CSE 115
Introduction To Computer Science for Majors I
Lecture C
Lecture Notes #26
Stuart C. Shapiro
Spring, 2007

1. Binary Coded Decimal (BCD). Each digit represented by its binary equivalent. For example, 35 in BCD is 0011 0101.

2. Binary Numbers

   Using bits to represent integers:

   Binary	Decimal
   000	0
   001	1
   010	2
   011	3
   100	4
   101	5
   110	6
   111	7

   Positional notation
   Decimal: ones, tens, hundreds, ...
   Binary: ones, twos, fours, ...
   Octal: ones, eights, sixtyfours, ...
   Hexadecimal: ones, sixteens, two hundred fiftysixes, ...

   The two binary digits: 0, 1
   The eight octal digits: 0, 1, 2, 3, 4, 5, 6, 7
   The ten decimal digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
   The fifteen hexadecimal digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, a, b, c, d, e, f

   For conversion:
   If divide a number n by a base b, the remainder is the right-most digit, and the result is the rest of the number.
   Examples: 3597/10 = 359 R7     11011/2 = 1101 R1

   It works even if n is not expressed in base b:

   1035 / 2 = 517 R 1
    517 / 2 = 258 R 1
    258 / 2 = 129 R 0
    129 / 2 =  64 R 1
     64 / 2 =  32 R 0
     32 / 2 =  16 R 0
     16 / 2 =   8 R 0
      8 / 2 =   4 R 0
      4 / 2 =   2 R 0
      2 / 2 =   1 R 0
      1 / 2 =   0 R 1
   

   So 1035₂ = 10 000 001 011

With 7 bits, can represent 128 numbers, e.g. 0, ..., 127.
But what about negatives: -0, ..., -127?

First idea (sign + magnitude):

 3 = 00000011     0 = 00000000
-3 = 10000011    -0 = 10000000

Second idea (biased):
use 00000000 for -127, 11111111 for 128.
That is, to represent n, use binary representation of 127 + n.

 3 = 10000010      0 = 01111111
-3 = 01111100     -0 = 01111111

Better idea: Two's Complement
To go from n to -n: complement bits, add 1, drop final carry

            0 = 00000000  1 = 00000001  3 = 00000011  4 = 00000100  7 = 00000111
complement      11111111      11111110      11111100      11111011      11111000
add 1       0 = 00000000 -1 = 11111111 -3 = 11111101 -4 = 11111100 -7 = 11111001

n + -n          00000000      00000000      00000000      00000000      00000000

                4 = 00000100        -3 = 11111101
               -1 = 11111111        -4 = 11111100
                    --------             --------
               sum  00000011 = 3   sum   11111001 = -7