New Ulm weekly review. [volume] (New Ulm, Minn.) 1878-1892, 1880-03-03
Three Rules Tor Abroviating Nultipli.
cation.

Three Rules for Abreviating Nultipl.
cation.
The "Talkhys ANAL al Hissab"
("Analytiesl Resume of Calculating
Processes, ~ written by Ibm al Banna,
Of Morocco, contains. in the chapter
devoted to the multiplication of in.
tegral numbers, some abbreviated
methods by means of which, in certain
particular cases, the product of the
multiplication Of two integral numbers
may be obtained very quickly. As
these processes deserves to be known,
and are not found in any arithmetical
treatise (although the '.Talkhys "
gave them nearly six centuries ago),
We publish them for the benefit of our
readers.
FIRST RULE.-Suppose it be required
to multiply by itself number COM
posed of figures, each equal to unity;
tor example, Hill by I'LL.
We say that the product will be 1885,
454.321.
To obtain this we write the number
Of figures contained in one of the fac
tors, and to the left and right of this
number we place symmetrically the
natural decreasing series of numbers
less than it. Thus, in the example pro-
posed we write down 8, that being the
number of figures in one of the factors,
and then we place on each side of that
number the natural decreasing series
of figures less than 5, that is, 4, 8, 2,
l, in the following form. 1884 5 1921.
another E1oneple.--Multiply ILL.,
Ill by I'LL, ILL., The product will be
at once obtained by writing to the left
and right of z (the number of figures
contained in either of the two factors)
the numbers 6, 5, 4, 8, 2, l, as follows:
l, 234, 567, 654,3 If we multiply II
by II, the application of the same rule
will give as result, 121. z,4.-s.
SECOND RULE.-To multiply by it.
self a number composed of figures,
each equal to 8; for example, SO, 999
by 90, 999. We say that the product
will be, 1893, 800, 0O1.
To obtain this result, we write down
the figure s, placing to its left as many
mines, and to its ri ht as many ciphcrs,
as there are figures less one, contained
in either of the two factors, aftcrward
adding to the extreme right of the re.
sulting number the figure l. Thus,
then, in the proposed example (99, 800
by 81, 999) we write the figure s, and
to it's left the figure 9 repeated four
times (5-D, and to it's right four
zeros OILY, giving as result 9999 8
O0O0; now annexing the figure l, we
obtain the product sought, 1899, $780,
0O1.
Another Example, we desired to
find the product of 8 by 9, we should
obtain, by applying the general rule, 81.
In fact, in this case, the number of, fig
ures of either factor, diminished by /,
gives zero as result. This explains
why the figure 8 does not appear ac-
companied by mines or cipher, but only
by the figure of the units.
THIRD RULE.-To multiply a number
composed of figures each equal to 9, by
another whose figures, although equal
to each other, arc different from 9; for
example, 999 by 666.
In this case we say the product will
be equal to 665, 1884.
To obtain this result, we first obtain
the product of a figure of the multiple
cand by that of the multplier; the figure
of the units of this preliminary product
will be the number of the units of the
product sought. To the left of the fig
ure of the tens Of the said preliminary
product write the figure of the multi
plier as many times as there are figures,
less one, in either of the two factors;
and to its right we place the same num-
Der of figures, each equal to the differ
ence between figure of the multiple
cand (O and figure of the multiplier
(6). TO the extreme left of the guam
tity thus obtained we annex the figure
of the unit of the preliminary product;
thus we have the product sought. To
make this clearer: in the proposed ex-
ample, 999xGC6, the preliminary pro-
duct will be 9x6=54; so that, to the
left of the figure (5) of the tens, we
place the figure of the multiplicr (6) az
many times, less one, as there are figures
in either factor, which in this case will
be twice (3--D, and to its right twice
the figure (tl,e difference between 9
and s., as follows, IS 5 33; and to COM
plete this number we annex to its right
the figure (4) of the units of the prelim-
mary product (54). We then have the
product sought, 665,334. l...~ ..
Another Example. it be re-
CURED to multiply 1899, 800 by 3, $685.
1888: The preliminary product is ~,
and the number of figures in each Of
the factors is 4; so that, writing to the
left of the figure (of the tens of the
preliminary product) six (7-D fig
urcs. each equal to one of those Of the
multiplier (3), we have 3333332. Now,
If to the right of the same figure (2) we
write six figures, each equal to the diff
ference between the figures of the two
factors (9-3), we have 33333333 666666;
and, finally, we obtain the definite pro-
duct by annexing to the right of the
foregoing quantity the number (the
unit figure of the preliminary product),
as follows: $685., 1926, 666, 667.
Anothcr Exan,ple.--H SO be multi
piled by %, as the preliminary product
is Is, and each factor contains two fig
ures, it will be sufficient to write, to the
left OF the figure of the tens ~, the
figure of the multiplier (2) but once,
and to its right the figure but onco
(the latter being the difference between
8 and p; we will thus have 217, and to
this we annex the number (the unit
of the preliminary product), and OBJ
tain the product sought, viz, 2,178.
This rule will hold good in all cases
except those in which the factors con-
tain each but a single figure. If, for
example, we should apply the rule to
the case 9x2, the preliminary pro-
duct, Is, would at the same time be the
final product. It is easy to see that the
second rule may be considered as q
particular case of the third in
which the difference between the fig
ure of the multiplicand and that of the
multiplier is zero.r-1 Translated from
Tie Cronica Scie,tifca, Of barcelona, IN
Scientific American.