Elementary arithmetic, with brief notices of its history... by Robert Potts.

20
5. A number is nmultiplied by 10, 100, 1000, -c., by annexing ones
two, three, Gc., ciphers to the right-hand figure of the multiplicand.
It will be seen that such multiplications are in fact the direct
consequence of the assumed notation. For when one, two, three, &c.,
ciphers are respectively annexed to any number, each of its digits is
removed one, two, three, &c., places respectively towards the left, and
in consequence of the assumed principle of the local value, each figure
becomes 10, 100, 1000 times respectively as great as it was before.
It may also be noted, if both multiplicand and multiplier end ia
one or more ciphers, the product will be found by multiplying the
significant figures of the two numbers, and annexing to the product
as many ciphers as are equal to those on the right of the multiplier
and multiplicand.
There are numerous devices whereby labour may in some instances
be avoided in finding the product of two numbers, some of which
are the following:
1. Any number is multiplied by 9, or 10 - 1, by subtracting the
given number from 10 times the number; and by 11, or 10 + 1, by
adding the given number to 10 times the number.
2. Any number can be multiplied by 99, 999, 9999, &c., by
annexing 2, 3, 4, &c., ciphers to the multiplicand, and subtracting
the multiplicand itself from this product.
And in a similar manner any number can be multiplied by another
composed of a repetition of the figure 9 with any other figure in the
highest place.1
6. The symbol x is assumed to indicate multiplication of numbers,
as 3 x 5 denotes 3 multiplied by 5, and 3 x 5 = 15 means the product
3 and 5 is equal to 15. Also, 3x5+2=17 denotes that 2 added to
the product of 3 and 5 is equal to 17, and 3x5-2=13, that 2
subtracted from the product of 3 and 5 is equal to 13. Also, when
two or more equal factors are to be multiplied together, the product
may be briefly expressed by writing at the upper part on the right of
one of them a small figure, denoting the number of equal factors.
Thus 3 x 3 is denoted by 32; 3 x 3 x 3 by 33; and 3 x 3 x 3 x 3 by 34.
Also 34 x 56 will express the product arising from 4 factors each equal
1 Ex. Find the product of 34578 by 999.
Here 34578000=1000 times 34578.
and 34578= 1,
34543422= 999 times 34578
Ex. Find the product of 34578 by 699.
Here 699=700-1.
And 24204600-700 times 34578
34578= 1,,
24170022=699 times 34578.