Binary

Objectives

We would like to :

Understand why we use binary in modern computers.
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