Gawis Theory of Equations

Part-fraction.— The so-called principle of undetermined coefficients has frequent applica tion in the solution of the problem, to decom pose a given fraction into part-fractions (com monly called partial fractions) whose sum shall be the given fraction. Any fraction whose terms are rational integral functions of x may be thus decomposed. The method of procedure may be made sufficiently clear by a few ex amples. It will be observed that the problem is in a sense the inverse of the problem of summing fractions.

2 — I For example, the sum of and x x)' 3(2+x) x+4x+4 is Or The inverse x( I —x) (2 +x) • problem is : given the latter fraction, to find its components. It is plain that the only frac tions whose denominators are linear and whose sum is a fraction of the proper denominator a b and a linear numerator are — — and x' 1—x' 2+x Hence we assume: x +4 = a b 2x— x ' 1— x 2+x whence x +4=a ( 1—x) (2 +x) +bx(2+x)-Fcx(1—x), which is to be valid for all values of x. Expand ing the right-hand member and equating corre sponding coefficients on right and left, we obtain: 4=2a, I = — a +2b +c, 0 = — c; whence a = 2, b =3, c — it; and the component 2 frac.

— I tions are seen to, be x' 3 —xr 3 (2+x). For another example, we may take F= — A little reflection suffices to show that the as • stunption to be made is F= a c -+ (x— Then -I-3x — I (x — (x— 1) (x +2)+c (x+2); equating coefficients and solving the resulting equations, it is found that a =1, b=3, c =2. In case a factor of the given denominator is repeated k times, as inrit the z+nk(gx +q)' assumption to be made is: given fraction a, ak + + If F is of the form (mxi-f-nx-1-1)k(px+q)8(gx+h)' then assume F= ax+b ax+b ax + etc., as before. If N is of + + nx +1)k degree equal to or higher than that of the given denominator, F is converted by division into an integral function + a fraction the degree of whose denominator exceeds that of its numera tor. The latter fraction is then decomposed by the methods above indicated.

Indeterminate (Undetermined, Evanescent, Illusory) case of a fraction, it may happen that both terms vanish for some (x) 0 ' value of x, as x = a, yielding the form 0 which, as division by zero is meaningless, is itself with out meaning and is commonly called indetermi nate. In such case we are free (logically) to give the form a meaning, any meaning or value whatever. But while all meanings (values) are allowable, not all are expedient. For ex ample, - has a definite value for every x-value except x=a. For this value the frac tion takes the form 2 . To this we might assign 0 the value of 5 or —3, or any other. But such a

choice would be motiveless. On the other hand, xi— a' =x + a for all values of x except a; x — a for this critical value a, the right member takes a definite value, 2a, which is accordingly sug gested as the value to be naturally assigned to the indeterminate form in this case. The de cisive motive for this choice lies yet deeper : it is that as x varies through a sequence of values, say a + 3, a + 3, a +1,—, having a as limit, the corresponding sequence of fraction-values, 2a + 3, 2a + 3, 2a +1,—, approaches 2a as limit. Accordingly, if 0(x) assumes the form 0 0 —for x = a, the value assigned to 4(a) is the limit value which the sequence of fraction-val ues approaches as x approaches a through any sequence x-values for each of which 4 (x) has a definite value. The fraction 4(x) may be such that as x approaches a, f(x) approaches a definite value other than zero and that F(x) approaches zero. Such a fraction is x—a +4 x — a In such case the fraction-value obviously be comes larger and larger, surpassing every pre scribed number, a fact- commonly expressed by saying that as x approaches a, q (x) approaches positive or negative infinity ( or—co ) accord ing as the numbers in the fraction sequence are positive or negative. If, as x approaches a, both f(x) and F(x) approach w, then, for x=a, q(x) assumes the indeterminate form Z. But it may be made to take the form-- 0 ' since 0 f (x) F (x)=(1 + F (x))+ (1 f (x)). Other indeterminate forms also reducible to the form 0 0 , are 0.rio , re, 0, co — co , co 0 For further treatment see CALCULUS.

The boundary of what is or should be called elementary algebra is ill defined alike in theory and in practice, and besides the topics dealt with in this article other subjects are briefly treated in some of the elementary textbooks. Of such additional subjects, the more important as chance or probability, the complex variable, and theory of numbers, series, and others, are subjects of special articles in this work. Rela tively meagre, merely, introductory, textbooks of algebra are sometimes quite absurdly de scribed in their titles as "complete;' and others that might be called advanced are improperly characterized as ((higher!" The better usage has appropriated the term higher algebra to the doctrine of invariants and covariants (q.v.).

Bibliography.— Textbooks of elementary algebra, good, bad and indifferent, are very numerous. The most scientific work on the subject is that of Weber and Wellstein: (Ele mentare Algebra and Analysis.' The most comprehensive elementary English textbook is Chrystal's 'Algebra> (2 vols.).