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Chapter 1

Binary Systems

September 2005

EEA051 - Digital Logic

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Chapter 1. Binary Systems

1-1 Digital Systems

1-2 Binary Numbers1-3 Number Base Conversions

1-4 Octal and Hexadecimal Numbers

1-5 Complements

1-6 Signed Binary Numbers

1-7 Binary Codes1-8 Binary Storage and Registers

1-9 Binary Logic

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Chapter 1. Binary Systems

Presents the various binary systems

suitable for representing information indigital systems

The binary number system is explainedand binary codes are illustrated

Examples are given for addition and

subtraction of signed binary numbers anddecimal numbers in BCD

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Analog vs. digital

analog continuous digital discrete

the real world is mainly analog

Why digital? digital systems are easier to design

information storage is easy

accuracy and precision is better operation can be programmed

digital circuits are less affected by noise

Example: digital camera

1-1 Digital Systems

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Digital age

Digital systems

telephone switching exchanges

digital camera

electronic calculators, PDA's

digital TV, digital broadcast

Digital computers

many scientific, industrial and commercial applications

Generality

Discrete information-processing systems

Digital Systems

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(ADC)

(DAC)

Typical Control System

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Representing Binary Quantities

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Digital Signals and Timing Diagrams

Signals: physical quantities, e.g. voltages and currents, torepresent discrete elements of information in a digital system predominately implemented by transistors most use just two discrete values, said to be binary

A binary digit, called a bit, has two values: 0 and 1Binary codes: groups of bitsWhy binary?

reliability: a transistor circuit is either on or off (two stable states)

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1-2 Binary Numbers

Numbers system: ana3a2a1a0.a-1a-2a-3 a-m

Decimal number (base or radix = 10) (10 digits)

7,392 = 7*103 + 3*102 + 9*101 + 2*100

Binary number (base = 2)

(11010.11)2 = (26.75)10

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Base-r System

Base-r system (coefficients multiplied by powers of r)

(4021.2)5, (127.4)8, (B65F)16

Base-r Decimal (4021.2)5 = (511.4)10

Octal (127.4)8 = (87.5)10

Hexadecimal (B65F)16=(46,687)10

Binary (110101)2=(53)10

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Binary Numbers

Powers of Two

K(kilo)=210, M(mega)=220, G(giga)=230, T(tera)=240

20, 21, 22, 23, 24, 25, 26, 27, 28,

110101, 100111

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Binary Arithmetic Operations

Arithmetic operations with numbers in base r follow

the same rules as for decimal numbers(discussed later)

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1-3 Number Base Conversions

Decimal Base-r: converting a decimal numberto a number in base r(four examples)

1. Convert decimal 41 to binary: (101001)2

2. Convert decimal 153 to octal: (231)8

3. Convert (0.6875)10 to binary: (0.1011)2

4. Convert (0.513)10 to octal: (0.406517)8

Combining:

(41.6875)10 = (101001.1011)2

(153.513)10 = (231.406517)8

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1-4 Octal and Hexadecimal Numbers

Binary to octal: 23=8

Binary to hexadecimal: 24=16

Octal to binary

Hexadecimal to binaryOctal or hexadecimal representation is more

desirable

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Binary

Octal

Hexadecimal

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1-5 Complements

Used for simplifying the subtraction operation

and for logical manipulation

Two types of complementdiminished radix complement: (r-1)s complement

(rn-1)-N

radix complement: rs complementrn-N

Decimal number

10s complement and 9s complement

Binary number2s complement and 1s complement

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Examples

Diminished Radix Complement

The 9s complement of 546700 is 999999 546700 = 453299

The 9s complement of 012398 is 999999 012398 = 987601The 1s complement of 1011000 is 0100111

The 1s complement of 0101101 is 1010010

Radix Complement

The 10s complement of 546700 is 453300

The 10s complement of 012398 is 987602The 2s complement of 1101100 is 0010100

The 2s complement of 0110111 is 1001001

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Subtraction with rs ComplementsSubtraction of two n-digit unsigned numbers M-N in

base r:

1. Add M to the rs complement of the subtrahend, N:M + (rn N) = sum

2. If M N, the sum will produce an end carry, rn,

which can be discardedM + (rn N) = sum = (M N) + rn,

so M-N = sum - rn

3. If M < N, the sum is the rs complement of (N-M)M + (rn N) = sum = rn (N-M),

so M-N = -(rn-sum)

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Subtraction with (r-1)s Complements

Subtraction of two n-digit unsigned numbers M-N in

base r: Add M to the (r-1)s complement of subtrahend N:

M + ((rn-1) N) = sum

If M N, the sum will produce an end carry, rn,which can be discarded

M + ((rn-1) N) = sum = (M N) + (rn-1),

so M-N = sum rn + 1 (end-around carry) If M < N, the sum is the rs complement of (N-M)

M + ((rn-1) N) = sum = (rn-1) (N-M),

so M-N = -((rn -1) - sum)

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Examples

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1-6 Signed Binary NumbersTable 1-3

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Arithmetic Addition

Arithmetic Subtraction(A) (+B) = (A) + (B)

(A) (B) = (A) + (+B)

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1.7 Binary Codes

n-bit binary code

2n distinct combinations

BCD Binary Coded Decimal (4-bit)

(185)10 = (0001 1000 0101)BCD = (10111001)2

(396)10 = (0011 1011 0110)BCD

BCD addition

Get the binary sum

If the sum > 9, add 6 to the sum

Obtain the correct BCD digit sum and a carry

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Number Systems and BCD Code

15

14

13

1211

10

9

8

7

6

5

4

32

1

0

Decimal

1111

1110

1101

11001011

1010

1001

1000

111

110

101

100

1110

1

0

Binary

0001 0011D15

0001 0100E16

0001 0101F17

0001 0010C140001 0001B13

0001 0000A12

1001911

1000810

011177

011066

010155

010044

001133001022

000111

000000

BCDHexadecimalOctal

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BCD Addition

184 + 576 = 760

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Other Decimal Codes

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Gray Code

only one bit change

between two consecutive

numbers

useful in Analog-to-DigitalConverter

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ASCIICharacter

Code

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Error-Detecting Code

Parity bit: an extra bit included with a message tomake the total number of 1s either even or odd

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1.8 Binary Storage and Registers

A binary cell two stable state

store one bit of information examples: flip-flop circuits, ferrite cores,

capacitor

A register a group of binary cells e.g. AX in x86 CPU

Register Transfer a transfer of the information stored in one

register to another one of the major operations in digital system

an example

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Transfer of information

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The other major component of a digital system circuit elements to manipulate individual bits of information

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1.9 Binary Logic

Binary Logic Boolean algebra consists of binary variables and logical operations

Binary variables two discrete values (true/false; yes/no; 1/0) Logical operations

Three basic operations: AND, OR, NOT

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Binary signalsElectrical signals: voltages or

currents

two separate voltage levels: logic-1and logic-0

the intermediate region is crossedonly during state transition

Logic circuits

circuits = logical manipulation paths

Computation and control

combinations of logic circuitsLogic gates

electronic circuits that operate onone or more input signals to produce

an output signal

Logic Gates

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SummaryChapter 1. Binary Systems

1-1 Digital Systems1-2 Binary Numbers

1-3 Number Base Conversions

1-4 Octal and Hexadecimal Numbers1-5 Complements

1-6 Signed Binary Numbers

1-7 Binary Codes1-8 Binary Storage and Registers

1-9 Binary Logic