The sum of the number of digits in the two factors = −3 + (−2) = −5,
but as the slide projects to the _right_, the number of digits will be
−5 − 1 = −6.
From the last two examples it will be seen that when the first
significant figure of a decimal factor does not immediately follow the
decimal point, the minus sign is to be prefixed to the number of digits,
counting as many digits _minus_ as there are 0’s following the decimal
point. Thus, 0·03 has −1 digit, 0·0035 has −2 digits, and so on. Some
little care is necessary to ensure these minus values being correctly
taken into account in determining the number of digits in the answer.
For this reason many prefer to treat decimal factors as whole numbers,
and to locate the decimal point according to the usual rules for the
multiplication of decimals. Thus, in the last example we take 184 × 5 =
920, but as by the usual rule the product must contain 6 + 3 = 9 decimal
places, we prefix six cyphers, obtaining 0·00000092. When both factors
consist of integers as well as decimals, the number of digits in the
product, and therefore the position of the decimal point, will be
determined by the usual rule for whole numbers.
Another method of determining the number of digits in a product deserves
mention, which, not being dependent upon the position of the slide, is
applicable to all calculating instruments.
GENERAL RULE FOR NUMBER OF DIGITS IN A PRODUCT.—_When the first
significant figure in the product is smaller than in_ EITHER _of the
factors, the number of digits in the product is equal to the_ SUM _of
the digits in the two factors. When the contrary is the case, the number
of digits is 1_ LESS _than the sum of the digits in the two factors.
When the first figures are the same, those following must be compared._
_Estimation of the Figures in a Product._—We have given rules for those
who prefer to decide the number of figures by this means, but experience
will show that to make the best use of the instrument, the result, as
read on the rule, should be regarded merely as the _significant figures
of the answer_, the position of the decimal point, if not obvious, being
decided by a very rough mental calculation. In very many instances, the
magnitude of the result will be evident from the conditions of the
problem—_e.g._, whether the answer should be 0·3 in., 3 in., or 30 in.;
or 10 tons, 0·1 ton, 100 tons, etc. In those cases where the magnitude
of the answer cannot be estimated, and the factors contain many figures,
or have a number of 0’s following the decimal point, the use of notation
by powers of 10 (page 8) is of considerable assistance; but more usually
it will be found, that a very rough calculation will settle the point
with comparatively little trouble. Considerable practice is needed to
work rapidly and with certainty, when using rules. Moreover, the
experience thus acquired is confined to slide-rule work. The same time
spent in practising the “rough approximation” method will enable
reliable results to be obtained rapidly, with the advantage that the
method is applicable to calculations generally. However, the choice of
methods is a matter of personal preference. Both methods will be given,
but whichever plan is followed, the student is strongly advised to
cultivate the habit of forming an idea of the magnitude of the result.