# Subtractor

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In electronics, a subtractor – a digital circuit that performs subtraction of numbers – can be designed using the same approach as that of an adder. The binary subtraction process is summarized below. As with an adder, in the general case of calculations on multi-bit numbers, three bits are involved in performing the subtraction for each bit of the difference: the minuend ($X_{i}$ ), subtrahend ($Y_{i}$ ), and a borrow in from the previous (less significant) bit order position ($B_{i}$ ). The outputs are the difference bit ($D_{i}$ ) and borrow bit $B_{i+1}$ . The subtractor is best understood by considering that the subtrahend and both borrow bits have negative weights, whereas the X and D bits are positive. The operation performed by the subtractor is to rewrite $X_{i}-Y_{i}-B_{i}$ (which can take the values -2, -1, 0, or 1) as the sum $-2B_{i+1}+D_{i}$ .

$D_{i}=X_{}\oplus Y_{i}\oplus B_{i}$ $B_{i+1}=X_{i}<(Y_{i}+B_{i})$ ,

where ⊕ represents exclusive or.

Subtractors are usually implemented within a binary adder for only a small cost when using the standard two's complement notation, by providing an addition/subtraction selector to the carry-in and to invert the second operand.

$-B={\bar {B}}+1$ (definition of two's complement notation)
{\begin{alignedat}{2}A-B&=A+(-B)\\&=A+{\bar {B}}+1\\\end{alignedat}} 