Understanding Number Systems and Codes for Digital Designers

EECE 320Digital Systems DesignLecture 2: Number Systems and Codes Ali Chehab

Number Systems and Codes • A digital designer must establish a correspondence between binary digits and real-life numbers, events and conditions

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

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

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

Result: 4610 = 1011102 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

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

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 • Binary to Hexadecimal: Similar, but in groups of four. • Example: 1 1 0 1 1 1 0 0 1 0 1 1 1 0 • 3 7 2 E • Result: 110111001011102 = 372E16

Addition and Subtraction of Binary Numbers 190 10111110 +141 10001101 331 101001011 229 11100101 - 46 00101110 183 10110111

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 = 01 0 1 and –5 = 11 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

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:

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.

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

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?

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

Representation of Negative Numbers +7 +6 +5 +4 +3 +2 +1 +0 -0 -1 -2 -3 -4 -5 -6 -7 -8

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.

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 Cin1 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

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

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

Next Lecture and Reminders • Next lecture • Reminders

Understanding Number Systems and Codes for Digital Designers