Elements of Algebra

24 ELEMENTS SECT. I. CHAP. VIII. *Of the Properties of Fractions.* 85. We have already seen, that each of the fractions, $ \frac{3}{1}, \frac{6}{2}, \frac{9}{3}, \frac{12}{4}, \frac{15}{5}, \ldots $ makes an integer, and that consequently they are all equal to one another. The same equality prevails in the following fractions, $ \frac{2}{1}, \frac{4}{2}, \frac{6}{3}, \frac{8}{4}, \frac{10}{5}, \frac{12}{6}, \ldots $ each of them making two integers; for the numerator of each, divided by its denominator, gives 2. So all the fractions $ \frac{3}{1}, \frac{6}{2}, \frac{9}{3}, \frac{12}{4}, \frac{15}{5}, \frac{18}{6}, \ldots $ are equal to one another, since 3 is their common value. 86. We may likewise represent the value of any fraction in an infinite variety of ways. For if we multiply both the numerator and the denominator of a fraction by the same number, which may be assumed at pleasure, this fraction will still preserve the same value. For this reason, all the fractions $ \frac{1}{2}, \frac{2}{4}, \frac{3}{6}, \frac{4}{8}, \frac{5}{10}, \frac{6}{12}, \frac{7}{14}, \frac{8}{16}, \frac{9}{18}, \frac{10}{20}, \ldots $ are equal, the value of each being $ \frac{1}{2} $. Also, $ \frac{1}{3}, \frac{2}{6}, \frac{3}{9}, \frac{4}{12}, \frac{5}{15}, \frac{6}{18}, \frac{7}{21}, \frac{8}{24}, \frac{9}{27}, \ldots $ are equal fractions, the value of each being $ \frac{1}{3} $. The fractions $ \frac{2}{5}, \frac{4}{10}, \frac{6}{15}, \frac{8}{20}, \frac{10}{25}, \frac{12}{30}, \frac{14}{35}, \frac{16}{40}, \ldots $ have likewise all the same value. Hence we may conclude, in general, that the fraction $ \frac{a}{b} $ may be represented by any of the following expressions, each of which is equal to $ \frac{a}{b} $; viz. merator, having 0 for its denominator, is, when expanded, precisely the same as $ \frac{a}{b} $. Thus, $ \frac{2}{0} = \frac{2}{2-2} $, by division becomes $ 2-2)2(1+1+1, \ldots $ ad infinitum $ 2-2 $ ---- $ 2 $ $ 2-2 $ ---- $ 2 $ $ 2-2 $ ---- $ 2, \ldots $