Ch1: Five Binary Concepts

Ch1: Five Binary Concepts

1. The Three Conversions

   • Stripping
     • Rightmost = Divide, Get Remainder
       • Each time you divide something by its base, you strip off the rightmost digit (as remainder)
         • Use for: decimal to BASE (dec to binary)
           • Keep dividing by BASE until the quotient is 0. Collect REMAINDERS in REVERSE order
             • Last result%(base) = (binary digit)
               • 5(dec) is ...... 5%2 = 1, 2%2 = 0, 1%2 = 1
             • NOTE: 1%2 = 1, 1/2 = 0
     • Leftmost = Multiply, Get Integer
       • Each time you multiply a FRACTIONAL part by its base, you strip off the leftmost digit (as integer)
         • Use for: decimal to BASE (Fractional part) (dec to binary)
           • Keep multiplying THE FRACTIONAL PART by BASE the fractional part becomes 0, collect REMAINDER in FORWARD order
   • Law Of Base^Order Sum (BOS)
     • Each digit is multiplied by Base^Order then summed.
       • Use for: binary to anything (binary to dec)
   • Groups
     • Use for: Binary to hex
       • Groups of 4
     • Use for: Binary to oct
       • Groups of 3

2. Complements

   • Radix (2's)
     • [N] = base^n - (N)
     • 2's Complement: Copy until first 1 (included) then complement
     • 2' Complement = Negative of original number
   • Diminished (1's)
     • [N]diminished = (base^n - 1)r - (N)
     • 1's Complement : Complement each bit

3. Codes & Registers

   • Overflow
     • Occurs when the sum falls outside the range of the register
       • Complement carry's do not count as overflow
     • Occurs when the addition of two positive numbers result in a negative number
       • Rule: leftmost digit 0 + leftmost digit 0 = leftmost digit 0
         • Otherwise, overflow
   • BCD
     • Each digit is converted into 4-bit representation then stringed back together
       • (123) = 0001 0010 0011

4. Signed binary numbers

   • Use left-most digit to represent signed binary numbers (hardware limitation)
     • 0 for positive
     • 1 for negative

5. Binary Arithmetic

   • Subtraction
     • 0 -1 = 1 (and borrow 1 from next)
     • Add 2's complement (add negative number)
       • Discard final carry
     • Add 1's complement
       • Add end carry back to sum
   • Multiplication
     • Multiply as if you would do decimals, same rules as AND
   • Divison

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