Hello World! An Invitation to Computer Science

Lecture Slides: Binary Arithmetic

Monday, February 18, 2008

Key ideas from last time

• don't panic!
• positional number systems
• base 2 is like base 10 with only 1s and 0s
• easy: binary → decimal
• easy: decimal → binary

Hexadecimal numbers: Base 16

                  hex                           hex
  -------------------------------------------------
       0      0    0             8      1000      8
       1      1    1             9      1001	  9
       2     10    2            10      1010	  A
       3     11    3            11      1011	  B
       4    100    4            12      1100	  C
       5    101    5            13      1101 	  D
       6    110    6            14      1110	  E
       7    111    7            15      1111	  F

compact, easy (for humans) to read
simple to convert binary ⇔ hex

Real computers are finite

• thus far: arbitrary binary #s, (with no leading 0s)
• real world: fixed-width (usually 8, 16, 32, or 64 bits)
• maximum and minimum values
• overflow (and underflow) and odometer effect
• pigeon-hole principle:
   only so many different things can be represented

Binary arithmetic

• can easily be expressed as algorithms
    operating on lists of digits/bits
    nowhere does computer have to "know" how
• addition: almost identical to base 10
• multiplication: even easier since only 1 or 0

Binary addition example

   13            1101
+  17        +  10001
-----        --------
   30           11110

Binary multiplication example

 
   5          101
x  6       x  110 
----       ------
  30            0
             1010
          + 10100
          -------
            11110

Shifting left and right

• appending 0 ⇔ multiplying by 2
• removing last bit ⇔ dividing by 2

examples:

2310	=	101112	=		1x24	+ 0x23	+ 1x22	+ 1x21	+ 1x20
4610	=	1011102	=	1x25	+ 0x24	+ 1x23	+ 1x22	+ 1x21	+ 0x20
1110	=	10112	=			1x23	+ 0x22	+ 1x21	+ 1x20

It's all 1s and 0s

• discrete (digital) information
    can always be thought of in terms of bits
• Boolean, integers (including negatives), fractions,
    text(ASCII, Unicode), still images, audio, video

Bitmaps

 ***    00111          
   *    00001
 ***    00111    →  0011100001001110000100111
   *    00001
 ***    00111