4 - 1 binary codes computers and other digital systems "work" with binary numbers. i/p...

4 - 1

Binary Codes

• Computers and other digital systems "work" with binary numbers.• I/P & O/P is usually done using decimal numbers, alphabetics,

special symbols.• Some way of representing alphanumerics with binary numbers is

required.• These representations are called codes.• Many codes are possible, and a few standard codes are used, such as:

ASCII - American Standard Code for Information Interchange

 EBCDIC - Extended Binary Coded Decimal

Interchange Code  BCD - Binary Coded Decimal. For numbers only. Hardware and/or software is required to convert coded numbers into binary numbers before any arithmetic operations can take place.

7-bit System

Used in Big Mainframe Systems

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Alphanumeric Character codes

Character 6-bit internal code ASCII code 8-bit EBCDIC

A 010 001 100 0001 1100 0001

B 010 010 100 0010 1100 0010

C 010 011 100 0011 1100 0011

D 010 100 100 0100 1100 0100

E 010 101 100 0101 1100 0101

F 010 110 100 0110 1100 0110

G 010 111 100 0111 1100 0111

H 011 000 100 1000 1100 1000

I 011 001 100 1001 1100 1001

J 100 001 100 1010 1101 0001

…………

…………………

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ASCII7-bit

Codes

Character ASCII Code Character ASCII Code

@ 1000000 FORM FEED 0001100 A 1000001 CARRIAGE RETURN 0001101 B 1000010 RUBOUT 1111111 C 1000011 SPACE 0100000 D 1000100 ! 0100001 E 1000101 " 0100010 F 1000110 # 0100011 G 1000111 $ 0100100 H 1001000 % 0100101 I 1001001 & 0100110 J 1001010 ' 0100111 K 1001011 ( 0101000 L 1001100 ) 0101001 M 1001101 * 0101010 N 1001110 + 0101011 O 1001111 , 0101100 P 1010000 - 0101101 Q 1010001 . 0101110 R 1010010 / 0101111 S 1010011 0 0110000 T 1010100 1 0110001 U 1010101 2 0110010 V 1010110 3 0110011 W 1010111 4 0110100 X 1011000 5 0110101 Y 1011001 6 0110110 Z 1011010 7 0110111 [ 1011011 8 0111000 \ 1011100 9 0111001 ] 1011101 : 0111010 | 1011110 ; 0111011

NULL 0000000 < 0111100 HORIZ TAB 0001001 = 0111101 LINE FEED 0001010 > 0111110 VERT TAB 0001011 ? 0111111

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Binary Codes for Decimal Numbers

Decimal

digit

8421

(BCD)

6311 Excess-3 2-out-of-5 Gray

0 0000 0000 0011 00011 0000

1 0001 0001 0100 00101 0001

2 0010 0011 0101 00110 0011

3 0011 0100 0110 01001 0010

4 0100 0101 0111 01010 0110

5 0101 0111 1000 01100 1110

6 0110 1000 1001 10001 1010

7 0111 1001 1010 10010 1011

8 1000 1011 1011 10100 1001

9 1001 1100 1100 11000 1000

Weighted codes: 8421, 6311, Excess-3 Non-weighted codes: 2-out-of-5, Gray

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Binary Codes for Decimal Numbers (cont.)

BCD - Convert decimal numbers to binary code, digit by digit

(at least bits required).

8421 code: 95.16

6311 code: 925

4 (for each decimal digit)

1001 0101 . 0001 0110

9 5 1 6

1100 0011 0111

By looking up the previous table

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The Meaning of Data

e.g.: Consider the following 16-cell register

If one assumes that the content of the register represents a binary integer, the decimal number is:

1100001111001001 = If one assumes an 8-bit EBCDIC code, the two characters are: In excess-3 code:  In BCD code: The same bit configuration may be interpreted differently for different types of elements of information. The computer must be programmed to process this information according to the type of information stored.

1 1 0 0 0 0 1 1 1 1 0 0 1 0 0 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

5012110

C I

9096

??? Meaningless, why???

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Boolean Algebra

• George Boole (1815-1864) applied a set of symbols to logical operations.

• Digital electronics applies his set theory and logic to (binary) switching networks.

• Binary number system is used to represent the two possible states of our systems.

 The symbols 0 & 1 are used to represent:  True or False  Flow or No Flow  Open or Closed  Voltage1 or Voltage2  etc.

word statementscurrents, fluidsswitches, doors, etc.

anything with 2 states

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Boolean Algebra

Deals with manipulation of Variables & Constants

Boolean Variables, such as X, Y, Z, A, B, C, etc. can have "values" of either 0 or 1.

 0 & 1 are constants & are symbols only, representing two different states of a quantity.

i.e.

F or T

Low voltage or high voltage, usually written L or H

Flow or not flow

e.g. 0V logical 0

+5V logical 1

or 0V 1

+5V 0

+ ve logic

- ve logic

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Basic Operations

Only 3

e.g. Not 1 is written as: 

Not :  X and Y :  X or Y :  1 or X : If the variables represent voltages of the I/P or O/P of a switching

network, we symbolically represent these operations by: 

NOT

If O/P is called C, we write:

X X '

NOT (compliment or invert)

AND

OR1´ or 1

X ´ or XX • YX + Y1 + X

C = X´

inversion symbol or “bubble

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Boolean Operations (cont.)

AND

OR

where values for X, A, B, C are . They actually correspond to two different voltage levels when realized electronically. 

e.g.

Characteristics of

an Inverter

0 or 1

0 & 5V; -12V & 0V, etc.

if X = 0 C = 1

if X = 1 C = 0

Truth Table

X C

0 1

1 0

AB

C = A + B+

B

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Boolean Operations (cont.)

AND gate A B C

OR gate A B C

0 0

0 1

1 0

1 1

0 0 00 1 01 0 01 1 1

0

111

LogicalMultiplication

Also called Inclusive OR

LogicalAddition

A

BC = A ● B

A

BC = A + B