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

5. PROP. To multiply one decimnal by another, and to dedtee the general;
rule for the place of the decimal point in the product.
To multiply 3-457 by 21-34:oere 3457 31057, and 21'34 _ 2134
1000 100
Then 3-457 x 21-34 = 3457 2134
1000 100
7377238
-= 73773' a decimal of five places,
= 73-77238
Hence the product of two decimals is found by multiplying the
decimals as integers, and pointing off from the right-hand figure of
the product as many decimal places as there are in the multiplicand
and the multiplier.
If, however, there are not as many figures in the product as the
number of decimal places in the multiplicand and multiplier together,
the required number of decimal places must be made up by prefixing
to the product as many ciphers as make up the defect, and the
product will be wholly a decimal.
6. POPo. A decimal is multiplied by 10, 100, 1000, &c., by removing the decimal point in the given decimal one, two, three, &c., places;
towards the right.
For any given decimal expressed as a decimal fraction is multiplied
by 10, 100, 1000, &c., by dividing the denominator by 10, 100, 1000,
&c., and thus the decimal fraction, when expressed as a decimal, will
consist of 1, 2, 3, &c., decimal places less than before.
7. PROP. To divide one decimal by another, and to deduce the general
rules for the place of the decimal point in the quotient.'
Three ciphers have been annexed to the decimal part of the larger number to
make the number of decimal places equal in the two numbers.
It is evident that as ten units make 1 ten, ten tens 1 hundred, and so on, so ten
tenths make 1 unit, ten hundredths 1 tenth, and so on; the same law obtains both.
in the integers and the decimal parts. The processes of addition, subtraction, multiplication, and division of decimals are effected in the same manner as in integral
numbers.
A decimal is said to be correct approximately for any number of places when any
number of figures on the right of the decimal have been omitted; but if the first of
the figures omitted be greater than 5, the last figure on the right must be increased
by 1. As an example 5'293 is nearer to 5'2929311 than 56292, for 5 293 exceeds:
5-2929311 by '0000689, and 5-292 is less than 5-2929311 by '0009311. Hence the
error is less in the former case than in the latter; the former is in excess and the
latter in defect of the truth.
In the operations of multiplication and division, as the terms dividend, divisor,.
and quotient in the latter correspond to product, multiplicand, and multiplier in.
the former; it is possible that the number of decimals given in a dividend may be.