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 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 301/27/2005Course ISM3230Dr. Simon Qiu


Data and numbering system conversion between numberings1

- 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 301/27/2005Course ISM3230Dr. Simon Qiu


Data and numbering system conversion between numberings

- Number System Conversion -1

Week 301/27/2005Course ISM3230Dr. Simon Qiu


Data and numbering system conversion between numberings

- 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 301/27/2005Course ISM3230Dr. Simon Qiu


Data and numbering system conversion between numberings

- 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 301/27/2005Course ISM3230Dr. Simon Qiu


Data and numbering system conversion between numberings

- Binary-Arithmetic Operations

  • Binary: Addition/multiplication/division

    1011 1101011

    + 1001 101/11110 X 1001

    10100 1011001

    101 101100

    101 1100011

    0

Week 301/27/2005Course ISM3230Dr. Simon Qiu


Data and numbering system conversion between numberings

- 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 301/27/2005Course ISM3230Dr. Simon Qiu


Data and numbering system conversion between numberings

- Binary-Arithmetic Operations

  • Subtraction by Complement Addition

    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 301/27/2005Course ISM3230Dr. Simon Qiu


Data and numbering system conversion between numberings

- 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 301/27/2005Course ISM3230Dr. Simon Qiu


Data and numbering system conversion between numberings

- 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 301/27/2005Course ISM3230Dr. Simon Qiu


Data and numbering system conversion between numberings

  • 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 301/27/2005Course ISM3230Dr. Simon Qiu


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