Data and Memory

• As we discussed, the basic block of memory only holds a 0 or a 1.
• A system where we use two characters to represent things is called binary. (Especially 0 and 1)
• So how do we represent things in memory?
• How do we represent things in "real life?"
  • We use 10 digits: 0, 1, 2, 3, 4, ..., 9
  • We use 26 letters: a, b, c, d, ... z
  • We use a shift to the letters to give us 26 more: A, B, C, ... Z
  • We have some special "characters": ., +, ....
• Overall we have about 100 or so symbols we use to represent things.
• We are going to have to find a way to "encode" these in binary.
• Base 10
  • The system we work in is called decimal or base 10.
  • What does 739 really mean?
  • $7\times10^2+3\times10^1+9\times10^0$.
  • or
    • 700
    • +30
    • + 9
  • Remember, $10^0$ = 1
  • The is a positional number system.
    • The positions of the numbers indicate the power of 10 they are multiplied by.
  • We have 10 different symbols, 0, 1, ... 9
  • Once we pass these, we go to the next power of 10
    • $9 + 1 = 10$
    • $7 + 3 = 10$
  • The number of symbols and the base are tied together.
• In binary we have two symbols.
  • So we will use base 2.
  • The numbers look strange as we only have 0 and 1.
  • The positions represent different powers of 2.
  • We will annotate base 2 numbers with the subscript 2.
  • The simple ones are 0 and 1.
    • $0_2 = 0$
    • $1_2 = 1$
  • The next two are
    • $10_2 = 1\times2^1 + 0\times2^0 = 1 + 0 = 2$
    • $11_2 = 1\times2^1 + 1\times2^0 = 1 + 1 = 3$
  • There are 4 at the next level, can we find the decimal equivalent?
    • $100_2$
    • $101_2$
    • $110_2$
    • $111_2$
  • There are 8 at the next level:
    • $1000_2$
    • $1001_2$
    • $1010_2$
    • $1011_2$
    • $1100_2$
    • $1101_2$
    • $1110_2$
    • $1111_2$
• This continues just like decimal.
• What is $1101101_2$ in decimal?
• You will work on this much more either in Logic and Switching Theory or in Computer Architecture.
• Given a n digit number in binary I expect you to be able to convert it to decimal.
• Binary numbers can get quite long, so we need another representation.
  • The natural extension is to go to base 8 or base 16.
  • Base 8 is called octal.
  • Base 16 is called hexadecimal.
  • For base 8 we need 8 symbols 0, 1, 2, ... 7
  • For base 16 we need 16 symbols
    • We have 10 : 0, 1, ... 9
    • We just add: a, b, c, d, e, f
      • $a_{16} = 10 = 1010_2$
      • $b_{16} = 11 = 1011_1$
      • $c_{16} = 12 = 1100_1$
      • $d_{16} = 13 = 1101_1$
      • $e_{16} = 14 = 1110_1$
      • $f_{16} = 15 = 1111_1$
  • So hex numbers will look something like
    • $af305_{16}$
    • $1032a_{16}$
  • But subscripts are difficult to represent in some places so we use
    • 0xaf305
    • 0x1032a
• Conversion between binary and hex
  • Binary to hex
    • Group the number in blocks of four from the right.
    • Pad on the left if necessary
    • Convert to hex digits.
  • Hex to binary
    • Replace the hex number with the four equivalent binary digits.

  • Decimal	Binary	Hex	Decimal	Binary	Hex
    0	0000	0	8	1000	8
    1	0001	1	9	1001	9
    2	0010	2	10	1010	a
    3	0011	3	11	1011	b
    4	0100	4	12	1100	c
    5	0101	5	13	1101	d
    6	0110	6	14	1110	e
    7	0111	7	15	1111	f

  • Convert $1011010101110_2$ to hex

    1011010101110 
    
    1 0110 1010 1110 
    
    0001 0110 1010 1110 
      1   6    a    e 
    
    0x16ae or $16a3_{16}$

  • Convert $2af_{16}$ to binary

      2      a     f
    0010   1010  1111
    
    $10101111_2$

• Memory and addresses
  • Memory is a bunch of boxes.
  • We distinguish between the boxes by giving each one an "address" or integer.
  • The address is really just the distance from the start of memory.
  • So the first memory address is 0, the next is 1
  • But we tend to have a bunch of memory
  • So addresses are given in hex.
  • 0x00000000
  • 0x00000001
  • 0x0000000f
  • 0x7affc30f
  • 0xffffffff
• These systems are great for positive integers, but we also need to encode
  • Negative integers
  • Non-integer numbers (some of the rationals to be exact)
  • Letter and other special symbols.
• We have different ways to represent each.
• Let's look at the characters.
  • Characters are the symbols, digits and letters.
  • We use a system called ascii
    • The America Standard Code for Information Interchange.
    • It use 8 bits or 2 hex digits.
  • Look at the Ascii table on Wikipedia.
  • Encode "Dan" in ascii.
  • What is 48 65 6c 6c 6f 20 57 6f 72 6c 64 21? (all values are in hex)
  • Given an ascii table, I would expect you to be able to encode and decode a message.
• There is a newer way to represent characters.
  • unicode.
  • But ascii is the first part.
  • It is somewhat complex.
  • But represents most known character sets.
  • IE check out the Egyptian Hieroglyphs unicode page on wikipedia.
• We need a little binary, hex and ascii to get through this semester.
• Please be able to
  • Convert binary numbers to decimal.
  • Convert between binary and hex.
  • Encode/Decode ascii