9. Division of Decimal Fractions
Finite Decimals with digits after the decimal point.
A ratio of decimal fractions is a compound fraction
in which the numerator and denominator are both decimals - proper or
improper fractions, both expressible as whole number over a power of 10.
Imagine for some reason, the question how many times 0.23 meters went
into the distance 4781.55 meters was met. The division algorithm above
applies only to whole number divisors. So to obtain whole numbers, we
ask how many times does 23 centimeters = 23 x (0.01 meters) go into
4781.55 meters = 478155 centimeters. The long division of 478155
by 23 gives 20789 times completely with a remainder of 8
centimeters = 0.08 meters leftover.
Shifting the Decimal Point
In general if M and N are given by decimal fractions then
for some whole or natural numbers P, Q, a and b. Here we assume a and b
give the number of digits after the decimal point in the finite decimal
notation for M and N respectively, and we may also assume the ones digit
in both P and Q are nonzero. Now
M
N
|
=
|
P
10a
---
Q
10b
|
=
|
P
10a
|
×
|
10b
Q
|
=
|
P×10b
10a
|
×
|
1
Q
|
=
|
P× 10b
10a
-----
Q
|
|
The foregoing justifies the long division method of shifting the decimal
point in the dividend and divisor by number of decimal places in
the divisor to obtain an integral (whole number) divisor Q
Another Twist:
M
N
|
=
|
P
10a
---
Q
10b
|
=
|
P
10a
|
×
|
10b
Q
|
=
|
P×10b
Q×10a
|
The twist yields
456.89 456.89 x 1000 456890
------ = -------------- = --------
34.567 34.567 x 100 34567
Long Division - Remainder Analysis and Convergence
The equalities
M
N
|
=
|
P
10a
|
=
|
P
10a
|
×
|
10b
Q
|
=
|
P×10b
10a
|
×
|
1
Q
|
=
|
P× 10b
10a
-----
Q
|
|
|
Q
10b
|
imply for the long division computation of M/N that we
can shift the decimal point in both the denominator or divisor N and
dividend or numerator M to obtain an equivalent fraction in which the
denominator is a whole number Q.
Theorem: If M and N are given by decimal fractions
then M has a or few places after the decimal point, N has b
or fewer places after the decimal point, and
Moreover if a = b then
Note: We can extend the decimal expansion of M or N with zeroes
after the decimal point to make a = b. The replacement of the
decimal compound fraction M/N by P/Q where P and Q are both whole
numbers provides a rule for moving the decimal point or eliminating it
in fractions involving decimals.
Note: The rule
M
N
|
=
|
M
|
×
|
10b
Q
|
=
|
M×10b
Q
|
provides a method for shifting the decimal point in the dividend M and
divisor N to obtain a whole number divisor Q. Then the long division
algorithm can be applied as is, or in continued form.
Now for any whole number k, long division
where 0 < r < Q is a natural number. Therefore
division by 10k Q gives
P
Q
|
=
|
s(k)
10k
|
+
|
1
10k
|
×
|
r
Q
|
Here
Therefore
provides at least the first k digits of the decimal expansion of
P
Q
beyond the decimal point. The result
P
Q
|
=
|
s(k)
10k
|
+
|
1
10k
|
×
|
r
Q
|
can also (I presume) be obtained by continuing the long division process
as well.
Remark: The foregoing implies the decimal expansion of P/Q will
either terminate or provide an increasing Cauchy sequence of decimal
fractions s(k)/10k which converge to a limit.
Exercise: Show computing P/Q to k decimal after the decimal
point gives
to k+b-a places after the decimal point.
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Teachers & Tutors: Site pages offer better or best practices for providing skills -
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Calculus Starter Lessons
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Unsolicited Advice
Learning to do and high marks if it comes to easy is often
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