Algebra

(al'je-brah.) [Ar. al, the, and gebr, resolution.] (Math.) A science the object of which is to abridge and generalize the resolution of all questions relating to quantities. The symbols it employs are of three kinds: those of quantity, known or unknown, which consist of ordinary numbers and letters of the alphabet; those of operations several of which are borrowed from arithmetic,as +, -, X,, /, .;- and mere abbreviations for ordinary words. The combinations of these symbols according to fixed laws led to algebraic al expre ions or form ulæ, in which actual computations are indicated rather than performed. The universality of A. as compared with arithmetic consists in the fact that, in the latter, computations being effected as they arise, all traces of their intermediary steps are obliterated, and the result is applicable to a single case only; whereas, in A., the form ulæ contain implicitly the answers to an unlimited number of questions. Again, to the equivalence of two algebraic al form ulæ always corresponds a general theorem, which arithmetic can only verify in particular cases. Thus, from the algebraic al identity (a + b) (a - b) =a² -b², we learn that the product of the sum and difference of any two numbers is equal to the difference of their squares. [s. 19]