- Data and Numbering System - Conversion Between Numberings

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- Data and Numbering System - Conversion Between Numberings. Data Digitally represented information in a form suitable for communication, interpretation, or processing by human or automatic means. Data include constants, variables, arrays, and character strings. Number System

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### - Data and Numbering System- Conversion Between Numberings

Data

Digitally represented information in a form suitable for communication, interpretation, or processing by human or automatic means.

Data include constants, variables, arrays, and character strings.

Number System

A numbering system is specific notation for representing numbers.

Week 3 01/27/2005 Course ISM3230 Dr. Simon Qiu

### - Data and Numbering System- Conversion Between Numberings

System Base:

The base represents the number of symbols which are used

in the system:

Ten-base system (Decimal): 0, 1, 2, …, 9;

Two-base system (Binary): 0 and 1;

Eight-base system (Octal): 0, 1, 2, …, 8;

16-base (Hex): 0, 1, 2, …, 9, A, B,…,E , F.

Week 3 01/27/2005 Course ISM3230 Dr. Simon Qiu

- Number System Conversion -1

Week 3 01/27/2005 Course ISM3230 Dr. Simon Qiu

- Number System Conversion -2

• Converting Number to Base 2

2|283 0.5764 *2 = 1.1568  0.1

2|141 | 1 1.1568 *2  0.10

2| 70 | 1 =0.3056 *2  0.100

2|35 | 0 =0.6112 *2  0.1001

2|17 | 1 =1.2224 *2  0.10010

2| 8 | 1 = 0.4448 *2  0.100100

2| 4 | 0 = 0.8896 *2  0.1001001

2 | 2 | 0 = 1.7792 *2  0.10010011

2 | 1 | 0 = 1.5584 *2  0.10010011

2 | 0 | 1 =>100011011 =1.1168

Week 3 01/27/2005 Course ISM3230 Dr. Simon Qiu

- Number System Conversion -2

• Converting Number to Base8 (or Base16)

Grouping Method from Bits of Base2 into Base8

(or Base 16)

101,0010,11102 <=> 52E16;

10,100,101,1102 <=> 24568

Week 3 01/27/2005 Course ISM3230 Dr. Simon Qiu

- Binary-Arithmetic Operations

1011 110 1011

+ 1001 101/11110 X 1001

10100 101 1001

101 101100

101 1100011

0

Week 3 01/27/2005 Course ISM3230 Dr. Simon Qiu

- Binary-Arithmetic Operations

• Negative Number

-8 =1111,111,111,1000 = 0xFFF8

( = - ( 4 + 2 + 1 )

OR –

32768+16384+8192+4096+1024+512+256+128+64+32+16+8 )

Week 3 01/27/2005 Course ISM3230 Dr. Simon Qiu

- Binary-Arithmetic Operations

1101-1001 => 1101 + (~(1001)) +1 = 1101 + 0110 + 1 = 0100

• Two’s Complement - Representing negative numbers.

The way twos complement works is by

defining the representation for the negative numbers as following:

1 byte: -X = 28 – X; 2 byte: -X = 216 – X; 4 byte: -X = 232 – X

It is almost always easiest to work in Hexadecimal System:

1 byte: -X = 0X100– X; Examples:

2 byte: -X = 0X10000– X; -74 = -0x 4A (Hex) = 0x B6

4 byte: -X = 0X10000000– X -3 = -0x 03 (Hex) = 0x FD

-8 = -0x 0008 (Hex) = 0x FFF8

Week 3 01/27/2005 Course ISM3230 Dr. Simon Qiu

- Binary-Arithmetic Operations

• Two’s Complement - Representing negative numbers.

The way twos complement works is by defining the representation for the negative numbers as following:

1 byte: -X = 28 – X;

2 byte: -X = 216 – X;

4 byte: -X = 232 – X

Week 3 01/27/2005 Course ISM3230 Dr. Simon Qiu

- Binary-Arithmetic Operations

• It is almost always easiest to work in Hexadecimal System:

1 byte: -X = 0X100– X;

2 byte: -X = 0X10000– X;

4 byte: -X = 0X10000000– X

Examples:

-74 = -0x 4A (Hex) = 0x B6

-3 = -0x 03 (Hex) = 0x FD

-8 = -0x 0008 (Hex) = 0x FFF8

Week 3 01/27/2005 Course ISM3230 Dr. Simon Qiu

Assignments2

1. What are the decimal (base ten) values of

1012 , 2547 , 4F16 , 20A12 ,20A16

2. Convert the following numbers from base 10:

25 to base 2; 103 to base 4; 435 to base 16.

3. Convert the hex number 12EFA to binary

4. Convert the binary number 11110100101101111100 to hexadecimal.

5. What are the twos complement equivalents of the following decimal numbers?

-74 = - 0x_ _ (Hex) = 0x_ _ (two complement)

-128 = - 0x_ _ (Hex) = 0x_ _ (two complement)

-3 = - 0x_ _ _ _ (Hex) = 0x_ _ _ _ (two complement)

Week 3 01/27/2005 Course ISM3230 Dr. Simon Qiu