Number system

                                  Number system

Number system:-

•Decimal (base 10)     {0 1 2 3 4 5 6 7 8 9}–Place value gives a logarithmic               representation of the number–Ex. 4378   means• 4 X 103 = 4000• 3 X 102 =   300• 7 X 101 =     70• 8 X 100 =       8–The place also gives the exponent of the base

Example:-

•432,600

4  3  2  6  0  0

Powers of ten:-

100 = 1  102 = 100  104 = 10000

101 = 10  103 = 1000  105 = 100000

        

          Binary (base 2)       {0 1}

Example:-

                             

Decimal Equivalent:-

•1101 1001

        1 X 27 =  128   

  +    1 X 26  =   64

  +    0 X 25  =     0

  +    1 X 24  =   16

  +    1 X 23  =     8

  +    0 X 22  =     0

  +    0 X 21  =     0

  +    1 X 20  =     1

                        217

Notice how powers of two stand out:

20 = 1

21 = 10

22 = 100

23 = 1000

Decimal to Binary Conversion:-

•Ex.  575

–Find the largest power of two less than the number

  •29 = 512

–Subtract that power of two from the number

   •575 – 512 = 63

–Repeat steps 1 and 2 for the new result until you reach 

   zero.

•25 = 32             63 – 32 = 31

•24 = 16             31 – 16 = 15

•23 = 8               15 – 8 = 7

•22 = 4                7 – 4 = 3

•21 = 2                3 – 2 = 1

•20 = 1                1 – 1 = 0

–Construct the number

•1000111111

Another Example:-

•144

–27 = 128                144 – 128 = 16

–24 = 16                  16 – 16 = 0

•Result     10010000

Hexadecimal (base 16):-

• {0 1 2 3 4 5 6 7 8 9 A B C D E F}

• Assignments

Example:-

  3 X 163 = 12288

11 X 162 =   2816

  6 X 161 =       96

14 X 160 =       14

=> 15214

Hexadecimal is Convenient for Binary Conversion:-

Binary to Hex Conversion:-

• Group binary number by fours (nibbles)

–1101  1001  0110

•Convert each nibble into hex equivalent

–1101  1001  0110

     D        9        6

Decimal to Hex Conversion:-

• Ex.  284

–162 = 256           284 – 256 = 28

–161 =   16             28 -    16 = 12  (Hex C)

–Result   1 1 C