Solution Of Cubic Equation Research Articles

Introduction to Complex Numbers

The purpose of this article is to present an elementary introduction to complex numbers with a glance at some applications. The intent is to present the concepts intuitively, more or less as they probably originated, rather than formally by -a rigorous procedure of definitions and postulates. Inevitably this leads to some repetitions in statements but in an elementary presentttion this is not a pedagogical fault. A brief review of the history of complex numbers is interesting and relevant. The first introduction of imaginary numbers occurred in connection with the solution of cubic equations. During the 16th century Italian mathematicians of the then famous university of Bologna discovered that sometimes correct answers could be obtained moreexpeditiously if they assumed 1) a symbol, i, 2) i2 = -1, and 3) in other respects treated i as an ordinary number. One of QCardano's problems was Divide 10 into two parts whose product is 40. Cardano was intrigued by the answer but at the same time he was very doubtful of the method. For several hundred years mathematicians continued to play -with the concept. Euler (1707-1783) achieved brilliant results by the use of complex numbers but the fundamental principles of their logic was either not deemed important or was completely misunderstood. Struik ways that in the 18th century 'There was experimentation with infinite series, with infinite products, ... With the use of symbols such as 0, OD, 0,j?, ... Much of the work of the leading mathematicians impresses us as wildly enthusiastic experimentation. Wessel, a Danish mathematician, probably furnished the first logical foundation for complex numbers, but his paper 'On the Analytical Representation of Direction, presented in 1797, was unknown to mathematicians until republished one hundred years later. Argand, a Paris accountant, also presented in 1806 a logical foundation but his work appears to have received very little attention. Actually Gauss (17771855) first used the phrase complex number, the symbol i forthe imaginary unit, and introduced mathematicians to the true theory of these numbers. Gauss' work was followed by that of Cauchy (1789-1857) and that of Riemann (1826-1866). From their works has arisen the complex function theory which is basic and indispensable for advanced mathematics, and even for a full comprehension of the more elementary theorems of algebraic analysis. The point to be emphasized is this. More than 200 years passed before mathematicians placed complex numbers upon a firm logical basis.

On the Graphical Solution of Cubic Equations

On the Graphical Solution of Cubic Equations

The Rapid Solution of Cubic Equations

THE problem of finding approximate solutions of algebraic equations arises frequently in engineering practice; perhaps the most important or frequent instance is that of the ‘characteristic equation’ of a linear differential equation describing the behaviour of some electrical or mechanical system.

The congruence ax3+by3+c≡0 (modxy), and integer solutions of cubic equations in three variables

The congruence ax3+by3+c≡0 (modxy), and integer solutions of cubic equations in three variables

A New Method for the Solution of Cubic Equations

Cubic equations can be solved by a method similar to that used in the extraction of a cube root.

A New Method for the Solution of Cubic Equations

A New Method for the Solution of Cubic Equations

On the Solution of Cubic Equations

On the Solution of Cubic Equations

Corrigendum

Corrigendum

991. [A. 3. k.] The Numerical Solution of Cubic Equations by Circular Functions.

991. [A. 3. k.] <i>The Numerical Solution of Cubic Equations by Circular Functions.</i>

Complete groups of points on a plane cubic curve of genus one

1. Of the great niumber of papers which have been published which deal with rational solutions of cubic equations in two non-homogeneous or three homogeneous variables, the majority have been concerned with special numerical examples. (See Dickson's History of the Theory of Nuimbers, vol. 2, 1920, Chap. 21, for an account of these papers.) Some few, however, have dealt with the general problem of classifying cubics with rational coefficients with reference to rational solutions. Of these, the following have been consulted in the preparation of this paper:

The Numerical Solution of Cubic Equations

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Application of the Pell Equation to the Problem of Extraction of the Cube Root of a Binomial Surd

In the solution of cubic equations and in certain other cases it is desirable to have a somewhat general method for extracting the cube root of a binomial surd. This may be accomplished by the aid of the Pell Equation.

Příspěvek ku řešení rovnice kubické

Příspěvek ku řešení rovnice kubické

O nejjednodušším řešení rovnic kubických

O nejjednodušším řešení rovnic kubických

Graphical Solution of Cubic Equations

Graphical Solution of Cubic Equations

XXVIII. A new and general solution of cubic equations

XXVIII. A new and general solution of cubic equations