Binary Arithmetic James A. Rome Tennessee Governor's Academy August 2010.

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Presentation transcript:

Binary Arithmetic James A. Rome Tennessee Governor's Academy August 2010

Remember how the decimal system works Numbers are organized into columns. 983 is: This is in powers of 10 Binary numbers work the same way. 6 is: 389 OnesTensHundreds OnesTwosFours

Arithmetic works the same way = 12 12= 5 = 7 =

Negative numbers are trickier Computers do not have plus and minus signs!  One solution is to use a fixed number of bits for arithmetic and to use the left-most bit as the sign bit. Suppose we have 8 bits, 12 = and -12 =  Then the number range is ± = ±127  One's Complement: flip all the 0s and 1s; keep sign bit -12 =  Two's Complement: Start with 1's complement and add 1 if it is negative: =  Excess 2 (m-1) [m=7 for us]: Add 2 7 to each number 7 = = 135 = ; -7 = = 121 = Note that’s this is the same as the twos complement with the left bit flipped. You need to know which representation was used!

Examples Using the regular algorithm for binary addition, add (5+12), (-5+12), (-12+-5), and (12+-12) in each system. Then convert back to decimal numbers. Signed Magnitude: __________ ________ ________ _________ One' Complement: _________ ________ ________ ________ Two's Complement: ________ ________ ________ ________ Excess 2^m: ________ ________ ________ ________ Two's complement works!

How to do two's complement fast Flip all the bits including the sign bit and add one = 12; ———————— = -12 in two's complement It is critically important to remember that the place of the negative-weight bit must be already determined before any two's complement conversions can be done.