Number System

bit  - a binary digit, the smallest increment of data on a computer

        - can hold only one of two values: 0 or 1, corresponding to the electrical values of off
          or on, respectively.

 

 

byte  - the smallest addressable unit for a CPU.

word  - the natural size with which a processoris handling data(the register size).

            - most common word sizes encountered today are 8, 16, 32 and 64 bits, but other sizes are possible.

nibble  - sometimes spelled "nibble," is a set of four bits.

               - since there are eight bits in a byte, a nybble is half of one byte.                
              - in the computer world, two nybbles always equal one byte.

Types of Number System

1. Binary System
It is a number system with base digits 0 or 1.It can be written as subscripted B or 2.For example:(1001)B or (1001)2

….2⁰ = 1, 2¹ = 2, 2² = 4, 2³ = 8, 2⁴ = 16.....

2. Decimal Number System
It is number system with base 10 and use digits 0 to 9.It can be written as subscripted D or 10.For example:(158)10 or (268)D.

.....10⁰ = 1, 10¹ = 10, 10² = 100, 10³ = 1000.....

3. Hexadecimal  Number System
It is a number system with base 16 and use digits 0 to 9 and symbols A to F.It can be written as subscripted 16 or H.For example:(2AB)H or (ABC)16.

..... 16⁰=1,16¹ =16,16²=256,16³=4096…..

 

Converting from One System to Another

Other bases into decimal

Decimal	Binary		Hexadecimal
(base 10)	(base 2 )	(base 16)
0	0	0
1	1	1
2	10	2
3	11	3
4	100	4
5	101	5
6	110	6
7	111	7
8	1000	8
9	1001	9
10	1010	A
11	1011	B
12	1100	C
13	1101	D
14	1110	E
15	1111	F

 

Decimal:

45297 =

4 * 10⁴ = 4 * 10,000 = 40,000

5 * 10³ = 5 *  1,000  =   5,000

2 * 10² = 2 *    100   =     200

9 * 10¹ = 9 *     10    =       90

7 * 10⁰ = 7 *      1     =         7

                                     45297

Other bases into the decimal system:

Binary	Hex
11011101 = 1 * 27 = 1 * 128 = 128 1 * 26 = 1 *   64 =  64 0 * 25 = 0 *   32 =   0 1 * 24 = 1 *   16 =  16 1 * 23 = 1 *     8 =   8 1 * 22 = 1 *     4 =   4 0 * 21 = 0 *     2 =   0 1 * 20 = 1 *     1 =   1                            221	285BCE = 2 * 165 =   2 * 1,048,576 = 2,097,152 8 * 164 =   8 *     65,536 =    524,288 5 * 163 =   5 *       4,096 =      20,480 B * 162 = 11 *          256 =       2,816 C * 161 = 12 *           16 =          192 E * 160 = 14 *             1 =           14                                          2,644,942

·         Convert 101100101₂ to the corresponding base-ten number.

I will list the digits in order, and count them off form the RIGHT, starting with zero:

 

 digits : 1 0 1 1 0 0 1 0 1

                  numbering: 8 7 6 5 4 3 2 1 0

 

(1 x 2⁸)+(0 x 2⁷)+(1 x 2⁶)+(1 x 2⁵)+(0 x 2⁴)+(0 x 2³)+(1 x 2²)+(0 x 2¹)+(1 x 2⁰)

= (1 x 256)+(0 x 128 )+(1 x 64)+(1 x 32)+(0 x 16)+(0 x 8)+(1 x 4)+(0 x 2)+(1 x 1)
=256 + 64 + 32 + 4 + 1
= 357                                                                                                                                                                                                          Then 101100101₂ = to 357₁₀.

·         Convert 165₁₆ to the corresponding decimal number.

List the digits, and count them off from the RIGHT, starting with zero:

digits:	1  6   5
numbering:	2  1   0

Remember that each digit in the hexadecimal number represents how many copies you need of that power of sixteen, and convert the number to decimal:

1×16² + 6×16¹ + 5×16⁰
     = 1×256 + 6×16 + 5×1
     = 256 + 96 + 5
     = 357

Then 165₁₆ = 357₁₀.

 

Decimal to Binary Or Hex

 

let's convert 221₁₀ into binary.

   110 r 1

2)221

The remainder becomes the rightmost digit in our answer; the quotient is used for the next dividend. Because we are converting to binary, 2 is our divisor.
 

    55 r 0

2)110

Now 0 is pre-pended to the 1 we got before. And we repeat our divisions until the quotient is 0.
 

I find it easier to start at the bottom and stack up my divisions, keeping track of my remainders. Read this example as a series of divisions from the bottom up, but read the conversion as the list of remainders from the top down.

      0     r 1

  2 )1     r 1  
  2 )3     r 0
  2 )6     r 1  
  2 )13   r 1  
  2 )27   r 1  
  2 )55   r 0  
  2 )110 r 1   
  2 )221

The answer is read from the top to the bottom using the remainders: 11011101.

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To convert from decimal to hex, divide by 16 and remember to convert the remainder into hex digits (0-F). For example, let's convert 2644942₁₀ into hex.
 

          ­­­­ 0          r 2

    16 )_2         r 8     
    16 )40         r 5    
    16 )645       r 11 (B)     
    16 )10331   r 12 (C)    
    16 )165308 r 14 (E)    
    16 )2644942

The answer is read from the top to the bottom: 285BCE

 

 

·         Convert 357₁₀ to the corresponding hexadecimal number.

 Here, I will divide repeatedly by 16, keeping track of the remainders as I go. (You might want to use some scratch paper for this.)

      1      r 6

16)22     r 5
16)357    

 

   

Reading off the digits, starting from the top and wrapping around the right-hand side, I see that 357₁₀ = 165₁₆

 

 

·         Convert 63933₁₀ to the corresponding hexadecimal number

I will divide repeatedly by 16, keeping track of my remainders:

­­­­            15 r 9

16 )    249 r 11 (B)   
16 )  3995 r 13 (D)   
16 )63933     

 

 

I cannot write the hexadecimal number as “1591113”, because this would be confusing and imprecise. So I will use the letters for the “digits” , let “F” = “15”, “B” = “11”, and “D” =”13”.

Then 6393310 = F9BD16.