Fractions

One of the leaves in the notebook is missing. The leaf that follows begins midsentence describing mixed numbers (for example, 4⅔ is a mixed number) and then moves on to define mixed fractions. These are fractions or mixed numbers expressed as fractions (for example, 5⅜ / 4⅔). This is followed by a definition of a compound fraction. These are factions or mixed numbers connected by a multiplication sign (×) or the word “of” (for example, 3 × ⅞ of 9½, which is the same as 3 × ⅞ × 9½).

The section concludes with some rules of arithmetic for fractions.

1.  Addition or subtraction of fractions. Add or subtract the numerators of the fraction when the denominators are the same.

For example, .

2.  Multiplication of a fraction. Either multiply the numerator or divide the denominator.

For example, , or .

3.  Division of a fraction. Either multiply the denominator or divide the numerator.

For example, or .

4.  To change the terms of a fraction without changing its value. Multiply the numerator and denominator by the same amount.

For example, .

5.  Inverse. Take any fraction and invert it (i.e. the numerator becomes the denominator and the denominator becomes the numerator). Then the product of the two fractions has the value 1. Thus the reciprocal of any fraction is 1 divided by the fraction.

For example, . is the reciprocal of .

6.  To multiply by a fraction, divide by its reciprocal and to divide by a fraction multiply by its reciprocal.

7.  For several factions of the same value, the sum of the numerators divided by the sum of the denominators will have the same value as the original fractions. For two fractions of the same value, the difference in the numerators divided by the difference in the denominators will have the same value as the original fractions.

For example, for , , and , .