Reduction

The
material related to reduction is expressed in the form of problems.

1. Take a number that is expressed as a whole
number plus a fraction and express it as a fraction, i.e. express a mixed
number as a fraction.

The
numerator in the expression is found by multiplying the denominator in the
fraction by the whole number and adding it to the numerator.

2. Reduce a given fraction to its lowest
terms.

This
is done by finding the greatest common divisor between the numerator and
denominator. Then divide the numerator and denominator by this number.

3. Reduce a mixed fraction to a simple
fraction.

There
are two cases. When the mixed fraction is composed of fractions, invert the
fraction in the denominator. Multiply two numerators together and multiply the
two denominators together. When the mixed fraction is composed of mixed
numbers, express the mixed numbers as fractions and proceed as in the first
case.

4. Reduce a compound fraction to an equivalent
simple one.

Reduce
all the numbers, be they fractions or mixed fractions,
to simple fractions. Then multiply the numerators together and the denominators
together.

5. Reduce any number or fraction to a fraction
that shall have any proposed denominator.

Express the number as a fraction and multiply it by
the proposed denominator.

6. Reduce any number of fractions to
equivalent simple fractions that have a common denominator.

Express
all the fractions as simple fractions. Then multiply the numerator and denominator
of each fraction by the denominators of the others.

7. Any fraction in its lowest terms being
given, to find a series of a simple fraction in terms still lower that shall
each express the value of the given fraction as nearly as possible.

The
phrase “in its lowest terms” means that the faction has been reduced to the
point that the numerator and denominator are relatively prime.
The solution is expressed as a continued fraction. To find the continued
fraction, divide the numerator into the denominator. This yields a number plus
a remainder. The remainder and its divisor is now a fraction. Repeat the
procedure until the remaining fraction is 1 divided by some number.

No
reason is given in the notebook for looking at continued fractions. It is just
a numerical exercise.