Placement of Decimal Points

In any whole number in the decimal system, there
is understood to be a terminating mark, called
a decimal point, at the right-hand end of the
number. Although the decimal point is seldom shown except in numbers involving
decimal fractions (covered in chapter 5 of
this course), its location must be known. The
placement of the decimal point is
automatically taken care of when the end O’s
are correctly placed. Practice problems.
Multiply in each of the following problems:

DIVISION METHODS

Just as multiplication can be considered as repeated
addition, division can be considered as repeated
subtraction. For example, if we wish to
divide 12 by 4 we may subtract 4 from 12 in successive
steps and tally the number of times that the
subtraction is performed, as follows:

As indicated by the asterisks used as tally marks,
4 has been subtracted 3 times. This result is
sometimes described by saying that "4 is
contained in 12 three times."

Since successive subtraction is too cumbersome for rapid, concise
calculation, methods which treat division as
the inverse of multiplication are more useful. Knowledge of the multiplication
tables should lead us to an answer for a
problem such as 12 + 4 immediately, since we
know that 3 x 4 is 12. However, a problem such
as 84 + 4 is not so easy to solve by direct reference
to the multiplication table.

One way to divide 84 by 4 is to note that 84 is the same as 80 plus 4. Thus 84 + 4 is the same
as 80 + 4 plus 4 + 4. In symbols, this can be
indicated as follows:

(When this type of division symbol is used, the quotient
is written above the vinculum as shown.) Thus,
84 divided by 4 is 21.

From the foregoing example, it can be seen that
the regrouping is useful in division as well as
in multiplication. However, the mechanical procedure
used in division does not include writing
down the regrouped form of the dividend. After becoming familiar with the
process, we find that the division can be performed directly,
one digit at a time, with the regrouping taking
place mentally. The following example illustrates
this:

The thought process is as follows: "4 is contained in 5 once"
(write 1 in tens place over the 5); "one
times 4 is 4" (write 4 in tens place under
5, take the difference, and bring down 6); and
"4 is contained in 16 four times" (write 4 in
units place over the 6). After a little practice, many people can do the work
shown under the dividend mentally and write
only the quotient, if the divisor has only 1 digit. The
divisor is sometimes too large to be contained
in the first digit of the dividend. The following
example illustrates a problem of this kind:

Since 2 is not large enough to contain 7, we divide
7 into the number formed by the first two digits,
25. Seven is contained 3 times in 25; we write
3 above the 5 of the dividend. Multiplying, 3
times 7 is 21; we write 21 below the first two digits
of the dividend. Subtracting, 25 minus 21 is
4; we write down the 4 and bring down the 2 in
the units place of the dividend. We have now formed
a new dividend, 42. Seven is contained 6
times in 42; we write 6 above the 2 of the dividend.
Multiplying as before, 6 times 7 is 42; we
write this product below the dividend 42. Subtracting,
we have nothing left and the division is complete.

Estimation

When there are two or more digits in the divisor,
it is not always easy to determine the first
digit of the quotient. An estimate must be made,
and the resulting trial quotient may be too
large or too small. For example, if 1,862 is
to be divided by 38, we might estimate that 38
is contained 5 times in 186 and the first digit of
our trial divisor would be 5. However, multiplication reveals that the product
of 5 and 38 is larger than 186. Thus we would
change the 5 in our quotient to 4, and the
problem would then appear as follows:

On the other hand, suppose that we had estimated that 38 is contained in 186
only 3 times.

We would then have the following:

Now, before we make any further moves in the division
process, it should be obvious that something is wrong. If our new dividend is
large enough to contain the divisor before
bringing down a digit from the original
dividend, then the trial quotient should have
been larger. In other words, our estimate is
too small.

Proficiency in estimating trial quotients is gained
through practice and familiarity with number
combinations. For example, after a little
experience we realize that a close estimate can be made in the foregoing problem
by thinking of 38 as "almost 40."
It is easy to see that 40 is contained 4
times in 186, since 4 times 40 is 160. Also,
since 5 times 40 is 200, we are reasonably
certain that 5 is too large for our trial
divisor.