Ch 2. Number Systems and Codes

2.2 Octal and Hexadecimal Numbers

10 ~ 15 : Alphabet

Ex) A number D of the form has the value

𝐷=∑𝑖=−𝑛

𝑝− 1

𝑑𝑖𝑟𝑖

• p digit to the left of the point and n digits to the right of the point

𝑑𝑝−1𝑑𝑝− 2𝑑𝑝−3…𝑑1𝑑0 .𝑑−1𝑑−2𝑑−3…𝑑−𝑛

p n

2.3 General Positional-Number-System Conversions

𝟐𝟑 .𝟓𝟔𝟖=010011.101110𝟐𝑨𝑩 .𝟓𝑪𝟏𝟔=10101011.01011100𝟐

𝟏𝟏𝟎𝟎𝟏𝟏𝟎𝟏𝟎𝟎𝟏𝟏𝟏𝟎𝟐=315168𝟏𝟎𝟏𝟎𝟏𝟏𝟏𝟏𝟎𝟎𝟏𝟏𝟏𝟎𝟏𝟏𝟎𝟏𝟎𝟏𝟏 𝟐=15 𝐸76 𝐵16

𝟏𝟎𝟏𝟎𝟏𝟏𝟎𝟎 .𝟏𝟏𝟏𝟏𝟎𝟎𝟏𝟏𝟐=254.7468𝟏𝟎𝟏𝟎𝟏𝟏𝟎𝟎 .𝟏𝟏𝟏𝟏𝟎𝟎𝟏𝟏𝟐=𝐴𝐶 .𝐹 316

• Number conversion example

123416

7716

4 13

2 𝟏𝟐𝟑𝟒𝟏𝟎=𝟒𝑫𝟐𝟏𝟔

5678

708

8 6

7

8

1 0

𝟓𝟔𝟕𝟏𝟎=𝟏𝟎𝟔𝟕𝟖

• Number conversion example (decimal to hexadecimal, octal)

0.78

6.24

8×

8×

1.92

8×

7.36

8×

𝟎 .𝟕𝟖𝟏𝟎=𝟎 .𝟔𝟏𝟕⋯𝟖

• Number conversion example (decimal to octal)

Carry in

11+

01

10+

01Carry out

1

Carry out

01-

11

1 01-

11

Burrow out

1 Burrow in

Burrow out

: Carry in

: Burrow in

: Input data 1

: Input data 2

: Carry out

: Sum

: Burrow out

: Difference

2.4 Addition and Subtraction of Nondecimal Numbers

+

𝐹 9𝐵𝐵

Hexadecimal addition

• Signed-Magnitude System– Magnitude and Symbol ( ‘+’, ‘-’ )– Applied to binary number by using ‘sign bit’– Ex)

• Complement System– Negates a number by taking its complement– More difficult than changing the sign bit– Can be added or subtracted directly

2.5 Representation of Negative Numbers

Sign bit

𝒓𝒏−𝑫 :

: (𝒓 ¿¿𝒏−𝟏)−𝑫 ¿

(𝒓 ¿¿𝒏−𝟏)−𝑫 ¿(𝒓 ¿¿𝒏)−𝑫 ¿

Number :

• Conversion example

1000010184910−

815110

999910184910−

815010𝒓𝒏−𝑫 (𝒓 ¿¿𝒏−𝟏)−𝑫 ¿

+1Easy to complement

2.6 Two’s-Complement Addition and Subtraction

: (𝒓 ¿¿𝒏−𝟏)−𝑫 ¿

[ -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7 ]

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

2.7 One’s-Complement Addition and Subtraction

+6 (0110) -3 (1100) +

10010

1

0011

End-around carry

One’s complement

2.8 Binary Multiplication

Shifted and negated multiplicand

2.8 Binary Division

2.10 Binary Codes for Decimal Numbers

2.11 Gray Code

(0) 1 1 0

1 0 1

Binary to Gray Code

If different, ‘1’else (same) ‘0’

(0) 1 0 1

1 1 0

Gray Code to Binary

If different, ‘1’else (same) ‘0’

12

3

12 3

2.13 Codes for Actions, Conditions, and States

2.14 n-Cubes and Distance

• Hamming Distance– Distance between two vertices, the number of difference

bits in each position EX) D(010, 111) = 2

2.15 Codes for Detecting and Correcting Errors

Parity-bit

At least two non codes between each pair of code words

If minimum distance = 2C+1, up to C-bits can be correctedIf 2C+D+1, then C-bits can be corrected, and d bits can be detected

4= 2C+D+1, (a) C=1, D=1(b) 1 bit can be corrected(c) D=3, 3 bit errors can be detected

111 110 101 100 011 010 001

LSB is 1 if all 7 bits are odd

LSB is 0 if all 7 bits are even

k = # of parity bitsm = # of info bits

𝟐𝐤≥𝒎+𝒌+𝟏 , m=4,3,2,1 , m=11,10,9,…,2,1

Undetectable Error

An important application of 2-D codes

2.16 Codes for Serial Data Transmission and Storage

NRZ : Non-Return to Zero

NRZI : Non-Return to Zero Invert on 1s

BPRZ : Bipolar Return to Zero