The Binary Number System

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					The Binary Number System


   Data Representation
What is a number?
What is a number?

A number is a unit of an abstract
  mathematical system subject to the laws
  of arithmetic.
The Laws of Arithmetic

   Succession

   Addition

   Multiplication
Number Categories

   Natural (Whole)
       The counting numbers
   Negative
       Less than 0
   Rational
       An integer, or the quotient of 2 integers
Succession
Positional Notation

   The Decimal system is based on the number of digits we
    have.


                               
   Positional Notation allows us to count past 10 by organizing
    numeric digits in columns.

   Each column of a number represents a power of the base.
       The base is 10.
       The exponent is the order of magnitude for the column.
Positional Notation

              3               2               1                0
        10              10              10              10
              1               1               1                1
  1000               100                10                 1
   •The exponent is the order of magnitude for the column.
   •The Least Significant digit is in the right-most column.
   •The Most Significant digit is in the left-most column.
Positional Notation

           3          2         1        0
       10            10       10    10
           1              1     1        1
  1000          100           10     1
   The base is 10.
Positional Notation

           3          2          1                  0
      10          10         10            10
           1           1         1                  1
  1000         100           10               1
                                         exponent
   The magnitude of the column is base
Positional Notation

         104 103 102 101                      100
      10000 1000 100 10                        1
          2    7    9   1                      6
      20000+7000 +900 +10                     +6
               =27916
     Consider a number like the one above.
     How many does it represent?
Positional Notation

         104 103 102 101                    100
      10000 1000 100 10                      1
          2    7    9   1                    6
      20000+7000 +900 +10                   +6
               =27916
     The size of a number is determined by
      multiplying the magnitude of the column by the
      digit in the column and summing the products.
Positional Notation

         104 103 102 101                    100
      10000 1000 100 10                      1
          2    7    9   1                    6
      20000+7000 +900 +10                   +6
               =27916
     The columns are labelled with their exponents.
Positional Notation

         104 103 102 101              100
      10000 1000 100 10                1
          2    7    9   1              6
      20000+7000 +900 +10             +6
               =27916
     The base of the system is 10.
Positional Notation

         104 103 102 101                     100
      10000 1000 100 10                       1
          2    7    9   1                     6
      20000+7000 +900 +10                    +6
               =27916
                                            exponent
     The magnitude of the column is base
Positional Notation

         104 103 102 101                     100
      10000 1000 100 10                       1
         *2   *7   *9 *1                     *6
      20000+7000 +900 +10                    +6
               =27916
     Multiply the magnitude of the column by the digit
      in the column.
Positional Notation

         104 103 102 101    100
      10000 1000 100 10      1
         *2   *7   *9 *1    *6
      20000+7000 +900 +10   +6
 27 thousand, 9 hundred, sixteen
     Sum the products.
Binary Numbers

The binary number system is a means of
  representing quantities using only 2 digits:
                   0 and 1.

Like other number systems it’s based on
              Positional Notation.
Positional Notation

In Binary, the columns have the expected exponents,




        23            22           21            20


        81            41           21            11
Positional Notation

In Binary, the columns have the expected exponents,
but the base of the system is 2.




        23            22           21            20


        81            41           21            11
Positional Notation

In Binary, the columns have the expected exponents,
but the base of the system is 2.
So the column magnitudes are powers of 2.




        23            22           21            20


        81            41           21            11
Binary

Rather than referring to each of the numbers as
 a binary digit, we shorten the term to bit.

To facilitate addressing, binary values are
  typically stored in units of 8 bits, which is
  called a byte.

Large values occupy multiple bytes.
A Single Byte

  27 26 25 24 23 22       21 20
128 64 32 16 8 4          2   1
  1   1   1   1 1 1       1   1
128 +64 +32 +16 +8 +4 +   2 + 1
            =255
A Single Byte

  27 26 25 24 23 22       21 20
128 64 32 16 8 4          2   1
  1   1   1   1 1 1       1   1
128 +64 +32 +16 +8 +4 +   2 + 1
            =255
A Single Byte

  27 26 25 24 23 22       21 20
128 64 32 16 8 4          2   1
  1   1   1   1 1 1       1   1
128 +64 +32 +16 +8 +4 +   2 + 1
            =255
A Single Byte

  27 26 25 24 23 22                               21 20
128 64 32 16 8 4                                  2   1
  1   1   1   1 1 1                               1   1
128 +64 +32 +16 +8 +4 +                           2 + 1
            =255
is the largest decimal value that can be expressed in 8 bits.
   How many different patterns are there?
A Single Byte

  27 26            25     24     23 22           21 20
128 64             32     16      8 4            2   1
  0   0             0       0     0 0            0   0
  0 +0             +0     +0     +0 +0 +         0 + 0
                           =0
There is also a representation for zero, making 256 (28)
  combinations of 0 and 1, in 8 bits.
Natural Numbers in Binary

  Consider the pattern:


                  10010101
  To calculate the Decimal equivalent:
  1.  multiply each digit by the value of the column
  2.  sum the products.
Natural Numbers in Binary

        27 26   25   24 23   22   21   20
      128 64    32   16 8    4    2     1
        1   0    0    1 0    1    0     1
Natural Numbers in Binary

        27 26   25   24 23   22   21   20
      128 64    32   16 8    4    2     1
        1   0    0    1 0    1    0     1
Natural Numbers in Binary

        27 26   25   24 23   22   21   20
      128 64    32   16 8    4    2     1
        1   0    0    1 0    1    0     1
Natural Numbers in Binary

        27 26 25 24 23 22       21 20
      128 64 32 16 8 4          2   1
        1   0   0   1 0 1       0   1
      128 + 0 + 0 +16 +0 +4 +   0 + 1
Natural Numbers in Binary

        27 26 25 24 23 22       21 20
      128 64 32 16 8 4          2   1
        1   0   0   1 0 1       0   1
      128 + 0 + 0 +16 +0 +4 +   0 + 1

                =149
Natural Numbers in Binary

Conversion from Decimal to Binary uses the
 same technique, in reverse.

Consider the value 73.

This is 7 units of 10, plus 3 units of 1.
Natural Numbers in Binary

We need to express the value in terms of
  powers of 2.

     27    26   25    24   23    22   21   20
   128    64    32   16     8     4    2   1
      0     1
Natural Numbers in Binary

What is the largest power of 2 that is included
  in 73?

     27    26    25    24    23    22   21    20
    128    64    32   16      8     4    2        1
      0     1
Natural Numbers in Binary

64 is the largest power of 2 that is included in
    73, so a 1 is needed in that position

      27    26    25    24    23    22    21       20
    128    64    32    16      8     4     2       1
       0     1
Natural Numbers in Binary

Subtracting 64 from 73 leaves 9, which cannot include
   32, or 16, but does include 8.


      27    26     25    24     23    22    21     20
    128     64    32     16      8     4     2      1
       0     1      0     0      1
Natural Numbers in Binary

Subtracting 8 from 9 leaves 1, which cannot include 4,
   or 2, but does include 1.


      27     26    25     24    23     22    21     20
    128     64     32    16      8      4     2      1
       0      1     0      0     1      0     0      1
Natural Numbers in Binary

So the 8 bit binary representation of 73 is:
                    01001001
Short Forms
Longer Numbers

Since 255 is the largest number that can be
  represented in 8 bits, larger values simply
  require longer numbers.
For example, 27916 is represented by:

              0110110100001100
Longer Numbers

Since 255 is the largest number that can be
  represented in 8 bits, larger values simply
  require longer numbers.
For example, 27916 is represented by:

              0011011010000110

Can you remember the Binary representation?
Short Forms for Binary

Because large numbers require long strings of
 Binary digits, short forms have been
 developed to help deal with them.

An early system was called Octal.

It’s based on the 8 patterns in 3 bits.
Short Forms for Binary - Octal
111   7         0011011010000110
110   6
           can be short-formed by
101   5
           dividing the number into 3 bit
100   4    chunks (starting from the
011   3    least significant bit) and
010   2    replacing each with a single
001   1    Octal digit.
000   0
Short Forms for Binary - Octal
111   7      000011011010000110
110   6
101   5
                   0   3   3   2   0   6
           added
100   4
011   3
010   2
001   1
000   0
Short Forms for Binary - Hexadecimal
0111   7   1111   F   It was later determined that
0110   6   1110   E   using base 16 and 4 bit
                      patterns would be more
0101   5   1101   D
                      efficient.
0100   4   1100   C
                      But since there are only 10
0011   3   1011   B   numeric digits, 6 letters
0010   2   1010   A   were borrowed to complete
0001   1   1001   9
                      the set of hexadecimal
                      digits.
0000   0   1000   8
Short Forms for Binary - Hexadecimal
0111   7   1111   F    0011011010000110
0110   6   1110   E
                      can be short-formed by
0101   5   1101   D   dividing the number into 4-
0100   4   1100   C   bit chunks (starting from the
                      least significant bit) and
0011   3   1011   B
                      replacing each with a single
0010   2   1010   A   Hexadecimal digit.
0001   1   1001   9
0000   0   1000   8
Short Forms for Binary - Hexadecimal
0111   7   1111   F   0011011010000110
0110   6   1110   E
0101   5   1101   D
                        3   6   8   6
0100   4   1100   C
0011   3   1011   B
0010   2   1010   A
0001   1   1001   9
0000   0   1000   8
Short Forms for Binary

Octal and Hexadecimal are number systems.
It is possible to perform arithmetic in both.
                  2
There are 64 (8 ) rules of octal addition, and
           2
   256 (16 ) rules of hexadecimal addition.
But why design a machine with so many rules
   when conversion to Binary is simple and
   there are only 4 rules of Binary addition?

				
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posted:4/2/2013
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