EECE 320 Lecture 2, Fall 2011

EECE 320 L2: Number Systems and Codes 1 Chehab, AUB, 2003 EECE 320 Digital Systems Design Lecture 2: Number Systems and Codes Ali Chehab EECE 320 L2: Number Systems and Codes 2 Chehab, AUB, 2003 Number Systems and Codes  A digital designer must establish a correspondence between binary digits and real-life numbers, events and conditions EECE 320 L2: Number Systems and Codes 3 Chehab, AUB, 2003 Positional Number Systems  A number is represented by a string of digits  Each digit has an associated weight  Number = weighted sum of digits  EX1: 534 = 5x100 + 3x10 + 4x1 - The weights are power of 10: 100, 10, 1  EX2: 85.28 = 8x10 + 5x1 + 2x0.1 + 8x0.01 - We have positive and negative powers - This system had a base (radix, r) of 10  General form of a number is: - dp-1dp-2 … d1d0.d-1d-2 … d-n  The vule of the number is: - dp-1 = Most significant digit and d-n = least significant digit       1 2 p n i i i where r d r D EECE 320 L2: Number Systems and Codes 4 Chehab, AUB, 2003 Positional Number Systems  Binary: radix = 2  Used to represent numbers in a digital system  Reliable since only 2 values need to be distinguished  EX: 110.01 = 1x4 + 1x2 + 0x1 + 0x0.5 + 1x0.25 = 6.2510  In general the value is given by:  Octal and Hexadecimal represent a shorthand notation for the binary system • Octal (Base 8): { 0, 1, 2, 3, 4, 5, 6, 7 } – Example: 576 = 6*80 + 7*81 + 5*82 = 38210 • Hexadecimal (Base 16): { 0, …, 9, A, B, C, D, E, F } – Example: A2F = F*160 + 2*161 + A*162 = 15*160 + 2*161 + 10*162 = 260710      1 2 p n i i ib B EECE 320 L2: Number Systems and Codes 5 Chehab, AUB, 2003 General Positional Number System Conversions  To convert a number D to decimal, we convert each digit to radix-10 equivalent and expand the weighted-sum formula using radix-10 arithmetic  EX1: 1CE8.116 = 1x163 + 12x162 + 14x16 + 8x1 + 1x16-1  = 7400.062510  EX2: F1AC16 = ((15x16 + 1)x16 + 10)x16 + 12 = 6186810 EECE 320 L2: Number Systems and Codes 6 Chehab, AUB, 2003 General Positional Number System Conversions  Decimal to Binary: It is done by successive divisions until a zero quotient is obtained. The remainders taken in reverse order constitute the binary representation Example: Convert 46 Decimal into Binary 46 / 2 = 23 and Remainder = 0 23 / 2 = 11 and Remainder = 1 11 / 2 = 5 and Remainder = 1 5 / 2 = 2 and Remainder = 1 2 / 2 = 1 and Remainder = 0 1 / 2 = 0 and Remainder = 1 Result: 4610 = 1011102 EECE 320 L2: Number Systems and Codes 7 Chehab, AUB, 2003 General Positional Number System Conversions  Decimal fraction to binary: It is done by successive multiplications taking out each time the integer part of the result.  Example: convert 0.3125 to binary  0.3125 x 2 = 0.625  0  0.625 x 2 = 1.25  1  0.25 x 2 = 0.5  0  0.5 x 2 = 1  1  0.312510 = 01012 EECE 320 L2: Number Systems and Codes 8 Chehab, AUB, 2003 General Positional Number System Conversions  Binary to Octal: Bits are grouped in “Threes” from right to left, and the binary value of every group represents the corresponding Octal Digit Example: 1 0 0 1 1 1 0 0 1 0 1 2 3 4 5 Result: 100111001012 = 23458 Example: 1 1 0 1 1 1 0 0 1 0 1 1 1 0 3 7 2 E Result: 110111001011102 = 372E16  Binary to Hexadecimal: Similar, but in groups of four. EECE 320 L2: Number Systems and Codes 9 Chehab, AUB, 2003 Addition and Subtraction of Binary Numbers 190 10111110 +141 10001101 331 101001011 229 11100101 - 46 00101110 183 10110111 Cin or Bin X Y Cout S Bout D 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 1 0 0 1 0 1 0 1 1 1 0 0 0 1 0 0 0 1 1 1 1 0 1 1 0 1 0 1 1 0 1 0 0 0 1 1 1 1 1 1 1 EECE 320 L2: Number Systems and Codes 10 Chehab, AUB, 2003 Representation of Negative Numbers  Negative integers can be represented using sign/magnitude, one’s complement or two’s complement notations  Sign and Magnitude: Leftmost bit is used to indicate the sign - 0  positive and 1  negative - Example using 4 bits: +5 = 0 1 0 1 and –5 = 1 1 0 1 - Range: from -7 to +7 (from 1111 to 0111) (-(2N-1-1) to 2N-1-1) - Disadvantages – two representations of 0 (0000, 1000) – The process of addition needs a relatively complex adder EECE 320 L2: Number Systems and Codes 11 Chehab, AUB, 2003 Complements  Complements are used in digital systems to negate a number and to simplify the subtraction operation.  Two types of complements for any base-r system:  Radix complement: called r’s complement  Diminished radix complement: called (r-1)’s complement  In base ten: 10’s complement, 9’s complement  In base 2: 2’s complement, 1’s complement  Diminished radix: Given a number N in base r having n digits, the (r-1)’s complement is defined as follows: EECE 320 L2: Number Systems and Codes 12 Chehab, AUB, 2003 Complements  EX: r=10, what is 10n?  It is a single ‘1’ followed by n 0s. EX: 104 = 1,0000(ten)  EX: r=10, what is 10n - 1?  It is n 9s. 104 -1 = 9999  EX: What is 9’s complement of 546700?  EX: r=2, what is 24? 10000  EX: r=2, what is 24-1? 1111  1’s complement: subtract each digit from 1. But 1-0=1, 1-1=0 => to get 1’s complement of a number, flip all its digits. EECE 320 L2: Number Systems and Codes 13 Chehab, AUB, 2003 Complements  Radix complement: Given a number N in base r having n digits, the r ’s complement is defined as follows:  Since  r’s complement is obtained by taking (r-1)’s complement and adding 1.  EX: 2’s complement of 101100 is  010011 + 1 = 010100  EX: 10’s complement of 012398 is  987601 + 1 = 987602 EECE 320 L2: Number Systems and Codes 14 Chehab, AUB, 2003 Complements  Short cuts:  10’s complement: leave all least significant 0s unchanged, subtract first nonzero least significant digit from 10, and subtract all higher significant digits from 9.  2’s complement: leave all least significant 0s unchanged, leave the first 1 unchanged, flip all other higher significant digits.  What is the r’s complement of the r’s complement of a number? EECE 320 L2: Number Systems and Codes 15 Chehab, AUB, 2003 Representation of Negative Numbers  Two’s Complement: A negative # is represented by complementing each bit of the corresponding positive # and adding 1.  MSB has a weight of -2N-1 instead of 2N-1  A negative number has MSB = 1  Example using 8 bits: +9 = 0 0 0 0 1 0 0 1  -9 = 1 1 1 1 0 1 1 1  Range is from –128 to +127 (10000000 to 01111111)  For N bits, the range is: -(2N-1) to +(2N-1-1)  Overflow and underflow occur when result is outside the range  1 representation of 0. Using 4 bits: 0000  Addition can be easily carried out: A carry will be ignored. - Example: to add –1 and –2 using 4 bits: 1 1 1 1 1 1 1 0 ---------- Ignored 1 1 1 0 1 = -3 EECE 320 L2: Number Systems and Codes 16 Chehab, AUB, 2003 Representation of Negative Numbers +7 +6 +5 +4 +3 +2 +1 +0 -0 -1 -2 -3 -4 -5 -6 -7 -8 EECE 320 L2: Number Systems and Codes 17 Chehab, AUB, 2003 2’s Complement Addition: Overflow  An overflow is said to occur if the result of the addition exceeds the range.  Addition of 2 numbers with different signs cannot produce an overflow  Examples  -3 1101 +5 0101  + -6 1010 + +6 0110  ---- -------- ----- -------  -9 1 0111 = +7 +11 1011 = -5  A simple rule to detect overflow is to check if the sign of the result is different from the sign of the addends. EECE 320 L2: Number Systems and Codes 18 Chehab, AUB, 2003 Subtraction Rules  Most subtraction circuits for 2’s complement numbers do not perform subtraction. Instead, we take the two’s complement of the subtrahend and add the 2 numbers: A – B = A + (-B)  We can complement bit by bit the subtrahend, added to the minuend and add a carry of 1  Examples  1  Cin 1  Cin  +4 0100 0100 +3 0011 0011  - +3 - 0011 + 1100 - +4 - 0100 + 1011  ---- -------- ------ ----- ------- ------  +1 1 0001 -1 1111  We can use the same rules of addition to detect overflow  We can subtract -8 as long as the result does not overflow  (-3) – (-8) = (1101) – (1000) = (1101) + (0111) + 1 = 1 0101 = +5 EECE 320 L2: Number Systems and Codes 19 Chehab, AUB, 2003 Binary Codes for decimal Numbers  Some digital devices process decimal numbers  The interface of a digital system may read or display decimal #s  A decimal number is represented by a string of at least 4 bits EECE 320 L2: Number Systems and Codes 20 Chehab, AUB, 2003 Binary Codes for decimal Numbers  Gray Code  Only 1 bit changes between each pair of consecutive codewords  It is a reflective code - For a 1 bit code  0 and 1 - For (N+1)-bit code, the first 2N are the same of N-bit code with a leading zero - The last 2N codewords are the reverse of N-bit code with a leading 1 EECE 320 L2: Number Systems and Codes 21 Chehab, AUB, 2003 Next Lecture and Reminders  Next lecture  Reminders