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Page 37
_Example in addition by two notations_
77,823,876
14,348,907
8,654,912
5,764,801
4,635,857
1,594,323
6,417,728
4,782,969
83,886,075
34,012,224
2,903,111
48,828,125
1,724,826
7,529,536
43,344,817
10,000,000
8,334,712
1,953,125
11,308,417
759,375
21,180,840
9,765,625
18,643,788
1,000,000
44,739,243
1,889,568
2,517,471
40,353,607
4,438,414
1,679,616
23,708,715
11,890,625
945,754
823,543
15,308,805
60,466,176
30,685,377
10,077,696
19,416,381
43,046,721
===========
740,685,681
[Illustration]
Eight volunteer observers to whom this example has already been
submitted showed wide difference in arithmetical skill. One of them
took but a few seconds over two minutes, in the best of six trials, to
add by the usual figures, and set down the sum, but one figure in all
the six additions being wrong; another added once in ten minutes
fifty-seven seconds, and once in eleven minutes seven seconds, with
half the figures wrong each time. The last-mentioned observer had had
very little training in arithmetical work, but perhaps that gave a
fairer comparison. In the binary figures she made three additions in
between seven and eight minutes, with but one place wrong in the
three. With four of the observers the binary notation required nearly
double the time. These observers were all well practiced in
computation. Their best record, five minutes eighteen seconds, was
made by one whose best record was two minutes forty seconds in
ordinary figures. The author's own best results were two minutes
thirty-eight seconds binary, and three minutes twenty-three seconds
usual. He thus proved himself inferior to the last observer, as an
adder, by a system in which both were equally well trained; but a
greater familiarity (extending over a few weeks instead of a few
hours) with methods in binary addition enabled him to work twice as
fast with them. Of the author's nine additions by the usual figures,
four were wrong in one figure each; of his thirty-two additions by
different forms of binary notation, five were wrong, one of them in
two places. One observer found that he required one minute
thirty-three seconds to add a single column (average of five tried) by
the usual figures, and fifteen seconds to count the characters in one
(average of six tried) by the binary. Though these additions were
rather slow, the results are interesting. They show, making allowance
for the greater number of columns (three and a third times as many)
required by the binary plan, a saving of nearly half; but they also
illustrate the necessity of practice. This observer succeeded with the
binary arithmetic by avoiding the sources of delay that particularly
embarrass the beginner, by contenting himself with counting only, and
not stopping to divide by two, to set down an unfamiliar character, or
to recognize the mark by which he must distinguish his next column.
One well-known member of the Washington Philosophical Society and of
the American Association for the Advancement of Science, who declined
the actual trial as too severe a task, estimated his probable time
with ordinary figures at twenty minutes, with strong chances of a
wrong result, after all.
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