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6.0 BINARY ADDITION

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So we all know how to add decimal numbers (im presuming), its easy

 

 1 1 1

 + 1 + 2 + 3

 --- --- ---

 = 2 = 3 = 4



But how do we add in binary? Easy watch this



 0 0 1 1

 + 0 + 1 + 0 + 1

 --- --- --- ---

 = 0 = 1 = 1 = 10



Thats fine but with 1+1 we get a carry over that we have to 

deal with. So we add an extra space to handle the carry.



 0 0 1 1

 + 0 + 1 + 0 + 1

 --- --- --- ---

 = 00 = 01 = 01 = 10



Lets write a truth table to handle this data.



 A B | CO Q

 -------+--------

 0 0 | 0 0

 0 1 | 0 1

 1 0 | 0 1

 1 1 | 1 0



Notice the new column on the truth table for the carry out (CO).

We now notice that CO and Q and familiar, C0 is the same as an AND

gate and Q is the same as an XOR.



 A & B = Q

 

 (A & ~B) | (~A & B) = CO



Thats fine for adding single bit numbers but what if we want to

 add 2 8-bit numbers? In this case were going to need a component

 called a full binary adder. Once we create the full adder we 

can put 8 of them together to form an 8-bit or byte-wide adder 

and move the carry bit from one adder to the next.



The main difference for us between the first adder and this 

full adder is that we now need a third input called 

a Carry-In(CI).



Heres the truth table for the full adder.





 A B CI | CO Q

 -------------+--------

 0 0 0 | 0 0

 0 0 1 | 0 1

 0 1 0 | 0 1

 0 1 1 | 1 0

 1 0 0 | 0 1

 1 0 1 | 1 0

 1 1 0 | 1 0

 1 1 1 | 1 1



If we examine the output youll realise that the top 4 numbers 

for C0 look like an AND gate for A and B and the bottom 4 look 

like an OR gate for A and B, while top 4 of Q look like an XOR 

and the bottom 4 like an XNOR gate.



By putting those gates together we could form a form a more 

complicated and useful circuit such as the adder and thats 

how a computer builds up from a series of 1's and 0's to addition.