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Binary Arithmetic is carried out by combinational logic circuits, the simplest of which is the half adder, shown in Fig. 4.1.1. This circuit consists, in its most basic form of two gates, an XOR gate that produces a logic 1 output whenever A is 1 and B is 0, or when B is 1 and A is 0. The AND gate produces a logic 1 at the carry output when both A and B are 1. The half adder truth table is shown in Table 4.1.1 and describes the result of binary addition.
1 plus 0 = 1_{2} (1_{10})
and
1 plus 1 = 10_{2} (2_{10})
The half adder is fine for adding two 1bit numbers together, but for binary numbers containing several bits, a carry may be produced at some time (as a result of adding 1 and 1) that must be added to the next column. As the half adder has only two inputs it cannot add in a carry bit from a previous column, so it is not practical for anything other than 1bit additions.
When 2 or more bits are to be added, the circuit used is the Full Adder, shown in Fig 4.1.2, (blue background) together with its simplified block diagram symbol. This circuit simply comprises two half adders, the sum of A and B from the first half adder is used as input A on the second half adder, which now produces a sum of the first half adder sum (S_{1}) plus any ‘carry in’ from the C_{IN} terminal. Any carries produced by the two half adders are then ‘ORed’ together to produce a single C_{OUT}. output. The truth table for the circuit is given in Table 4.1.2.
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