Root (mathematics)

“ But an equation with real coefficients may have as well imaginary as real roots, and we have further the theorem that for any such equation the imaginary roots enter in pairs, viz. α + βi being a root, then α − βi will be also a root. It follows that if the order be odd, there is always an odd number of real roots, and therefore at least one real root. ”

“ I observed that the numbers which are required in calculating by Completion and Reduction are of three kinds, namely, roots, squares, and simple numbers relative to neither root nor square.
A root is any quantity which is to be multiplied by itself, consisting of units, or numbers ascending, or fractions descending. ”

“ The fundamental result in this subject is that every rational integral equation has a root, a theorem which it is by no means easy to prove. It follows without difficulty that an equation of degree n has exactly n roots, real or imaginary. ”

“ It has already been shown how the several roots of the equation xn − 1 = 0 can be expressed in the form cos 2sπ/n + i sin 2sπ/n, but the question is now that of the algebraical solution (or solution by radicals) of this equation. There is always a root = 1 ”

“ Very little has been done in regard to the calculation of the imaginary roots of an equation by approximation; and the question is not here considered.
10. A class of numerical equations which needs to be considered is that of the binomial equations xn − a = 0 (a = α + βi, a complex number) .
The foregoing conclusions apply, viz. there are always n roots, which, it may be shown, are all unequal. And these can be found numerically by the extraction of the square root, and of an nth root, of real numbers, and by the aid of a table of natural sines and cosines. ”

“ We thus come to the modern distinction between axiom and postulate, and say that a general statement admitted to be true without proof is an axiom, while a postulate in geometry is a geometric statement admitted to be true, without proof. For example, when we say "If equals are added to equals, the sums are equal," we state an assumption that is taken also as true in arithmetic, in algebra, and in elementary mathematics in general. ”

“ For in this case both addition and subtraction may be employed, which will not answer in any other of the three cases in which the number of the roots must be halved. And know, that, when in a question belonging to this case you have halved the number of the roots and multiplied the moiety by itself, if the product be less than the number of dirhems connected with the square, then the (8) instance is impossible; [19] but if the product be equal to the dirhems by themselves, then the root of the square is equal to the moiety of the roots alone, without either addition or subtraction. ”

“ But the existence of a double root implies a certain relation between the coefficients; the general case is when the roots are all unequal.
We have then the theorem that every rational symmetrical function of the roots is a rational function of the coefficients. This is an easy consequence from the less general theorem, every rational and integral symmetrical function of the roots is a rational and integral function of the coefficients. ”

“ Hence it follows that the roots of the proposed equation are generally expressed by the formula
y = √a + √b + √c; where a, b, c denote the roots of this cubic equation,
But to find each particular root, we must consider, that as the square root of a number may be either positive or negative, so each of the quantities √a, √b, √c may have either the sign + or − prefixed to it; and hence our formula will give eight different expressions for the root. It is, however, to be observed, that as the product of the three quantities √a, √b, √c must be equal to √R or to −1/8 q ”

“ When the root is an odd composite number, as 9, 15, &c., it will be found that in some paths, different from the two along which the p1 and q1 were placed, instead of having each of the p’s and q’s, some will be wanting, while some are repeated. ”

“ If, then, I know that b is the square root of a, an individual embodiment of this simple system of ideas is not such that the determination of one object A in an individual embodiment predetermines absolutely the other individual. In this case, two alternatives are left open; and when I exemplify the system by the individual quantity A, B is free to be either one of the two square roots of A. The n different roots of an algebraic equation of the nth degree illustrate a still more complex instance of this sort of ambiguity. ”

“ The condition in order that an equation of a given prime order n may be solvable by radicals was in this way obtained—in the first instance in the form (scarcely intelligible without further explanation) that every function of the roots x1, x2 . . . xn, invariable by the substitutions xak + b for xk, must be rationally known; and then in the equivalent form that the resolvent equation of the order 1.2. . . (n − 2) must have a rational root. ”

“ In reality, in the present condition of the human mind, there are directly but two great general classes of phenomena, whose equations we constantly know; these are, firstly, geometrical, and, secondly, mechanical phenomena. Thus, then, the concrete part of mathematics is composed of Geometry and Rational Mechanics. ”

“ His first published paper—a thesis for the Doctorate of Philosophy, in 1799—was devoted to the demonstration that every equation has a root; and of this theorem he made two other distinct demonstrations in 1815 and 1816. ”

“ It does not follow that a number has a square root because it has a square; thus, though 5 can be multiplied by itself, there is no number which multiplied by itself will produce 5. It is proved in algebra, that no fraction [22] multiplied by itself can produce a whole number, which may be found true in any number of instances; therefore 5 has neither a whole nor a fractional square root; that is, it has no square root at all. ”

“ On the other hand, important as is the influence of the calculus in our modern geometry, various solutions obtained without algebra may sometimes manifest the peculiar character which distinguishes it from the ancient geometry, although analysis is generally indispensable. ”

“ The more reason that we have here to consider geometry as being at the present day essentially analytical, the more necessary was it to guard against this abusive exaggeration of mathematical analysis, according to which all geometrical observation would be dispensed with, in establishing upon pure algebraical abstractions the very foundations of this natural science. ”

“ All traditional pure mathematics, including analytical geometry, may be regarded as consisting wholly of propositions about the natural numbers. That is to say, the terms which occur can be defined by means of the natural numbers, and the propositions can be deduced from the properties of the natural numbers—with the addition, in each case, of the ideas and propositions of pure logic. ”

“ The mathematical questions which have given birth to the Calculus of Variations consist generally in the investigation of the maxima and minima of certain indeterminate integral formulas, which express the analytical law of such or such a phenomenon of geometry or mechanics, considered independently of any particular subject. ”

“ The second power and third power are usually called the square and cube respectively. If we know p and N, n is called the pth root of N, so that n is the second (or square) root of n2, the third (or cube) root of n3, the fourth root of n4, . . . If we know n and N, then p is the logarithm of N to base n.
The calculation of powers (i.e. of N when n and p are given) is involution; the calculation of roots (i.e. of n when p and N are given) is evolution; the calculation of logarithms (i.e. of p when n and N are given) has no special name. ”

“ A last important class of applications of the calculus of finite differences, which deserves to be distinguished from the preceding, consists in the eminently useful employment made of it in geometry for determining by approximation the length and the area of any curve, and in the same way the cubature of a body of any form whatever. ”

“ After the general exposition of the philosophical character of concrete mathematics, compared with that of abstract mathematics, given in the introductory chapter, it need not here be shown in a special manner that geometry must be considered as a true natural science, only much more simple, and therefore much more perfect, than any other. ”

“ It is true that in the greatest number of important cases its use requires to be preceded and prepared by that of the Calculus of indirect Functions, which is intended to facilitate the establishment of equations. But, although algebra has then only a secondary office to perform, it has none the less a necessary part in the complete solution of the question, so that the Calculus of direct Functions must continue to be, by its nature, the fundamental base of all mathematical analysis. ”

“ Many of these operations consist of two parts, the direct and the inverse. To extract a root is the inverse, with reference to that of raising any number to a higher power. So the integral is the inverse of the differential calculus; and the same observation applies to finite differences. In all these cases the inverse method is by far the more difficult of the two. ”

“ From this last consideration we must infer that, although the geometry which I have called general must be now regarded as the only true dogmatical geometry, and that to which we shall chiefly confine ourselves, the other having no longer much more than an historical interest, nevertheless it is not possible to entirely dispense with special geometry in a rational exposition of the science. ”

“ When square root of minus one comes to you, behave reasonably about him. Treat him logically, exactly as if he were six or nine; only always remember to keep well in front of you the fact of your own ignorance. ”

“ It is nevertheless evident, to any one who examines with attention the character of geometrical reasonings, even in the present state of abstract geometry, that, although the facts which are considered in it are much more closely united than those relating to any other science, still there always exists, with respect to every body studied by geometers, a certain number of primitive phenomena, which, since they are not established by any [Pg 180] reasoning, must be founded on observation alone, and which form the necessary basis of all the deductions. ”

“ The complete solution of every question of the calculus, from the most elementary up to the most transcendental, is necessarily composed of two successive parts, whose nature is essentially distinct. In the first, the object is to transform the proposed equations, so as to make apparent the manner in which the unknown quantities are formed by the known ones: it is this which constitutes the algebraic question. ”

“ This necessary perfection of geometry, obtained essentially by the application of mathematical analysis, which it so eminently admits, is apt to produce erroneous views of the real nature of this fundamental science, which most minds at present conceive to be a purely logical science quite independent of observation. ”

“ The Calculus of Values, or Arithmetic, would appear, at first view, to present a field as vast as that of algebra, since it would seem to admit as many distinct questions as we can conceive different algebraic formulas whose values are to be determined. ”

“ Computation: You say, ten less thing, multiplied by itself, is a hundred plus a (35) square less twenty things, and this is equal to eighty-one things. Separate the twenty things from a hundred and a square, and add them to eighty-one. It will then be a hundred plus a square, which is equal to a hundred and one roots. Halve the roots; the moiety is fifty and a half. Multiply this by itself, it is two thousand five hundred and fifty and a quarter. Subtract from this one hundred; the remainder is two thousand four hundred and fifty and a quarter. Extract the root from this ”

“ The abstract part of mathematics is then purely instrumental, and is only an immense and admirable extension of natural logic to a certain class of deductions. On the other hand, geometry and mechanics, which, as we shall see presently, constitute the concrete part, must be viewed as real natural sciences, founded on observation, like all the rest, although the extreme simplicity of their phenomena permits an infinitely greater degree of systematization, which has sometimes caused a misconception of the experimental character of their first principles. ”

“ It is composed of what is called the Calculus, [2] taking this word in its greatest extent, which reaches from the most simple numerical operations to the most sublime combinations of transcendental analysis. The Calculus has the solution of all questions [Pg 34] relating to numbers for its peculiar object. Its starting point is, constantly and necessarily, the knowledge of the precise relations, i.e., of the equations, between the different magnitudes which are simultaneously considered; that which is, on the contrary, the stopping point of concrete mathematics. ”

“ Thus, for example, in the case of rectifications, it will be first necessary to calculate dy, or dy and dz, by differentiating the equation or the equations of each curve proposed; after eliminating these expressions, the general differential formula above enunciated will then contain only ds and dx; having arrived at this point, the elimination of the infinitesimals can be completed only by the integral calculus. ”