If then a be very small, 1/a is very large, and if a be very large, 1/a is very small. When a becomes zero, the fraction a/b assumes the form 0/6 and, by the proposition of Art. 127, is itself zero. But when b is zero, the fraction assumes the form a/0 and, regarded as a quotient, has no meaning; for division by zero is an impossible operation. It becomes merely symbolic. Yet it frequently makes its appearance in algebraic operations, and since it may be regarded as having arisen by virtue of a decrease of the denominator from a finite quantity to zero, that is, by an increase of the fraction itself beyond measurable limits, it is called an infinite quantity, or infinity, and for convenience the special symbol is used to represent it. But this nomenclature must be regarded as conventional, and the symbol must be used in algebraic operations with extreme caution. A complete study of its legitimate use requires more extended discussion than is at present appropriate. [See Treatise on Algebra, Art. 217.] 169. We will now give some examples of more complex fractional expressions. α a b b a a b a + x a - x α a + х 4) / ( х a + x α - X a (a + x) (a + x)-(ax) (ax) (α − x)(a + x) (a + x) (a + x) + ( a − x) ( a − x) _ 2 a2 + 2 x2 Hence the given fraction is equal to 170. The following theorems are of importance. Theorem I. If the fractions a/b, c/d, e/f, etc., be all equal to one another, then will each be equal to the fraction pa + qc+re +... Let each of the equal fractions be equal to x. Then, since a/b = x, c/d = x, e/ƒ = x, etc., ... a = bx, c=dx, e=fx, etc.; .. pa=pbx, qc=qdx, re=rfx, etc. |