Principles Of Algebra Research Articles

History of the Theory of Imaginary and Complex Quantities

It is worthy of note that the imaginary was introduced into mathematics in the course of strictly non-utilitarian researches on the theory of quadratic equations, being for a long time rejected by the mathematicians as an entirely useless conception. This formal condemnation of the imaginary seems less extraordinary when we recollect that at the time of its inception, and indeed for long afterwards, serious scruples were experienced by some mathematicians even regarding the admission of negative quantities in their results. The last writers who objected to the use of negative quantities were Baron Maseres, a Fellow of Clare College, Cambridge, and William Frend, a second wrangler. Both these men had written extensively in condemnation of the use of negative quantities; their attitude in this matter is clearly evinced by the following extract from the preface to Frend’s Principles of Algebra, published in 1796, at which period, strangely enough, the notion of the imaginary was firmly eatablished.

Live Problem Material in Algebra

Verbal problems in algebra have been, and, one might venture to say, still are the object of much destructive criticism from both students and teachers. By some inexperienced instructors, verbal problems are sometimes omitted from the outline of study. Other instructors often place undue emphasis upon problems and continue to drill on problems until the nerves of the parties of the first and second parts are exhausted. The teacher with a well balanced outline of work will use live verbal problems in driving home the principles of algebra. Verbal problems, then, should be a means to an end, not an end in themselves. They should assist in causing mathematics to function in solving real life situations.

Analytical Geometry and Principles of Algebra

Analytical Geometry and Principles of Algebra

(1) (a) Plane and Solid Geometry (b) Solid Geometry (2) A School Course in Geometry (3) Analytic Geometry and Principles of Algebra (4) Slide-Rule Notes (5) A First Numerical Trigonometry (6) Elementary Graphic Statics (7) Models to Illustrate the Foundations of Mathematics (8) Exercises in Mathematics (9) A School Statics (10) Proceedings of the London Mathematical Society

(1) (a) Plane and Solid Geometry (b) Solid Geometry (2) A School Course in Geometry (3) Analytic Geometry and Principles of Algebra (4) Slide-Rule Notes (5) A First Numerical Trigonometry (6) Elementary Graphic Statics (7) Models to Illustrate the Foundations of Mathematics (8) Exercises in Mathematics (9) A School Statics (10) Proceedings of the London Mathematical Society

Analytic Geometry and Principles of Algebra.

Analytic Geometry and Principles of Algebra.

Algebraical Examples

IT will be a convenience to many teachers to possess this collection of algebraical exercises to supplement those given in Hall and Knight's “Algebra for Beginners” and “Elementary Algebra,” up to quadratic equations. The exercises are carefully graduated, and are classified so that the teacher can easily select those referring to the subject with which he is dealing. In addition, there are a number of test-papers containing miscellaneous examples to test the pupil's grasp of the principles of algebra in which he has been exercised.

Book Review: Principles of the Algebra of Physics

Book Review: Principles of the Algebra of Physics

Our Book Shelf

IN this book the principles of algebra as far as quadratic equations only are dealt with. The authors have explained them in plain and simple language, and have worked out numerous examples in order to illustrate the methods adopted. In an elementary treatise like this, intended for those who are preparing for public school entrance examinations, great importance must be attached to the working out, briefly and neatly, of examples, and here we have a good and copious collection, together with some papers set at former entrance examinations.

On the Tradition Question in Probabilities

In a book entitled Principles of the Algebra of Logic, which deals with the Sciences of Necessity and Probability as one whole, I have considered as an example the question in probabilities discussed in vol. xxi (p. 204) of this Journal, namely—“ A says “ that B says that a certain event took place : required the pro- “ bability that the event did take place, p1 and p2 being A’s and “ B’s respective probabilities of speaking the truth.”

On the Principles of the Logical Algebra; with Applications. Part II

The equationx2 = xexpresses the condition that the symbol x be single and positive. The equationx2 = −xexpresses the condition that the symbol x be single and negative. The principle of contradictionx(1−x)=0can be legitimately deduced with the help of the fundamental axioms of the science.

3. On the Principles of the Logical Algebra; with Applications. Part I

AbstractIn this memoir I examine the principles of the logical calculus of Boole, as laid down in his celebrated treatise on the “Laws of Thought,” and also the criticisms which have been published concerning these principles. I bring forward a new theory of the operation of the mind, founded upon an analysis of language and the nature of mathematical reasoning, which enables me to correct these principles, to place them on a clear, rational, and generalised basis, and to show that there is a logical algebra which coincides with the ordinary algebra when its symbols are integral, but is a generalised form of the ordinary algebra when its symbols are fractional.

XVII Algebraical researches, containing a disquisition on Newton’s rule for the discovery of imaginary roots, and an allied rule applicable to a particular class of equations, together with a complete invariantive determination of the character of the roots of the general equation of the fifth degree, &c

(1) This memoir in its present form is of the nature of a trilogy; it is divided into three parts, of which each has its action complete within itself, but the same general cycle of ideas pervades all three, and weaves them into a sort of complex unity. In the first is established the validity of Newton’s rule for finding an inferior limit to the number of imaginary roots of algebraical equations as far as the fifth degree inclusive. In the second is obtained a rule for assigning a like limit applicable to equations of the form Σ( ax + b ) m =0, m being any positive integer, and the coefficients a , b real. In the third are determined the absolute invariantive criteria for fixing unequivocally the character of the roots of an equation of the fifth degree, that is to say, for ascertaining the exact number of real and imaginary roots which it contains. This last part has been added since the original paper was presented to the Society. It has grown out of a foot-note appended to the second, itself an independent offshoot from the first part, hut may be studied in a great measure independently of what precedes, and constitutes, in the author’s opinion, by far the most valuable portion of the memoir, containing as it does a complete solution of one of the most interesting and fruitful algebraical questions which has ever yet engaged the attention of mathematicians (1). I propose in a subse­quent addition to the memoir to resume and extend some of the investigations which incidentally arise in this part. The foot-notes are numbered and lettered for facility of reference, and will be found in many instances of equal value with the matter in the text, to which they serve as a kind of free running accompaniment and commentary. 2) In the ‘Arithmetica Universalis,’ in the first chapter on equations, Newton has given a rule for discovering an inferior limit to the number of imaginary roots in an equation of any degree, without proof or indication of the method by which he arrived at it, or the evidence upon which it rests(²). Maclaurin, in vol. xxxiv. p. 104, and vol. xxxvi. p. 59 of the Philosophical Transactions, Campbell (³) in vol. xxxviii. p. 515 of the same, and other authors of reputation have sought in vain for a demonstration of this marvellous and mysterious rule ( 4 ). Unwilling to rest my belief in it on mere empirical evidence, I have investigated and obtained a demonstration of its truth as far as the fifth degree inclusive, which, although presenting only a small instalment of the desired result, I am induced to offer for insertion in the Transactions in the hope of exciting renewed attention to a subject so intimately bound up with the fundamental principles of algebra.

XXXII. On the differential coefficients and determinants of lines, and their application to analytical mechanics

I propose in these pages to prove the principal theorems of dynamics in a manner which appears to me both simpler and more methodical than that in which they are generally proved; and I believe that I shall be able, by applying a few conceptions which spring naturally from the principles of higher algebra and statics, to give a clear interpretation to most of the more complicated formulæ in dynamics, as well as to the several analytical steps which lead to those formulæ. 2. There are many reasons why the diagonal AD, which is constructed on the straight lines AB, AC, should be considered as the sum of those two lines. Those reasons may be found developed in De Morgan’s 'Double Algebra,’ in Warren 'On Imaginary Quantities,’ and in the Tract of Benjamin Gompertz 'On Imaginary Quantities.'

XIX. —Consideration of the objections raised against the geometrical representation of the square roots of negative quantities

Some years ago my attention was drawn to those algebraic quantities, which are commonly called impossible roots or imaginary quantities: it appeared extraordinary, that mathematicians should be able by means of these quan­tities to pursue their investigations, both in pure and mixed mathematics, and to arrive at results which agree with the results obtained by other independent processes; and yet that the real nature of these quantities should be entirely unknown, and even their real existence denied. One thing was evident re­specting them; that they were quantities capable of undergoing algebraic operations analogous to the operations performed on what are called possible quantities, and of producing correct results: thus it was manifest, that the operations of algebra were more comprehensive than the definitions and funda­mental principles; that is, that they extended to a class of quantities, viz. those commonly called impossible roots, to which the definitions and funda­mental principles were inapplicable. It seemed probable, therefore, that there was a deficiency in the definitions and fundamental principles of algebra ; and that other definitions and fundamental principles might be discovered of a more comprehensive nature, which would extend to every class of quantities to which the operations of algebra were applicable; that is, both to possible and impossible quantities, as they are called. I was induced therefore to examine into the nature of algebraic operations, with a view, if possible, of arriving at these general definitions and fundamental principles: and I found, that, by considering algebra merely as applied to geometry, such principles and definitions might be obtained. The fundamental principles and definitions which I arrived at were these: that all straight lines drawn in a given plane from a given point, in any direction whatever, are capable of being algebra­ically represented, both in length and direction; that the addition of such lines (when estimated both in length and direction) must be performed in the same manner as composition of motion in dynamics; and that four such lines are proportionals, -both in length and direction, when they are proportionals in length, and the fourth is inclined to the third at the same angle that the second is to the first. From these principles I deduced, that, if a line drawn in any given direction be assumed as a positive quantity, and consequently its oppo­site, a negative quantity, a line drawn at right angles to the positive or nega­tive direction will be the square root of a negative quantity, and a line drawn in an oblique direction will be the sum of two quantities, the one either posi­tive or negative, and the other, the square root of a negative quantity.

XIV. On the Resolution of Indeterminate Problems

It is a fundamental principle in Algebra, that a problem admits of solution, when the number of independent equations is equal to that of the unknown quantities. If simple expressions only occur, the answers will always be found in numbers, either whole or fractional. But if the higher functions be concerned, the values of the unknown quantities will commonly be involved in surds, which it is impossible to exhibit on any arithmetical scale, and to which we can only make a repeated approximation. Hence the origin of that branch of analysis which is employed in the investigation of those problems, where the number of unknown quantities exceeds that of the proposed equations, but where the values are required in whole or fractional numbers. The subject is not merely an object of curiosity; it can be applied with advantage to the higher calculus. Yet the doctrine of indeterminate equations has been seldom treated in a form equally systematic with the other parts of algebra. The solutions commonly given are devoid of uniformity, and often require a variety of assumptions. The object of this paper is to resolve the complicated expressions which we obtain in the solution of indeterminate problems, into simple equations, and to do so, without framing a number of assumptions, by help of a single principle, which, though extremely simple, admits of a very extensive application.