Q. TSE: Math: About finding roots of linear (1st), quadratic (2nd), cubic (3rd), quartic (4th), quintic (5th) equations in 1 variable / history of complex numbers / real and imaginary numbers / includes complex number operations (add, multiply, integer power)
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knud van eeden
Jun 5, 2025, 5:32:18 PM6/5/25
to SemWare TSE Pro Text Editor
Q. TSE: Math: About finding roots of linear (1st), quadratic (2nd), cubic (3rd), quartic (4th), quintic (5th) equations in 1 variable / history of complex numbers / real and imaginary numbers / includes complex number operations (add, multiply, integer power)
===
Included also the 5th degree 'quintic' polynomial.
This quintic works on Microsoft Windows, but does not work on Linux WSL or non-WSL
(issue with directories and execute permissions which has to be resolved further).
All other degrees 1, 2, 3 and 4 work on both Microsoft Windows and Linux WSL.
===
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'.
1494: Luca Pacioli, an Italian mathematician, in his book 'Summa de
Arithmetica, Geometria, Proportioni et Proportionalita' stated
that solving cubic equations was not possible, and even
questioned whether a general solution would ever be found.
His belief spurred mathematicians, notably Scipione del Ferro and
Niccolo Tartaglia, to seek a solution, ultimately leading to the
development of the first general solution of the cubic equation,
which was later published by Gerolamo Cardano.
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cubic to be found, (which was not publised yet) it could not be
published immediately.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 de Regulis Algebraicis' (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".
In the book he solved equations using the method of del
Ferro/Tartaglia.
He is generally regarded as the inventor of complex
numbers, as no one before him had made rules for dealing with
such numbers, and no one believed that working with imaginary
numbers would have useful results.
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'.
He also conceived methods for the general resolution of equations
of the second, third and fourth degrees different from those of
Scipione dal Ferro and Lodovico Ferrari, with which he had not
been acquainted.
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: Carl Friedrich Gauss already mentions in 1799 in his dissertation
about the impossibility to resolve the equations of all fifth
degree polynomials.
1799: Paolo Ruffini, an Italian mathematician, attempted a proof of the
impossibility of solving the quintic and higher equations.
1823: Niels Henrik Abel, a Norwegian mathematician believed that he had
found a general exact solution of the 5th degree equation. But it
showed to be a mistake.
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.
The proof appeared in the 'Journal f<u..>r die reine und
angewandte Mathematik' from the German publisher from Berlin
A. L. Crelle.
Evariste Galois and Niels Henrik Abel showed that there is NO
general formula, using only radicals, for degrees greater than 4
(though special cases, like x^N=1 have general solutions in
radicals. Such cases can be found from group theory). Hermite
showed that there is a general solution, if you allow elliptic
integrals (but you have to approximate this elliptic integrals
also, e.g. from tables, so you still are far from home).
Other approaches are using numerical solutions, using e.g. an
extended method of Newton (method of Bairstow, method of Graef,
...)
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 Louis 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.
Evariste Galois was unaware of Niels Henrik Abel's work in the
first stages of his investigation, although he did learn of it
later. This was perhaps fortunate because Evariste Galois
actually had launched himself on a much more ambitious study,
while yet a student, at about age 16, he sought, by what is now
called the "Galois theory," a deeper understanding of the
essential conditions that an equation must satisfy in order for
it to be solvable by radicals. His method was to analyze the
"admissible" permutations (a change in an ordered arrangement) of
the roots of the equation. That is, in today's terminology, he
formed the "group" of automorphisms (a particular kind of
transformation) of the "field," obtained by adjoining the roots
of the equation. His key discovery, brilliant and highly
imaginative, was that solvability by radicals is possible if and
only if the group of automorphisms is solvable, which means
essentially that the group can be broken down into prime-order
constituents (prime numbers are positive numbers greater than 1
divisible only by themselves and 1) that always have an easily
understood structure. The term solvable is used because of this
connection with solvability by radicals. Thus Galois perceived
that solving equations of the quintic and beyond required a
wholly different kind of treatment than that required for the
quadratic, cubic, and quartic.
1846: Joseph Liouville, a French mathematician.
He founded the 'Journal de Mathematiques Pures et Appliquees' which
retains its high reputation up to today, in order to promote
other mathematicians' work.
He was the first to read, and to recognize the importance of, the
unpublished work of Evariste Galois which appeared in his journal
in 1846.
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.
For higher degrees than 5 using this elliptic functions method is
also not possible.
1870: Marie Ennemond Camille Jordan, a French mathematician,
further developed the Galois theory.
He published his findings on Galois theory in 1870
with his book
"Traite des substitutions et des equations algebriques"
(=Treatise on Substitutions and Algebraic Equations).
This treatise, which included a comprehensive study of Galois
theory, was also the first group theory book ever written.
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.
===
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. -To solve the 5th degree quintic equation a numerical
approximation is calculated. This by applying the
method of Bairstow (it tries to approximate the
given Nth (e.g. 5) degree equation by a product of 2 or
more 2nd degree quadratic equations.
It also uses a variation of the method of Laguerre,
becaue the Baistow method might not converge sometimes.
5. -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:
1 x^4 + 2 x^3 + 3 x^2 + 4 x + 5 = 0
but these negative roots were largely ignored in the beginning.
6. -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.
7. -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.
===
3. See also:
How Imaginary Numbers Were Invented
Quadratic equation:
Cubic equation:
Quartic equation:
Quintic equation:
History of group theory:
===
4. Description:
1. -Attached are the latest TSE program versions of solving linear,
quadratic, cubic, quartic and quintic equations in 1 variable x.
2. -Input all the coefficients of the equation from highest
to lower power, in that order.
3. -It calculates also if the roots found make the equation indeed 0
4. -To fast run the program, skip checking if the polynomial reaches
zero, so select 'no' when asked.
5 -It will take a longer time of approximately 10 minutes totally
when solving e.g. quartic equations and checking for zero.
6. -The quintic equation solves fast (because it runs an
internal PowerShell program).
But the checking if close to zero using the complex number
calculations takes very long time (20 minutes_.
Thus also expected somewhere many minutes waiting here.
If skipping checking for zero then it takes about 1 minute
waiting time.
7. -It should find numeric solutions for respectively 1 root, 2
roots, 3 roots, 4 roots and 5 roots for the linear, quadratic,
cubic, quartic and quintic equation.
8. -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.
9. -It optionally also opens automatically a WolframAlpha webpage in
the browser with your polynomila equation.
So you can see graphical representations and exact solutions of
the equation (e.g. the N roots shown in the complex number plane).
10. -It uses Microsoft PowerShell to do the numerical calculations.
11. -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.
12. -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 distributions).
But not tested or tried that yet on any Linux non-WSL distribution.
===
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 equations): getstrft.s
5. 5th degree (=quintic equations): getstrfu.s
6. Inline screenshot of the equation 1 x^5 + 2 x^4 + 3 x^3 + 4 x^2 + 5 x + 6 = 0
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