Building an Computer Adder Building an Adder No matter how complex the circuit or how complex the task being solved at the base level computer circuits are made up of three basic components These basics components or gates are AND OR NOT Building an Adder Examine the following binary addition problem 1010 111 Copyright 2008 by Helene G Kershner Building an Adder Examine the following binary addition problem 1010 111 1 Copyright 2008 by Helene G Kershner Building an Adder Remember that in binary addition there are only five possible numeric combinations 1 Carry In 0 0 0 0 1 1 1 0 1 1 1 10 and 1 1 11 Carry Out to next digit Copyright 2008 by Helene G Kershner Building an Adder Let s put this information in a table If we rotate this table placing the A B and Sum at the top we get what looks like a standard turth table A 0 0 1 1 B 0 1 0 1 Sum 0 1 1 10 A B Sum 0 0 0 1 0 1 1 0 1 1 1 10 Building an Adder A computer will treat each column of digits in our binary addition as one operation 1010 111 1 Essentially the computer uses the circuit designed to implement the table below to solve each column A 0 0 1 1 B 0 1 0 1 S 0 1 1 1 0 Ignore the 1 which carries to next column for the moment Copyright 2008 by Helene G Kershner Building an Adder Comparing this truth table to the ones for AND OR and NOT it is clear this is none of the above A 0 0 1 1 B 0 1 0 1 S 0 1 1 0 This truth table is close to being a table for OR but the final set of values doesn t match Copyright 2008 by Helene G Kershner Building an Adder Using basic logic gates computer architects designed the following circuit to match the truth table for adding two binary digits Copyright 2008 by Helene G Kershner Building an Adder If A 0 and B 0 then Sum 0 Copyright 2008 by Helene G Kershner Building an Adder If A 0 and B 1 then Sum 1 Copyright 2008 by Helene G Kershner Building an Adder If A 1 and B 0 then Sum 1 Copyright 2008 by Helene G Kershner Building an Adder If A 1 and B 1 then Sum 0 Copyright 2008 by Helene G Kershner Building an Adder This circuit has proved so important that it is given its own name This gate is called an eXclusive OR and is give the symbol in a logic statement Copyright 2008 by Helene G Kershner Building an Adder Complete the binary addition problem 1010 111 Copyright 2008 by Helene G Kershner Building an Adder Examine the following binary addition problem 1010 111 1 Copyright 2008 by Helene G Kershner Building an Adder Examine the following binary addition problem 1010 111 01 Copyright 2008 by Helene G Kershner Building an Adder Examine the following binary addition problem 1 1010 111 01 Copyright 2008 by Helene G Kershner Building an Adder Examine the following binary addition problem 1 1 1010 111 001 Copyright 2008 by Helene G Kershner Building an Adder Examine the following binary addition problem 1 1 1010 111 10001 Copyright 2008 by Helene G Kershner Building an Adder When reviewing the example the addition is not quite as simple as A B Sum In many cases a carry impacts our addition 10 10 1 11 10 1 Copyright 2008 by Helene G Kershner Building an Adder Looking at our example one column at a time we notice the following Carry A B Sum 0 0 0 0 0 1 1 0 1 0 1 0 1 1 0 1 Out Sum A B A XOR B Carry Out AB A AND B Copyright 2008 by Helene G Kershner Building an Adder To accurately reflect single digit binary addition our circuit is Checking this circuit it matches the truth table Copyright 2008 by Helene G Kershner Building an Adder This is still not quite accurate Going back to our binary addition problem we find that in reality we are not adding two bits and getting a two bit answer Rather our Carry Out from one column become the Carry In to the next column Copyright 2008 by Helene G Kershner Building an Adder carry in 1 0 1 0 1 0 1 0 0 1 1 0 1 0 1 A B S carry out Copyright 2008 by Helene G Kershner Building an Adder 0 1 0 1 0 1 0 1 0 0 1 1 0 1 1 0 0 1 carry in A B S carry out Copyright 2008 by Helene G Kershner Building an Adder 1 0 1 0 1 0 1 0 1 0 0 1 1 0 1 0 1 0 1 0 1 carry in A B S carry out Copyright 2008 by Helene G Kershner Building an Adder 1 1 1 0 1 0 1 0 1 0 1 0 0 1 1 0 1 1 0 0 0 1 0 0 1 1 1 0 1 carry in A B S carry out Copyright 2008 by Helene G Kershner Building an Adder Adding two 1 bit numbers together turns out to really require that we add 3 bits together including our Carry In and producing an answer that is 2 bits including our carry out A complete or full adder 3 bits Cin A B and produces a Sum and a Carry Out Copyright 2008 by Helene G Kershner Building an Adder The truth table that represents all the possible outcomes of these bits would look like this This table can be represented by the following logic statements Sum Cin A B Cout A B Cin AB Cin A B Sum Cout 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 1 1 0 0 1 0 1 0 1 0 1 Copyright 2008 by Helene G Kershner Building an Adder The circuit that represents this truth table is called a Full Adder Copyright 2008 by Helene G Kershner Building an Adder Another way to look at this is Copyright 2008 by Helene G Kershner Building an Adder A Full Adder then looks like this Copyright 2008 by Helene G Kershner Building an Adder Going back to our binary addition we find that a Full Adder adds one column at a time 1 1 1 0 1 0 0 1 0 1 0 1 0 0 1 1 0 1 1 0 0 0 1 0 0 1 1 1 0 1 Copyright 2008 by Helene G Kershner Building an Adder To perform real binary addition a series of binary adders are linked together The number of linked adders is determined by the manufacturer and reflects the finite length a particular computer can handle This is usually 8 bits 16 bits 32 bits or more Copyright 2008 by Helene G Kershner Building an Adder A series of …
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