shift and add much faster than multiply u 2k2k * u · 2 k True Product: w+k bits Operands: w bits Discard k bits: w bits UMult w (u, 2 k ) k 000 TMult w (u, 2 k ) 000

– 32 – Unsigned Power-of-2 Divide with Shift Quotient of Unsigned by Power of 2 u >> k gives  u / 2 k  Uses logical shift u 2k2k / u / 2 k Division: Operands: k 0  u / 2 k  Result:. Binary Point 0

– 33 – Signed Power-of-2 Divide with Shift Quotient of Signed by Power of 2 x >> k gives  x / 2 k  Uses arithmetic shift Rounds wrong direction when u < x 2k2k / x / 2 k Division: Operands: k 0 RoundDown(x / 2 k ) Result:. Binary Point 0

– 34 – Correct Power-of-2 Divide Quotient of Negative Number by Power of 2 Want  x / 2 k  (Round Toward 0) Compute as  (x+ 2 k -1)/ 2 k  In C: (x + (1 > k Biases dividend toward 0 Case 1: No rounding Divisor: Dividend: u 2k2k /  u / 2 k  k Binary Point k +– Biasing has no effect

– 35 – Correct Power-of-2 Divide (Cont.) Divisor: Dividend: Case 2: Rounding x 2k2k /  x / 2 k  k Binary Point k +–1 1 Biasing adds 1 to final result Incremented by 1

– 36 – Fractional Binary Numbers Representation Bits to right of “binary point” represent fractional powers of 2 Represents rational number: bibi b i–1 b2b2 b1b1 b0b0 b –1 b –2 b –3 b–jb–j i–1 2i2i 1/2 1/4 1/8 2–j2–j

– 37 – Frac. Binary Number Examples ValueRepresentation 5-3/ / / Observations Divide by 2 by shifting right Multiply by 2 by shifting left Numbers of form … 2 just below 1.0 1/2 + 1/4 + 1/8 + … + 1/2 i + …  1.0

– 38 – Representable Numbers Limitation Can only exactly represent numbers of the form x/2 k Other numbers have repeating bit representations ValueRepresentation 1/ [01]… 2 1/ [0011]… 2 1/ [0011]… 2