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Number & codes (1)

- Digital vs. Analog
- Numbering systems
- Decimal (Base 10)
- Binary (Base 2)
- Hexadecimal (Base 16)
- Octal (Base 8)
- Number conversion
- Binary arithmetic
- 1’s and 2’s complements of binary numbers

- Signed/Unsigned numbers
- Arithmetic operations with signed numbers
- Coded
- Binary-Coded-Decimal (BCD)/ 8421
- ASCII
- Gray
- Excess-3)
- Error Detecting and Correction Codes
- Floating Point Numbers

Digital vs. Analog

- Two ways of representing the numerical values of quantities :

i) Analog (continuous)

ii) Digital (discrete)

- Analog : a quantity represented by voltage, current or meter movement that is proportional to the value that quantity.
- Digital : the quantities are represented not by proportional quantities but by symbols called digits (0/1).

Digital vs. Analog (cont.)

- Digital system:
- combination of devices designed to manipulate logical information or physical quantities that are represented in digital forms
- Analog system:
- contains devices manipulate physical quantities that are represented in analog forms

Digital vs. Analog (cont.)

- Why digital ?
- Problem with all signals – noise
- Noise isn\'t just something that you can hear - the fuzz that appears on old video recordings also qualifies as noise. In general, noise is any unwanted change to a signal that tends to corrupt it.
- Digital and analogue signals with added noise:

Digital : easily be recognized even

among all that noise : either 0 or 1

Analog : never get back a perfect copy of the original signal

Digital Techniques

- Advantages:
- Easier to design
- Information storage is easy
- Accuracy and precision are greater
- Operation can be programmed - simple
- Digital circuits less affected by noise
- More digital circuitry can be fabricated on IC chips
- Limitations:
- In real world there are analog in nature and these quantities are often I/O that are being monitored, operated on, and controlled by a system. Thus, conversion and re-conversion in needed

Introduction to Numbering Systems

- We are familiar with decimal number systems

for daily used such as calculator, calendar,

phone or any common devices use this

numbering system :

Decimal = Base 10

- Some other number systems:
- Binary = Base 2
- Octal = Base 8
- Hexadecimal = Base 16

Hex

Octal

Binary

000000010010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111

0123456789ABCDEF

0123456789101112131415

00010203040506071011121314151617

Numbering Systems (cont.)N

U

M

B

E

R

S

Y

S

T

E

M

S

Significant Digits

Binary : 1 0 1 1 0 1

Most Significant Bit Least Significant Bit

(MSB) (LSB)

Hexadecimal: 1 D 6 3 A 7

Most Significant Digit Least significant Digit

(MSD) (LSD)

Decimal numbering system (base 10)

Weights for whole numbers are positive power of ten that increase from right to left , beginning with 100

- Base 10 system: (0,1,2,3,4,5,6,7,8,9)
- Example : 39710

3 9 7

+

7 X 100

3 X 102

9 X 101

+

=>

300 + 90 + 7

39710

=>

Binary Number System (base 2)

- Base 2 system: (0 , 1)
- used to model the series of computer electrical signals represent the informations.
- 0 represents the no voltage or an ‘off’ state
- 1 represents the presence of voltage or an ‘on’ state
- Example: 1012

1 0 1

Weights in a binary number are based on power of two, that increase from right to right to left, beginning with 20

+

0 X 21

+

1 X 20

1X 22

=>

4 + 0 + 1

=>

510

Octal Number System (base 8)

- Base 8 system: (0,1,………,7)
- multiplication and division algorithms for conversion to and from base 10
- example: 7568 convert to decimal

7 5 6

Weights in a binary number are based on power of eight that increase from right to right to left, beginning with 80

+

7X 82

5 X 81

+

6 X 80

=>

448 + 40 + 6

49410

=>

- Readily converts to binary
- Groups of three (binary) digits can be used to represent each octal number
- example : 7568 convert to binary
- 7 5 6

1111011102

Hexadecimal Number System (base 16)

- Base 16 system
- Uses digits 0 ~ 9 &

letters A,B,C,D,E,F

- Groups of four bits represent each base 16 digit

Hexadecimal Number System (2)

- Base 16 system
- multiplication and division algorithms for conversion to and from base 10
- example : A9F16 convert to decimal

A 9 F

Weights in a hexadecimal number are based on power of sixteen that increase from right to right to left,beginning with 160

10X 162

9 X 161

+

15 X 160

+

=>

2560 + 144 + 15

271910

=>

- Readily converts to binary
- Groups of four (binary) digits can be used to represent each hexadecimal number
- example : A9F16 convert to binary
- A 9 F

1010100111112

Number Conversion

- Any Radix (base) to Decimal Conversion

Number Conversion (BASE 2 –> 10)

- Binary to Decimal Conversion

Binary to Decimal Conversion

- Convert (10101101)2 to its decimal equivalent:

Binary 1 0 1 0 1 1 0 1

Positional Values

x

x

x

x

x

x

x

x

27

26

25

24

23

22

21

20

Products

128 + 0 + 32 + 0 + 8 + 4 + 0 + 1

= 17310

Octal to Decimal Conversion

- Convert 6538 to its decimal equivalent:

Octal Digits

6 5 3

x

x

x

Positional Values

82 81 80

Products

384 + 40 + 3

= 42710

Hexadecimal to Decimal Conversion

- Convert 3B4F16 to its decimal equivalent:

Hex Digits

3 B 4 F

x

x

x

x

Positional Values

163 162 161 160

12288 + 2816 +64 +15

Products

= 15,18310

Number Conversion

- Decimal to Any Radix (Base) Conversion
- INTEGER DIGIT:

Repeated division by the radix & record the remainder

- FRACTIONAL DECIMAL:

Multiply the number by the radix until the answer is in integer

- example:

25.3125 to Binary

Decimal to Binary Conversion

Remainder

2 5 = 12 + 1

2

1 2 = 6 + 0

2

6 = 3 + 0

2

3 = 1 + 1

2

MSB LSB

1 = 0 + 1

2 2510 = 1 1 0 0 1 2

Decimal to Binary Conversion

MSB

LSB

Carry . 0 1 0 1

0.3125 x 2 = 0.625 0

0.625 x 2 = 1.25 1

0.25 x 2 = 0.50 0

0.5 x 2 = 1.00 1

Answer: 1 1 0 0 1.0 1 0 1

Decimal to Octal Conversion

Convert 42710 to its octal equivalent:

427 / 8 = 53 R3 Divided by 8; R is LSD

53 / 8 = 6 R5 Divide Q by 8; R is next digit

6 / 8 = 0 R6 Repeat until Q = 0

6538

Decimal to Hexadecimal Conversion

Convert 83010 to its hexadecimal equivalent:

830 / 16 = 51 R 14

51 / 16 = 3 R3

3 / 16 = 0 R3

= E in Hex

33E16

Decimal to Octal Conversion

- Binary to Octal Conversion (vice versa)
- Grouping the binary position in groups of three starting at the least significant position.

Octal to Binary Conversion

- Each octal number converts to 3 binary digits

To convert 6538 to binary, just substitute code:

6 5 3

110 101 011

Example : Number Conversion

- Convert the following binary numbers to their octal equivalent (vice versa).
- 1001.11112
- 47.38
- 1010011.110112

Answer:

- 11.748
- 100111.0112
- 123.668

Binary to Hexadecimal Conversion

- Binary to Hexadecimal Conversion (vice versa)
- Grouping the binary position in 4-bit groups, starting from the least significant position.

Binary to Hexadecimal Conversion

- The easiest method for converting binary to hexadecimal is using a substitution code
- Each hex number converts to 4 binary digits

Number Conversion

- Example:
- Convert the following binary numbers to their hexadecimal equivalent (vice versa).
- 10000.12
- 1F.C16

Answer:

- 10.816
- 00011111.11002

Substitution Code (1)

Convert (010101101010111001101010)2 to hex using the 4-bit substitution code :

0101 0110 1010 1110 0110 1010

5 6 A E 6 A

= 56AE6A16

Substitution Code (2)

Substitution code can also be used to convert binary to octal by using 3-bit groupings:

010 101 101 010 111 001 101 010

2 5 5 2 7 1 5 2

= 255271528

Binary Addition

0 + 0 = 0 Sum of 0 with a carry of 0

0 + 1 = 1 Sum of 1 with a carry of 0

1 + 0 = 1 Sum of 1 with a carry of 0

1 + 1 = 10 Sum of 0 with a carry of 1

Example:

11001 111

+ 1101 + 11

100110 ???

Simple Arithmetic

Example:

5816

+ 2416

7C16

- Addition

Example:

100011002

+ 1011102

101 1 10102

- Substraction

Example:

10001002

- 1011102

101102

Binary Subtraction

0 - 0 = 0

1 - 1 = 0

1 - 0 = 1

10 -1 = 1 0 -1 with a borrow of 1

Example:

1011 101

- 111 - 11

100 ???

Binary Multiplication

0 X 0 = 0

0 X 1 = 0 Example:

1 X 0 = 0 100110

1 X 1 = 1 X 101

100110

000000

+ 100110

10111110

Binary Division

- Use the same procedure as decimal division

1’s complements of binary numbers

- Changing all the 1s to 0s and all the 0s to 1s

Example:

1 1 0 1 0 0 1 0 1 Binary number

0 0 1 0 1 1 0 1 0 1’s complement

****** same as applying NOT gate ******

2’s complements of binary numbers

- 2’s complement
- Step 1: Find 1’s complement of the number

Binary # 11000110

1’s complement 00111001

- Step 2: Add 1 to the 1’s complement

00111001

+ 1

00111010

Signed Magnitude Numbers

110010..

…00101110010101

Sign bit

31 bits for magnitude

0 = positive

1 = negative

***** This is your basic

Integer format

Sign numbers

- Left most is the sign bit
- 0 is for positive, and 1 is for negative
- Sign-magnitude

0 0 0 1 1 0 0 1 = +25

sign bit magnitude bits

- 1’s complement
- The negative number is the 1’s complement of the corresponding positive number
- Example:

+25 is 00011001 -25 is 11100110

Sign numbers

- 2’s complement
- The positive number – same as sign magnitude and 1’s complement
- The negative number is the 2’s complement of the corresponding positive number.

Example:

Express +19 and -19 in

i. sign magnitude

ii. 1’s complement

iii. 2’s complement

Digital Codes (1)

- BCD (Binary Coded Decimal) / 8421 Code
- Represent each of the 10 decimal digits (0~9) as a 4-bit binary code.

Example:

- Convert 15 to BCD.

1 5

0001 0101

- Convert 10 to binary and BCD.

Digital Codes (2)

- ASCII (American Standard Code for Information Interchange) Code
- Used to translate from the keyboard characters to computer language
- A world standard alphanumeric code for microcomputers and computers
- A 7-bit code representing 27 (128) diff. characters (26 upper case, 26 lower case, 10 numbers, 33 special characters/symbol, 33 ctrl characters
- 8-bit version ASCII (USACC-II 8 or ASCII-8) represent max. of 256 characters.

Digital Codes (3)

- The Gray Code
- Only 1 bit changes
- Can’t be used in arithmetic circuits
- Can convert from Binary to Gray Code and vice versa.
- How to convert ?????

Digital Codes (4)

- Excess-3 Code
- Used to express decimal numbers.
- The code derives its name from the fact that each binary code is the corresponding 8421 code plus 3

Digital Codes (6)

- Error Detecting and Correction Code
- Required for reliable transmission and storage of digital data.
- Error Detecting Codes
- Parity (Even and Odd)
- Check sums
- Error Correcting Codes
- Hamming Code ????

**** Assingment#1: due date 10/01/11 ****

Digital Codes (7)

- EBCDIC (Extended Binary Coded Decimal Interchange) Code
- Mainly used with large computer systems like mainframe.
- An 8-bit code and accommodates up to 256 characters
- Divided into 2 portions:

4 zone bits (on the left) and 4 numeric bits (on the right)

Floating Point Numbers (FPN)

- A real number or FPN is a number which has both an integer and a fractional part.
- Examples:
- Real decimal numbers: 123.45, 0.1234, -0.12345
- Real binary numbers: 1100.1100, 0.1001, -1.001
- Generally, FPNs are expressed in exponential notation. Eg:
- 30000.0 can be written as 3 x 104
- 312.45 can be written as 3.1245 x 102
- 1010.001 can be written as 1.010001 x 103

mantissa exponent

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