196 finite

times 8” is computed as 60 + 2 ×2 = 64, “9 times 7” as

60 + 1 ×3 = 63, and “6 times 7” as 30 + 3 ×4 = 42.

Notice that one is never required to multiply two num-

bers greater than five. That the method works is

explained by the algebraic identity:

(5 + a)(5 + b) = 10(a+ b) + (5 – a)(5 – b)

The identity (N+ a)(N+ b) = 2N(a+ b) + (N– a)(N– b)

shows that we can extend this method to use of a differ-

ent number of digits on each hand. For example, with

N= 10, using fingers and toes, one can readily compute

17 ×18 as “seven raised fingers” and “eight raised

toes.” Counting each raised digit as 20 (2N) we have:

17 ×18 = 20 ×15 + 3 ×2 = 306.

See also E

GYPTIAN MULTIPLICATION

; E

LIZABETHAN

MULTIPLICATION

;

MULTIPLICATION

; N

APIER

’

S BONES

;

R

USSIAN MULTIPLICATION

.

finite Intuitively, a set is said to be finite if one can

recite all the elements of the set in a bounded amount

of time. For instance, the set {knife, fork, spoon} is

finite, for it takes only a second or two to recite the ele-

ments of this set. On the other hand, the set of natural

numbers {1, 2, 3, …} is not finite, for one can never

recite each and every element of this set.

Despite our intuitive understanding of the concept,

it is difficult to give a precise and direct mathematical

definition of a finite set. The easiest approach is to

simply define a finite set to be one that is not

INFINITE

,

since the notion of an infinite set can be made clear.

Alternatively, since there is a well-defined procedure

for mechanically writing down the string of natural

numbers 1, 2, 3, …, one can define a finite set to be

any set Swhose elements can be put in one-to-one cor-

respondence with a bounded initial segment of the

string of natural numbers. For instance, matching

“knife” with 1, “fork” with 2, and “spoon” with 3,

the set {knife, fork, spoon} is finite because its ele-

ments can be matched precisely with the string of nat-

ural numbers {1, 2, 3}.

finite differences To analyze the terms of a

SEQUENCE

, it can be useful to create a table of successive

differences (in the sense of “right minus left”) between

the terms of the sequence, and subsequent differences of

the differences. For example, for the sequence

1,2,4,8,15,26,42,64,… we obtain the difference table:

From the pattern that is now apparent, it is clear that

the next number in the original sequence will be

64 + 29 = 93.

The entries in the first row under the original

sequence are said to be the “first finite differences”; the

second row under the sequence depicts the “second

finite differences,” and so forth. All the terms that

appear in a table of finite differences are completely

determined by the values that appear in the leading

diagonal. For instance, if the values a, b, c, … shown

below are known, then the remainder of the table must

appear as follows:

The coefficients that appear in the top row match

the entries in each row of P

ASCAL

’

S TRIANGLE

, which are

given by the

BINOMIAL COEFFICIENT

s. This suggests that

it would be enlightening to examine the finite difference

tables of the sequences . We obtain: