arithmetic

• basics of arithmetic

  -> binary values are input into a processor and the ALU determines results using logic circuits

  
  

  -> the fundamental of operations is addition

  
  

  signed binary = positive or negative values, shown by the left most bit

  (1 = -ve and 0 = +ve)

  
  

  unsigned binary = only positive values
• binary addition

  rules:

  0 + 0 = 0

  0 + 1 = 1

  1 + 1 = 10 (the 1 carries to the next column)

  1 + 1 + 1 = 11

  
  

  eg 11110000 +

  10011111

  = 1 10001111 <- the extra bit is a result of overflow error

  (1 1 1) <- carrying
• two's complement

  two's complement allows us to use binary for both positive and negative values

  
  

  to find the binary for a negative number you flip the bits and add 1

  1 -> first find the positive 01001101 = 77

  2 -> flip the digits 10110010

  3 -> add 1 10110011 = -77

  
  

  the number of combinations available (for 8 bit) is still 256 but half are positive and half negative so the max value is 127 and the min value is -128

  
  

  (2^n-1 ... (2^n-1) -1)
• binary subtraction

  to subtract binary values we actually use addition and make use of two's complement.

  
  

  65 - 43 = 65 + - 43

  01000001 +

  11010101

  1 00010110 = 22 (you ignore the overflow bit)
• overflow errors

  when the result is too large for the allocated number of bits, overflow will occur.

  The number may therefore be presented incorrectly

  A '1' as the first digit indicates a negative number so overflow occurs for any number above 127 (in 8 bit binary)
• fixed point binary

  we can use binary to represent fractions using fixed points. using bits to the right of the units column (after the notional point ) introduces fractional values

  
  

  8 4 2 1 . 1/2 1/4 1/8 1/16

  0 1 0 1 . 1 1 0 0 = 5 3/4

  
  

  (fractional columns go up in negative powers of 2, 2^-2, 2^-3 etc)

  
  

  the placement of the binary point is fixed wih a specified number of bits each side
• binary multiplication

  rules:

  0 * 0 = 0, 0 * 1 = 0, 1 * 1 = 1

  
  

  steps:

  1 -> multiply by the right most digit of the 2nd number

  2 -> put in a 0 on the right and repeat for the next value

  3 -> continue until all digits of the 2nd number have been multipliers

  4 - add together all the multiplied numbers

  
  

  eg 1010 * 101 = (1010, + 00000, + 101000) = 110010