TSE: Math: About finding roots of linear, quadratic, cubic, quartic equations in 1 variable / history of complex numbers / complex number operations (add, multiply, integer power)

[knud van eeden]

 Q. TSE: Math: About finding roots of linear, quadratic, cubic, quartic equations in 1 variable / history of complex numbers / real and imaginary numbers / includes complex number operations (add, multiply, integer power)

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1. -History: Time line: About solving that type of equations

YEAR: DESCRIPTION

 800: Al-Khwarizmi, a Persian mathematician only allowed positive real roots

      for the polynomial equations in his book 'Al-jabr wa'l-muqabala'.

1526: Scipione del Ferro, an Italian mathematician, found a method for

      solving a class of cubic equations of the form

       x^3 + constant1 x = constant2

      (thus lacking the x^2 term) and informed his student Antonio del

      Fiore about it.

1535: Antonio del Fiore, an Italian mathematician, accepts the

      challenge of a athematical duel about who could solve most of

      some class of cubic polynomials within some time limit against

      Niccola Tartaglia. The latter won the contest.

1540: Lodovico Ferrari, an Italian mathematician and a student of

      Gerolamo Cardano discovers an exact solution for the quartic

      equation. But as its solution in turn involves the solution of a

      cubic to be found, (which was not publised yet) it could not be

      published immediately. The solution of the quartic was therefor

      published together with that of the cubic by Ferrari's mentor

      Gerolamo Cardano in his book Ars Magna (1545). It involves then

      also the solution of two quadratic equations to get the 4 roots.

1545: Gerolamo Cardano, an Italian mathematician, introduced the first

      general solution of the 3rd degree polynomial in his book 'Ars

      Magna'. He borrowed the idea of this solution of the cubic from

      Niccola Tartaglia, who was the first to find that general

      solution. This discovery was made in the course of studying a

      formula which gave the roots of a cubic equation. He was able

      to manipulate with his 'complex numbers' to obtain the right

      answer yet he in no way understood his own mathematics.

1555: Niccola Tartaglia, an Italian mathematician, introduced a general

      solution of the 3rd degree equation in his book 'general trattato

      di numeri et misure' and he was the first to find that solution

      thus.

1572: Raffaelle Bombelli, an Italian mathematician, was the first to

      accept imaginary numbers as a solution of the 2nd and 3rd degree

      equation in his book "l'algebra"

1591: Francois Viete, a French lawyer in the royal privy council,

      further developed solutions (e.g. the root formulas of Viete) of

      the polynomial equations in his book 'in artem analyticam

      isagoge'

1637: Rene Descartes, a French mathematician, introduces the terms

      'REAL' and 'IMAGINARY' in French in his book "La Geometrie".

1770: Joseph Louis Lagrange, a French mathematician, studied how

      permutations of the roots of a polynomial equation had effect on

      the values of expressions involving those roots. He also

      investigated symmetric functions, which are functions of

      variables that remain unchanged under permutations of those

      variables. It laid a basis of 'group theory' later by

      Evariste Galois.

1777: Leonhard Euler, a Swiss mathematician, introduced the symbol 'i'

      for the square root of '-1'. Here 'i' stands for 'I'maginary

      number, in a memoir presented to the Academy at St. Petersburg,

      Russia.

1799: Caspar Wessel, a Norwegian geometer, was the first to use a

      visualization of complex numbers (in the complex plane), when he

      presented his paper "On the Analytical Representation of

      Direction" (written in Danish) to the Royal Danish Academy of

      Sciences.

1799: Carl Friedrich Gauss, a German mathematician, proves as being the

      first in his doctoral PhD dissertation the fundamental theorem of

      algebra, that is that 'every polynomial equation of degree n with

      complex coefficients has n roots in the complex numbers'.

1799: Paolo Ruffini, an Italian mathematician, attempted a proof of the

      impossibility of solving the quintic and higher equations.

1824: Niels Henrik Abel, a Norwegian mathematician, proved that no

      general solution exists for all 5th degree (quintic) equations,

      using no more than the evaluation of Nth roots, which are also

      called 'radicals'. Among polynomials of degree 5 or more, some

      can be solved by radicals, but others can not.

1831: Carl Friedrich Gauss. a German mathematician, introduced the term

      'COMPLEX NUMBER' in his work titled "Theoria Residuorum

      Biquadraticorum, Commentatio secunda" and referred to

      these numbers as 'numeros integros complexos'.

1831: Augustine Cauchy. a French mathematician introduced the term

      'COMPLEX CONJUGATE'.

1832: Evariste Galois, a French mathematician, introduced his 'group

      theory' (with its origin in 'symmetry') which could determine

      which 5th degree equations and higher can be solved by radicals

      and those who can not. He showed that the mathematical symmetry

      behind the methods used to resolve lower-order polynomials became

      impossible for degree five and higher polynomials. Therefore, he

      figured, no general formula could solve them.

       https://en.wikipedia.org/wiki/History_of_group_theory

1858: Charles Hermite, a French mathematician, showed that solutions of

      the 5th degree (=quintic) equation are possible if one extends

      the possible operations with elliptic functions (generalizations

      of trigonometric functions (like e.g. the sine and the cosine)).

      The problem is that one has to numerically approximate those

      elliptic functions, so at the end one has only a numeric solution,

      thus still no exact solution.

2025: Norman Wildberger, an American mathematician, introduces a

      possibly new method to possibly solve Nth degree (N higher than

      4) equations using other types of operations than radicals, e.g.

      using algebraic series and Catalan numbers.

       https://www.unsw.edu.au/newsroom/news/2025/05/mathematician-solves-algebras-oldest-problem-using-intriguing-new-number-sequences

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2. History: About finding the roots of degree 1, 2, 3, 4 and higher 5, ... / About complex numbers (real numbers and imaginary numbers)

1. -The 'Fundamental Theorem of Algebra' (=FTA) states

     'Every polynomial equation of degree n with complex coefficients has n roots in the complex numbers'.

2. -General exact formulas exist for solving

    linear, quadratic, cubic and quartic equations.

3. -No such general exact formulas (using only radicals) exist for

    solving 5th, 6th, ..., Nth degree equations where N is 5 or

    higher).

4. -About imaginary numbers: Roots of a negative number pop up in

    history frequently when trying to solve polynomial equations. E.g.

    when trying to solve this fourth degree equation:

     x^4 + 2 x^3 + 3 x^2 + 4 x + 5 = 0

    but these negative roots were largely ignored in the beginning.

5. -In general complex numbers (a combination of a real number and

    imaginary number) op up when trying to find the roots of such

    polynomial equations.

6. -Note: In general complex numbers could pop up when describing

           ROTATIONS

          and because e.g. waves could be represented by some form of

          rotation these complex numbers and its equations also are

          present in e.g. quantum physics equations.

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3. See also more about the history of solving the 3rd degree equation before the 4th degree in 'How Imaginary Numbers Were Invented'

 https://www.youtube.com/watch?v=cUzklzVXJwo

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4. Description:

 1. -Attached are the latest TSE program versions of solving linear,

     quadratic, cubic and quartic equations in 1 variable x.

 2. -Input the coefficients of the equation from highest

     to lower power.

 3. -Note: It will take a longer time of approximately 10 minutes

           totally when solving e.g. quartic equations.

 4. -It finds numeric solutions for respectively 1 root, 2 roots, 3 roots and 4 roots for the

     linear, quadratic, cubic and quartic equation.

 5. -It calculates also if the roots found make the equation indeed 0.

 6. -For checking if it is indeed zero or close to zero it uses

     operations on complex numbers like complex number add, complex

     number multiply and complex number integer power.

 7. -It also opens automatically a WolframAlpha page in the browser

     with your equation. So you can see graphical representations and exact solutions of the

     equation (e.g. the roots are shown in the complex number plane).

 8. -It uses Microsoft PowerShell to do the numerical calculations.

 9. -It works on TSE for Microsoft Windows and TSE for Linux WSL, tested with the 4.50 official release versions, and

     also out of the box because Powershell is pre-installed there.

10. -Running on non-WSL Linux should be possible when installing

     PowerShell on that Linux distribution (that is possible

     for assumed many of the non-WSL Linux distributionsP.

     But not tested or tried that yet on any Linux non-WSL distribution.

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5. Attachments 

    (discard all previous versions)

    1. 1st degree (=linear equations): getstrfr.s

    2. 2nd degree (=quadratic equations): getstrfq.s

    3. 3rd degree (=cubic equations): getstrfs.s

    4. 4th degree (=quartic): getstrft.s

    5. Inline screenshot of the equation x^4 + 2 x^3 + 3 x^2 + 4 x + 5 = 0

with friendly greetings

Knud van Eeden