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Chapter 2 : Number Systems, Operations, and Codes, Addition:, Do 2^n-1ro…

- - - - Every number does have a binary representation but it is not necessarily unique.
      - For example $$1/2 = 1/4+1/8+1/16 +....$$ which means $$0.1 = 0.01+0.001 +0.0001+......=0.01111111.... $$
      - Binary Weights
        
        2^8
        
        256
        
        2^7
        
        128
        
        2^6
        
        64
        
        2^5
        
        32
        
        2^4
        
        16
        
        2^3
        
        8
        
        2^-1
        
        0.5
        
        2^-2
        
        0.25
        
        2^-3
        
        0.125
        
        2^-4
        
        0.0625
        
        2^-5
        
        0.03125?
      - For recurrsing integers
        
        We can repeat the process an arbitrary number of times and stop
- - - - 1->0, 0-> 1
    - - After 1s complement we add a 1.
      - Recommended method
        
        Starting at the LS, write all the digits as is until we reach the first 1.
        
        Take the 1's complements of the remaining bits
- - - - 7 bit long
    - - 0-> positive; 1-> negative
  - - - If the result goes beyond 128, incorrect :question:
      - 1. Convert the decimal number into the binary form with negative numbers expressed in the 2's complement form
        
        Question: Why call it 2's complement form :question:
        
        Question: Isn't 2's complement form supposed to be a procedure? :question:
      - 2. Perform binary addition
      - 3. Sign: If same sign -> same sign for result; Otherwise, overflow indicates the result is positive
  - - - Flip all numbers
      - Manual calculation: Negative of first
      - Recommended: Adding complements and subtracting 1
    - - Add +1 to the 1's complement
      - n bit numbers: [-2^(n-1), +2^(n-1)-1]
      - 8 bit numbers: [-128, 127]
      - Brutal approach
      - Textbook approach :!?:
        
        Question: "how are y and y' linked"
        
        -2^n + x?