Binary

Objectives

1. Understand why we use binary in modern computers.
2. Understand the basics of binary.

Notes

• This is mostly from Carpinelli
• Base 2 is called binary.
• Base 8 is called octal
• Base 16 is called hexadecimal, or hex.
• Computers natively use base 2.
  • In the past there have been attempts to use other base
  • In the future they may be computers that use other bases.
  • But for now, you would be very hard pressed to find a computer that does not use base 2.
• Why?
  • The electronics of base two are simpler
• Digital vs analog
  • In a digital system data is represented as digits.
    • Discrete, the data is broken into bins.
    • This is easier to read
  • In an analog system data is represented as positions.
    • Continuous, perhaps more accurate
    • The data is continuous wrt time.
    • The real world is probably continuous.
• So some form of continuous (we hope) electric power is driving our computers.
  • It may go from a low of 0 volts to a high of +5 volts.
  • But there will be noise
  • It is easy to draw one line (or perhaps two) and say that everything below the line is one value and everything above the line is another.
  • With two lines, the middle will be unknown/error/ .
  • 
  • Capanelli has a table that discusses various forms of logic.
    • ttl, the one we will be using is a 5v system.
    • 0V to .4V is logical 0
    • 2.7V to 5V is logical 1
• Also, we will see an entire set of algebra that deals well with base 2.
• In general we
  • Use 0/1 for our symbols in binary.
    • In logic we use F/T or false/true
  • For this class we will use 0/1
    • And I don't care what your discrete/formal logic instructor says.
    • But do what they ask in their class.
• In computing
  • A signal binary digit is called a bit.
  • 8 bits form together to make a byte.
  • A word is machine dependent, but usually between 2 and 8 bytes.
    • For intel assembly, we consider a word 2 bytes.
    • A double word is 4 bytes
    • A quad word is 8 bytes
  • A 32 bit machine is 32/8 or 4 bytes
  • A 64 bit machine is 64/8 or 8 bytes.
  • Memory is byte addressable.
• For larger quantities we use
  • The SI standard
    • 1000 or 10³ is a kilobyte (k)
    • 10⁶ is a megabyte (M)
    • 10⁹ is a gigabyte (G)
    • 10¹² is a terabyte (T)
    • 10¹⁵ is a petabyte (P)
    • 10¹⁸ is an exabyte
    • zetta, yotta, ronna and quetta are next.
    • See nist.
  • But we also h ave a binary prefix system
    • 1024 or 2¹⁰ is a kibibyte (k)
    • 2²⁰ is a mebiyte (M)
    • 2³⁰ is a gibiyte (G)
    • 2⁴⁰ is a tebiyte (T)
    • 2⁵⁰ is a pebiyte (P)
    • 2⁶⁰ is an exbite
    • zetta, yotta, ronna and quetta are next.
    • See wikipedia.
  • Rant for a moment.
• Some binary numbers
  • Count 0 to 16 in binary.
  • Convert 10110101₂ to decimal.
  • You need to be comfortable working with binary numbers.
• Converting from decimal to binary (integer)
  • We want to find the powers of two "in a number"
  • We can do this by repeated division.
  • Let's try this on something easy like 12

    12/2 = 6 r 0
    6/2  = 3 r 0
    3/2  = 1 r 1
    1/2  = 0 r 1 

  • Using this trick, we read the remainders backwards to get the binary representation 1100₂

  • DECIMAL-TO-BINARY
        Input: an integer in decimal d
        Output: an integer in binary b (you might think of this as a string)
    
     b = 0
     While d > 0
         r = d % 2
         d = d / 2
         b = cat(r,b)  (concatenation)
     return b