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2.9 BINARY ARITHMETIC 53

Direct subtraction of two binary numbers also parallels that for base 10 subtraction.
Now however, when the subtrahend bit is 1 when the minuend bit is 0, a borrow is required
from the next MSB. Thus, the borrowing process begins at the MSB and ends with the
LSB — the opposite of the carry process for addition. Remember that a borrow of \2 from
the next MSB, creates a 10 2 in the column being subtracted. The following 8-bit example
illustrates the subtraction process in base 2:
EXAMPLE 2.22

10 i
0 0 f0 10 0 10 f- Borrows
101,o 0 I Z 0 0 1 0 1 2 = Minuend
-58| 0 -00 1 1 1 0 1 0 2 = Subtrahend
43 m 001010 1 1 = Difference

Here, the notation 0 or I represents denial of the 0 or 1 when a borrow is indicated. Notice,
as in the example just given, that the borrowing process may involve more than one level
of borrowing as the process proceeds from right to left.

2.9.2 Two's Complement Subtraction

Computer calculations rarely involve direct subtraction of binary numbers. Much more
commonly, the subtraction process is accomplished by 2's complement arithmetic — a con-
siderable savings in hardware. Here, subtraction involves converting the subtrahend to 2's
complement by using Eq. (2.14) in the form $2 + 1 and then adding the result directly to
the minuend. For an n-bit operand subtraction, n + 1 bits are used where the MSB bit is
designated the sign bit. Also, the carry overflow is discarded in 2's complement arithmetic.
The following example illustrates the process for two four-bit numbers, A and B:
EXAMPLE 2.23
A o;noi o;iioi =+is 10
-B -0:0111 +i:i001 = -7 10
r-Q] 0:0110 = +6, 0
Discard L ~ Sign bit positive
overflow
Further illustration continues by interchanging the minuend and subtrahend so as to yield
a negative number:
EXAMPLE 2.24
A OiOlll OiOlll = +7 10
-B -OillOl
]0
Discard ~—Sign Bit negative
overflow