MATHEMATICAL RECREATIONS Yt

459. The theorem of § 448, that ab = (e + )10 + cd, has
an interesting special application as follows:

Th. If 5<a <10 and 5 < b < 10, excepting the cases
whena = 6 = b,a =6andb =7,a =7 and b = 6, then ab is
a two digit number whose first digit is the sum of the excesses of a
and b over 5 and whose second digit is the product of their com-
plements with respect to 10.

For in this case, since 10 < a 4+ b < 20,0 <e + f < 10.

Hence ¢ + f is a digit. Also by actual trial in the cases
included we find that cd is a digit.

In case ¢ =6 =0, ¢d = 16 111 case ¢ =06 and b = 1 o
g.= 1 and'd =6 cd'=1Th

460. If for our present purpose we agree to call © and [0
digits, as well as 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, the last theorem
may be stated simply as follows:

Th. The lazy man’s rule for multiplication (1).

The product of two numbers between 5 and 10 is a two digit
number whose first digit is the sum of the excesses of the two
numbers over 5 and whose second digit is the product of their
complements with respect to 10.

For 6 X 6 this rule gives twenty sixteen, for 6 X 7 and
7 X 6 thirty twelv, which of course are correct, but twelv and
sixteen are not ordinarily considerd digits in the decimal
system.

The advantage of this rule lies in the fact that it enables
one to get the product of two digits both greater than 5 by
means of addition, subtraction, and the multiplication of two
digits, the complements of the given digits, both less than 3.
As some persons find it hard to memorize the products of two
digits both greater than 5, the advantage of the rule is obvious.

461. The fact stated in § 451 will help to explain why in
the addition table up to 9 4+ 9, given on the next page with
the diagonal line of 10’s left out, two numbers in squares on
opposit sides of this diagonal line and equidistant from it
have complementary units digits.

() See Ball, 1. c., p. 194.