number systems and codes. cs 3402--digital logicnumber systems and codes2 outline number systems...

NUMBER SYSTEMS AND

CODES

CS 3402--Digital Logic Number Systems and Codes 2

Outline

• Number systems– Number notations– Arithmetic– Base conversions – Signed number representation

• Codes– Decimal codes– Gray code– Error detection code– ASCII code


Number Systems

The decimal (real), binary, octal, hexadecimal number systems are used to represent information in digital systems. Any number system consists of a set of digits and a set of operators (+, , , ).


Radix or Base

Decimal (base 10) 0 1 2 3 4 5 6 7 8 9

Binary (base 2) 0 1

Octal (base 8) 0 1 2 3 4 5 6 7

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

The radix or base of the number system denotes the number of digits used in the system.


Decimal Binary Octal Hexadecimal

00 0000 00 0

01 0001 01 1

02 0010 02 2

03 0011 03 3

04 0100 04 4

05 0101 05 5

06 0110 06 6

07 0111 07 7

08 1000 10 8

09 1001 11 9

10 1010 12 A

11 1011 13 B

12 1100 14 C

13 1101 15 D

14 1110 16 E

15 1111 17 F


Positional Notation

It is convenient to represent a number using positional notation. A positional notation is written as a sequence of digits with a radix point separating the integer and fractional part.

where r is the radix, n is the number of digits of

the integer part, and m is the number digits of the fractional part.

rmnnr aaaaaaaN 210121 .


Polynomial Notation

A number can be explicitly represented in polynomial notation.

where rp is a weighted position and p is the

position of a digit.

mm

nn

nnr rararararararaN

22

11

00

11

22

11


Examples

In binary number system In octal number system

In hexadecimal number system

321012342 2121202021202121011.11010

3210128 848281838786124.673

101216 16166160163.306 DD


Arithmetic

(101101)2 +(11101)2 : 1111 1

+ 101101 11101

1001010

Addition:

In binary number system,


Addition

(6254)8+(5173)8 : 1 1

+ 6254 5173

13447

In octal number system,

(9F1B)16 +(4A36)16 : 1 1

+ 9F1B 4A36

D951

In hexadecimal number system,


Subtraction

(101101)2 -(11011)2 : 10 10

- 101101 11011

10010



Subtraction

In octal number system,

In hexadecimal number system,

(6254)8 -(5173)8 : 8

- 6254 5173

1061

(9F1B)16 -(4A36)16 : 16

- 9F1B 4A36

54E5


Multiplication

(1101)2 (1001)2 :

1101 1001 1101 0000

00001101

1110101



Division

(1110111)2 (1001)2 :

1101

1001 1110111

1001

1011 1001

1011 1001

10



Base Conversions

Convert (100111010)2 to base 8

8

012

01112

0123456782

472828784

8281828484202120212121202021100111010

8

22

472274010111100100111010


Base Conversion


10

1010101010

0123456782

314281632256

202120212121202021100111010


Base Conversion


16

012

00112

0123456782

1316163161

162168161162161202120212121202021100111010

AA

16

22

1331101000110001100111010

AA


Base Conversion from base 8

• Convert (372)8 to base 2



2

8

11111010010111011273372

10

101010

0128

250256192828783372

16

28 10101111372FAAF


Base Conversion from base 16

• Convert (9F2)16 to base 2



2

16

1010011111000010111110012929

FF

82

16

4762)2

0106

1107

1114

100(0010111110012929

FF

10

101010

01216

254622402304 1621616929

FF


Binomial expansion (series substitution)

To convert a number in base r to base p.– Represent the number in base p in binomial

series.– Change the radix or base of each term to base p.– Simplify.


Convert Base 10 to Base rConvert (174)10 to base 8

Therefore (174)10 = (256)8

8 1 7 4 6 LSB 8 2 1 5 8 2 2 MSB 0 0


Convert Base 10 to Base rConvert (0.275)10 to base 8

Therefore (0.275)10 = (0.21463)8

8 0.275 2.200 MSD8 0.200 1.600 8 0.600 4.800 8 0.800 6.400 8 0.400 3.200 LSD


Convert Base 10 to Base rConvert (0.68475)10 to base 2

Therefore (0.68475)10 = (0.10101)2

2 0.68475 1. 3695 MSD2 0.3695 0.7390 2 0.7390 1.4780 2 0.4780 0.9560 2 0.9560 1.9120 LSD


Signed Number Representation

There are 3 systems to represent signed numbers in binary number system:– Signed-magnitude– 1's complement– 2's complement


Signed-magnitude system

In signed-magnitude systems, the most significant bit represents the number's sign, while the remaining bits represent its absolute value as an unsigned binary magnitude.– If the sign bit is a 0, the number is positive.– If the sign bit is a 1, the number is negative.


Signed-magnitude system


1's Complement system

• A 1's complement system represents the positive numbers the same way as in the signed-magnitude system. The only difference is negative number representations.

• Let be N any positive integer number and be a negative 1's complement integer of N. If the number length is n bits, then

__

N

.)12( NN n


Example of 1's Complement

For example in a 4-bit system, 0101 represents +5 and

1010 represents 5

101001011111

0101)000110000(0101000124



• A 2's complement system is similar to 1's complement system, except that there is only one representation for zero.

• Let be N any positive integer number and be a negative 2's complement integer of

N. If the number length is n bits, then

__

N

.2 NN n


Example of 2's Complement

For example in a 4-bit system, 0101 represents +5 and

1011 represents 5

1011010110000010124


Addition and Subtraction in Signed and Magnitude

(a) 5+2

0101+0010

7 0111

(b) -5-2

1101+1010

-7 1111

(c) 5-2

0101+1010

3 0011

(d) -5+2

1101+0010

-3 1011


Addition and Subtraction in 1’s Complement

(a) 5+2

0101+0010

7 0111

(b) -5-2

1010 +1101

-7 1 0111 1 1000

(c) 5-2

0101 +1101

3 1 0010 1 0011

(d) -5+2

1010 +0010

-3 1100


Addition and Subtraction in2’s Complement

(a) 5+2

0101+0010

7 0111

(b) -5-2

1011 +1110

-7 1 1001

(c) 5-2

0101 +1110

3 1 0011

(d) -5+2

1011 +0010

-3 1101


Overflow Conditions

Carry-in carry-out 0111 1000 5 0101 -5 1011 +3 +0011 -4 +1100 -8 1000 7 10111Carry-in = carry-out 0000 1110 +5 0101 -2 1110 +2 +0010 -6 +1010 7 0111 -8 11000


Addition and Subtraction inHexadecimal System

(9F1B)16 -(4A36)16 : 16 9F1B- 4A36 54E5

(9F1B)16 +(4A36)16 : 1 1 9F1B + 4A36

E951

Addition

Subtraction


Codes

• Decimal codes• Gray code• Error detection code• ASCII code


Decimal codesDecimal Digit BCD Excess-3 2421

84210 0000 0011 0000

1 0001 0100 0001

2 0010 0101 0010

3 0011 0110 0011

4 0100 0111 0100

5 0101 1000 1011

6 0110 1001 1100

7 0111 1010 1101

8 1000 1011 1110

9 1001 1100 1111


Gray CodeDecimal Equivalent Binary Code Gray Code

0 0000 0000

1 0001 0001

2 0010 0011

3 0011 0010

4 0100 0110

5 0101 0111

6 0110 0101

7 0111 0100

8 1000 1100

9 1001 1101

10 1010 1111

11 1011 1110

12 1100 1010

13 1101 1011

14 1110 1001

15 1111 1000


Error detection code Parity Bit (odd) Message

1 0000

0 0001

0 0010

1 0011

0 0100

1 0101

1 0110

0 0111

0 1000

1 1001

1 1010

0 1011

1 1100

0 1101

0 1110

1 1111


Error detection code Parity Bit (even) Message

0 0000

1 0001

1 0010

0 0011

1 0100

0 0101

0 0110

1 0111

1 1000

0 1001

0 1010

1 1011

0 1100

1 1101

1 1110

0 1111


ASCII Code

• ASCII: American Standard Code for Information Interchange.

• Used to represent characters and textual information• Each character is represented with 1 byte

– upper and lower case letters: a..z and A..Z– decimal digits -- 0,1,…,9– punctuation characters -- ; , . : – special characters --$ & @ / { – control characters -- carriage return (CR) , line feed (LF),

beep


Assignment 1

Page 74– 1.1: Only A+B and AB (a), (c), (f), and (g)– 1.2: Only A+B and AB (a), (c)– 1.3: Only A+B and AB (a), (c)– 1.4: (a), (c), (e)– 1.5: (a), (c), (e)– 1.6: (a), (e)– 1.7: (a), (b)– 1.8: (a), (b)– 1.10: (a), (c)– 1.11: (a), (c)– 1.12: (a), (c)– 1.13: (a), (b)

number systems and codes. cs 3402--digital logicnumber systems and codes2 outline number systems...

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