Lecture 1

DIGIDESREVIEW: NUMBER SYSTEMS

June 3, 2010

INTRODUCTION

� A digital computer manipulates discrete elements of information and these elements are represented in binary form.

� Operands used for calculations are expressed in binary number system. Operations are carried Operands used for calculations are expressed in binary number system. Operations are carried out by means of binary logic. Quantities are stored in binary storage elements.

� The purpose of this discussion is to review various binary concepts as a frame of reference for future detailed study of digital logic systems.

NUMBER SYSTEMNumber Representation

� General representation of a number from a base-R system

a4a3a2a1a0.a-1a-2a-3

where:

aj : coefficient with possible values of 0 to r-1

Number System

Number Base Conversion

Integer Representation

Binary Codes, aj : coefficient with possible values of 0 to r-1

i : place value

r : base/radix

� A number expressed in base-r system has coefficients multiplied by powers of r

anrn+an-1rn-1+..+a2r2+a1r+a0

+a-1r-1+a-2r-2+.. +a-(n-1)r-(n-1)+a-nr

-n

Binary Codes, Storage and Registers

Binary Logic

NUMBER SYSTEMCommonly Used Number Systems : Binary, Octal, Hexadecimal

Type of Number

System

Base/radix Possible Values

of Coefficients

Binary 2 0, 1

Octal 8 0, 1, 2, 3, 4, 5, 6,

Number System



Binary Codes, Octal 8 0, 1, 2, 3, 4, 5, 6, 7

Hexadecimal 16 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F


Binary Logic

4401004

3300113

2200102

1100011

0000000

Hexadecimal

(Base 16)

Octal

(Base 8)

Binary

(Base 2)

Decimal

(Base 10)

Number System



Binary Codes,

NUMBER SYSTEMCommonly Used Number Systems : Binary, Octal, Hexadecimal

F17111115

E16111014

D15110113

C14110012

B13101111

A12101010

91110019

81010008

7701117

6601106

5501015

4401004Binary Codes, Storage and Registers

Binary Logic

NUMBER SYSTEMArithmetic Operations

� Arithmetic operations with numbers in base- r follow the same rules as for decimal

numbers.

� Only the r allowable digits are used.

Number System



Binary Codes,

� Only the r allowable digits are used.Binary Codes, Storage and Registers

Binary Logic

NUMBER SYSTEMSample Problems

� Write the first 20 decimal digits in base-30, 1, 2, 10, 11, 12, 20, 21, 22, 100,

101, 102, 110, 111, 112, 120, 121, 122,

200, 201,…

Number System



Binary Codes,

� Add the following numbers in the given base without converting to decimal(1230)4 + (23)4 = (1313)4

(135.4)6 + (43.2)6 = (223.0)6


Binary Logic

NUMBER BASE CONVERSIONSample Problems

Base-10 to Base-2, -8, and -16

� (41.6875)10 = (101001.1011)2

� (153.513)10 = (231.406517)8

� (35.75)10 = (23.C)16

Number System



Binary Codes, � (35.75)10 = (23.C)16


� (1010.011)2 = (10.375)10

� (1101011.1111)2 = (153.74)8

� (1011001011.1111001)2 = (2CB.F2)16


Binary Logic

NUMBER BASE CONVERSIONSample Problems


� (673.124)8 = (110111011.001010100)2

� (630.4)8 = (408.5)10

� (261.12)8 = (B1.28)16

Number System



Binary Codes, � (261.12)8 = (B1.28)16


� (306.D)16 = (1100000110.1101)2

� (1A.B)16 = (26.6875)10

� (F2E.A)16 = (7456.5)8


Binary Logic

NUMBER BASE CONVERSIONChallenge

� Convert the decimal number 250.5 to base-3, base-4, and base-7.(100021.1111)3

(3322.2)4

(505.33333)7

Number System



Binary Codes, (505.33333)7

� Convert the following numbers to decimal(12121)3 = (151)10

(8.3)9 = (8.33333)10

(198)12 = (260)10


Binary Logic

NUMBER BASE CONVERSIONChallenge

� Convert the decimal number 250.5 to base-3, base-4, and base-7.(100021.1111)3

(3322.2)4

(505.33333)7

Number System



Binary Codes, (505.33333)7

� Convert the following numbers to decimal(12121)3 = (151)10

(8.3)9 = (8.33333)10

(198)12 = (260)10


Binary Logic

Integer Representation (Complements)

� Complements are used in digital computers to simplify subtraction and for logical manipulations.

Number System



Binary Codes,

� 2 types of complements for each base-r system:

� r’s complement

� (r-1)’s complement


Binary Logic

Integer Representation (Complements) The r’s Complement

� Given a positive number N in base-r with an integer part of n digits, the r’scomplement of N is defined as rn – N for N

≠ 0 and 0 for N = 0.

Number System



Binary Codes, ≠ 0 and 0 for N = 0.Binary Codes, Storage and Registers

Binary Logic

Decimal Numbers

10’s complement of 52520 is = 105 – 52520 = 47480

10’s complement of 0.3267 is

Binary Numbers

2’s complement of 101100 is = (26)10 – (101100)2= 1000000 – 101100= 010100


Number System



Binary Codes,

Integer Representation (Complements) The r’s Complement

10’s complement of 0.3267 is= 100 – 0.3267= 0.6733

10’s complement of 25.639 is= 102 – 25.639= 74.361

2’s complement of 0.0110 is= (20)10 – (0.0110)2= 1 – 0.0110= 0.1010

2’s complement of 10.001 is= (22)10 – (10.001)2= 100 – 10.001= 1.111


Binary Logic

� Given a positive number N in base-r with an integer part of n digits and a fraction part of m digits, the (r-1)’s complement of N is defined as rn – r-m – N.

Number System



Binary Codes,

Integer Representation (Complements) The (r-1)’s Complement

N is defined as r – r – N.Binary Codes, Storage and Registers

Binary Logic

Decimal Numbers

9’s complement of 52520 is = 105 – 100 – 52520 = 99999 – 52520 = 47479

9’s complement of 0.3267 is= 100 – 10-4 – 0.3267

Binary Numbers

1’s complement of 101100 is = (26)10 – (20)10 –

(101100)2= 111111 – 101100= 010011


Number System



Binary Codes,

Integer Representation (Complements) The (r-1)’s Complement

= 100 – 10-4 – 0.3267 = 0.9999 – 0.3267 = 0.6732

9’s complement of 25.639 is= 102 – 10-3 – 25.639 = 99.999 – 25.639= 74.360

1’s complement of 0.0110 is= (20)10 – (2-4)10 – (0.0110)2= 0.1111 – 0.0110= 0.1001

1’s complement of 10.001 is= (22)10 – (2-3)10 – (10.001)2= 11.111 – 10.001= 1.11


Binary Logic

Integer Representation (Complements) Subtraction with r’s Complement

� Add the minuend M to the r’s complement of the subtrahend N.

� Inspect the result obtained in step 1 for an end carry:

Number System



Binary Codes, end carry:

� If an end carry occurs, discard it.

� If an end carry does not occur, take the r’scomplement of the number obtained in step 1 and place a negative sign in front.


Binary Logic

Number System



Binary Codes,

Using 10’s complement, subtract 72532 – 3250

Using 10’s complement, subtract 3250 - 72532

Integer Representation (Complements) Subtraction with r’s Complement


Binary Logic

NUMBER BASE CONVERSIONSubtraction with r’s Complement

Number System



Binary Codes,

Using 2’s complement,

subtract

1010100 - 1000100

Using 2’s complement, subtract

1000100 - 1010100


Binary Logic

� Add the minuend M to the (r-1)’s complement of the subtrahend N.

� Inspect the result obtained in step 1 for an end carry:

Number System



Binary Codes,

Integer Representation (Complements) Subtraction with (r-1)’s Complement

end carry:

� If an end carry occurs, add 1 to the least significant digit (end-around carry).

� If an end carry does not occur, take the (r-1)’s complement of the number obtained in step 1 and place a negative sign in front.


Binary Logic

Number System



Binary Codes,


Using 9’s complement, subtract 72532 – 3250

Using 9’s complement, subtract 3250 - 72532


Binary Logic

Number System



Binary Codes,


Using 1’s complement,

subtract

1010100 - 1000100

Using 1’s complement, subtract

1000100 - 1010100


Binary Logic

� Binary codes

� used to represent any discrete element of information distinct among a group of quantities

� Binary codes merely change the symbols and

Number System



Binary Codes,

Binary Codes, Storage and RegistersBinary Codes

Binary codes merely change the symbols and

not the meaning of the elements of information that they represent.

� In representing a group of 2n distinct elements in a group, a minimum of n bits must be adopted.


Binary Logic

Number System



Binary Codes,

Binary Codes, Storage and RegistersBinary Codes

010 010000100110010100102

010 001000010111010000011

010 000100000000001100000

Biquinary

504 3210242184-2-1Excess-3

BCD

(8421)

Decimal

Digit


Binary Logic

101 000011111111110010019

100 100011101000101110008

100 010011011001101001117

100 001011001010100101106

100 000110111011100001015

011 000001000100011101004

010 100000110101011000113

Number System



Binary Codes,

Binary Codes, Storage and RegistersBinary Codes: Error Detection Code

010101

100100

010011

100010

100001

010000

Even ParityOdd ParityMessage


Binary Logic

011111

101110

101101

011100

101011

011010

011001

101000

100111

010110

010101

Number System



Binary Codes,

Binary Codes, Storage and RegistersBinary Codes: Gray Code (Reflected Code)

01115

01104

00103

00112

00011

00000

Reflected CodeDecimal Number


Binary Logic

100015

100114

101113

101012

111011

111110

11019

11008

01007

01016

01115

Number System



Binary Codes,

Binary Codes, Storage and RegistersBinary Codes: Alphanumeric Codes

� 6-Bit internal code“A” 010 001 “1” 000 001

“B” 010 010 “5” 000 101

� 7-Bit ASCII code“A” 100 0001 “1” 011 001


Binary Logic

“A” 100 0001 “1” 011 001

“B” 100 0010 “5” 011 101

� 8-Bit EBCDIC code“A” 1100 0001 “1” 1111 0001

“B” 1100 0010 “5” 1111 0101

� Register

� a group of binary cells used to store and hold binary information

� A register with n cells can store any discrete quantity of information that contains n-bits.

Number System



Binary Codes,

Binary Codes, Storage and RegistersRegisters

quantity of information that contains n-bits.

� The state of a register is an n-tuple number of 1’s and 0’s with each bit designating the state of one cell in the register.

� The content of a register is a function of the interpretation given to the information stored in it.


Binary Logic

Given a 16-cell register:Number System



Binary Codes,

Binary Codes, Storage and RegistersRegisters

16151413121110987654321

1001001111000011

� State: (16-tuple number) 1100001111001001

� Content:

Binary equivalent (50121)10

Alphanumeric code (EBCDIC) C, I

Excess-3 code 9096


Binary Logic

Number System



Binary Codes,

Binary Codes, Storage and RegistersRegisters: Register Transfer


Binary Logic

Number System



Binary Codes,

Binary Codes, Storage and RegistersRegisters: Register Transfer


Binary Logic

Number System



Binary Codes,

Binary Logic

� Binary Logic

� is used to describe, in a mathematical way, the manipulation and processing of binary information

Consists of binary variables and logical Binary Codes, Storage and Registers

Binary Logic

� Consists of binary variables and logical operations

� Variables are designated by letters of the alphabet such as A, B, C, x, y, z, etc. with each variable having only two distinct values: 1 and 0

Number System



Binary Codes,

Binary LogicDefinition

� Binary Logic

� is used to describe, in a mathematical way, the manipulation and processing of binary informationBinary Codes,

Storage and Registers

Binary Logic

� Consists of binary variables and logical operations

� Variables are designated by letters of the alphabet such as A, B, C, x, y, z, etc. with each variable having only two distinct values: 1 and 0

Number System



Binary Codes,

Binary LogicLogical Operations

� Three basic logical operations are: AND, OR, and NOT


Binary Logic

� Truth table

• a table of all possible combinations of the variables showing the relation between the values that the variables may take and the result of the operation

Lecture 1

Documents

binary base

binary number system

base r

registers binary logic

baser system

octal base

given base

hexadecimal base