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the sum of the digits now in the eights column is 1 1 1 1 1 1 10 1 11, hence we write down 1 and carry a 1 into the sixteens column, giving the answer 11000, as shown to the right above with decimal equivalents alongside.
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In the above we found the sum of just two binary numbers, but suppose the sum of, say, one thousand such numbers must be found. In the internal circuitry of a digital computer, this is most conveniently done by adding the binary numbers together two at a time until the nal sum is reached. As mentioned before, we must remember that a digital computer can execute millions of such routine operations per second. Next, let us consider the subtraction of one number from another number. Subtraction makes use of the borrow operation, illustrated rst in the following decimal-system example. PROBLEM 9 4 6 7 3 7 7 6 minuend subtrahend difference
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8 15 13 4 4 7 8 7 6
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Discussion. Beginning at the right-hand side in the PROBLEM we rst have 7 6 1, which presents no di culty because 7 is larger than 6. Continuing on, from right to left, we next must subtract 7 from 3 (actually, 70 from 30), which does present a di culty because 3 is smaller than 7. To get around this di culty we now, in the minuend, borrow a 1 from the 6, and transfer the borrowed 1 over to the 3; however, since a 1 in the third column has ten times the value of a 1 in the second column, this e ectively makes the 3 become 13, as shown in the SOLUTION. Since 13 is larger than 7 we now have 13 7 6, as shown. At this point we must not forget that the 6, in the third column, is now changed to 5 (because of the previous borrowing of the 1 from the 6). Therefore, continuing on in the PROBLEM, we must now subtract 7 from 5, which again presents a di culty because 5 is smaller than 7. To get around this di culty we now, in the minuend, borrow a 1 from the 9 and transfer the borrowed 1 to the 5, e ectively making the 5 become 15, as shown in the SOLUTION; thus we have 15 7 8, as shown. Because of the borrowing of the 1, the 9 becomes 8, as shown in the SOLUTION; hence the last step is to subtract 4 from 8, giving the nal answer 4861. We must remember that the same basic arithmetic procedures apply to all positional number systems, whatever the particular radix. Thus the borrow procedure, illustrated above for the decimal system, is used in the same way to subtract one binary number from another binary number, as the following example illustrates. PROBLEM 1 0 0 1 1 1 1 0 0 1 1 1 minuend subtrahend difference
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