The Real Number System by John M. H. Olmsted

Author:John M. H. Olmsted
Language: eng
Format: epub
Publisher: Dover Publications, Inc.
Published: 2018-10-14T16:00:00+00:00

Α few elementary properties of the correspondence that determines by means of (5) a rational function ϕ in terms of a given ordered pair f:g of polynomials (g ≠ 0) are given in the theorem:

Theorem V. If f and r are any polynomials, if g, h, and s are any nonzero polynomials, and if 0 and 1 denote the constant polynomials identically equal to the real numbers 0 and 1, respectively, then:

(i)[f:g] = [r:s] if and only if f:g ~ r:s;

(ii)[fh:gh] = [f:g];

(iii)[f:1] = f;

(iv)[h:h] = [1:1] = 1;

(v)[0:g] = [0:l]=0.

Proof. (i): If [f:g] = [r:s] = ϕ, then, by Theorem IV, the ordered pairs f:g and r:s are separately equivalent to the same ordered pair in canonical form, and hence are equivalent to each other : f:g ~ r:s. On the other hand, if f:g ~ r:s, then the unique ordered pair F:G in canonical form to which f:g is equivalent must be equivalent to r:s and hence must be the unique ordered pair R:S in canonical form to which r:s is equivalent. Therefore [f:g] = [r:s]. (ii): True by part (i), Theorem III. (iii): Since f:1 is in lowest terms and 1 is monic, if ϕ = [f: 1], then for every real number x (the polynomial 1 never vanishes) the following equation holds: ϕ(x) = f(x)/1 = f(x), and ϕ = f (iv) : By (ii) and (iii), [h:h] = [1:1] = 1. (v) : By (ii) and (iii),[0:g] = [0:1] = 0.

Finally, a useful criterion for (5) to hold, expressed in terms of the values of the functions concerned, is given in the theorem:

Theorem VI. The equation ϕ = [f:g], where ϕ is a rational function, holds if and only if the equation between real numbers:

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