History of Solving Polynomial Equations

MATH2001 Development of Mathematical Ideas
History of Solving Polynomial
Equations
29 March 2012
Dr. Tuen Wai Ng, HKU
What is a polynomial ?
anxn + · · · + a1x + a0
This modern notation was more or less developed by René Descartes
(1596-1650) in his book La Géométrie [Geometry] (1637).
Before the 17th century, mathematicians usually did not use any
particular notation.
Before Descartes, François Viète (1540-1603) has already developed
the basic idea of introducing arbitrary parameters into an equation and to
distinguish these from the equation’s variables.
François Viète (1540-1603)
René Descartes (1596-1650)
He used consonant (B, C, D, . . ..) to denote parameters and vowels
(A, E, I, . . .) to denote variables.
Certainly, letters had been used before Viète, but not in actual
computations; one letter would be used for a certain quantity, another
for its square, and so forth.
Let us give an idea of the notation used in Viète’s book Zététiques
[Introduction to the Analytic Art ](1591), the expression
F ·H +F ·B
=E
D+F
is written

F in H 
+F in B
æquabitur E.


D + F


Viète’s notation for powers of the unknown is very heavy: he writes “A
quadratum” for A2, “A cubus” for A3, “A quadrato-quadratum” for A4 ,
etc., and “A potestas,” “A gradum” for Am, An.
His notation was only partly symbolic. For example, an equation in the
unknown A, such as A3 + 3B 2A = 2C 3 would be expressed by Viète as
A cubus +B plano 3 in A aequari C solido 2.
Note that Viète required “homogeneity” in algebraic expressions: all
terms had to be of same degree (for example, all terms above are of the
third degree).
The requirement of homogeneity goes back to Greek antiquity, where
geometry reigned supreme. To the Greek way of thinking, the product
ab (say) denoted the area of a rectangle with sides a and b; similarly, abc
denoted the volume of a cube. An expression such as ab+c had no meaning
since one could not add length to area.
This condition of homogeneity was definitively abandoned only around
the time of Descartes.
Most of the shortcomings of Viète’s notations were overcome by
Descartes in his important book La Géométrie [Geometry], in which he
expounded the basic elements of analytic geometry.
2
Before Descartes, 3x2 was written as 3⌣ by Ralfaele Bombelli in 1572,
whereas Simon Stevin wrote 3⃝3 + 5⃝2 − 4⃝1 for 3x3 + 5x2 − 4x in 1585.
The exponential notation x2, x3, etc., came with Descartes, whose
formulas are actually written in a notation very close to our own. For
example, he used x, y, z, . . . for variables and a, b, c, for parameters.
Most importantly, he introduced an “algebra of segments.” That is, for
any two line segments with lengths a and b, he constructed line segments
with lengths a + b, a − b, a × b, and a/b.
Thus homogeneity of algebraic expressions was no longer needed. For
example, ab + c was now a legitimate expression, namely a line segment.
So there is no need of geometry in algebra and algebra had now become
considerably more abstract. It was well on its way to to play an important
role in mathematics.
Viète and Descartes’s notations and ideas proved indispensable in the
crucial developments of the seventeenth century: in analytic geometry,
calculus, and mathematized science.
Remarks on other notations.
The signs + and − were already in use around 1480 (+ was apparently
a deformation of the symbol &), but by the beginning of the 17th century,
they were used generally.
Multiplication was written as M by Michael Stifel (1545), and as in by
Viète (1591); our current notation dates back to William Oughtred (1637)
for the symbol ×, and to Wilhelm Leibniz (1698) for the dot.
The symbol = used by Michel Recorde (1557) came to replace the
symbol used by Descartes, an α written backward, toward the end of the
17th century, thanks to Leibniz.
√
Albert Girard (1595-1632) introduced the notation
3, which he
1
substituted for ⃝;
he also introduced the abbreviations for sine and tangent,
3
and used the symbols <, > like Harriot.
The symbol
∑
was introduced by Leonhard Euler (1707-1783).
These notations passed into general usage only during the 19th century.
What do we mean by solving an equation ?
Example 1. Solve the equation x2 = 1.
x2
x2 − 1
(x − 1)(x + 1)
x
=1
=0
=0
= 1 or = −1
• Need to check that in fact (1)2 = 1 and (−1)2 = 1.
Exercise. Solve the equation
√
√
x+ x−a=2
where a is a positive real number.
What do we mean by solving a polynomial
equation ?
Meaning I:
Solving polynomial equations: finding numbers that make the
polynomial take on the value zero when they replace the variable.
• We have discovered that x, which is something we didn’t know, turns out
to be 1 or −1.
Example 2. Solve the equation x2 = 5.
x2
2
x
−5
√
√
(x − 5)(x + 5)
x
• But what is
√
5 ? Well,
√
=5
=0
= 0√
√
= 5 or − 5
5 is the positive real number that square to 5.
• We have ”learned” that the positive solution to the equation x2 = 5 is
the positive real number that square to 5 !!!
• So there is a sense of circularity in what we have done here.
• Same thing happens when we say that i is a solution of x2 = −1.
What are “solved” when we solve these
equations ?
• The equations x2 = 5 and x2 = −1 draw the attention to an inadequacy
in a certain number system (it does not contain a solution to the equation).
• One is therefore driven to extend the number system by introducing, or
‘adjoining’, a solution.
• Sometimes, the extended system has the good algebraic properties of the
original one, e.g. addition and multiplication can be defined in a natural
way.
√
√
• These extended number systems (e.g. Q( 5) = {a + b 5|a, b ∈ Q} or
Q(i)) have the added advantage that more equations can be solved.
Consider the equation
x2 = x + 1.
• By completing√the square,
or by applying the formula, we know that the
√
solutions are 1+2 5 or 1−2 5 .
• It is certainly not true by definition that
equation.
√
1+ 5
2
is a solution of the
• What we have done is to take it for granted that we can solve the equation
x2 = 5 (and similar ones) and to use this interesting ability to solve an
equation which is not of such a simple form.
• When we solve the quadratic, we are actually showing that the problem
can be reduced to solving a particularly simple quadratic x2 = c.
What do we mean by solving a polynomial
equation ?
Meaning II:
Suppose we can solve the equation xn = c (i.e. taking roots), then try to
express the the roots of a degree n polynomial using only the usual algebraic
operations (addition, subtraction, multiplication, division) and application
of taking roots.
• In this sense, one can solve any polynomials of degree 2,3 or 4 and we
will see later that this is in general impossible for polynomials of degree 5
or above.
• The Babylonians (about 2000 B.C.) knew how to solve specific quadratic
equations.
Babyloniasn problem text on tablet YBC 4652
In fact, many Babylonian clay tablets have been preserved with lists of
mathematical problems on them.
• The equations were given in the form of “word problems.” Here is a
typical example and its solution:
I have added the area and two-thirds of the side of my square and it is
0;35 [35/60 in sexagesimal notation]. What is the side of my square?
• In modern notation the problem is to solve the equation x2 + (2/3)x =
35/60. The solution given by the Babylonians is:
You take 1, the coefficient. Two-thirds of 1 is 0;40. Half of this, 0;20,
you multiply by 0;20 and get [the result] 0;6,40. Add 0;6,40 to 0;35 and
[the result] 0;41,40 has 0;50 as its square root. The 0;20, which you
have multiplied by itself, you subtract from 0;50, and 0;30 is [the side
of] the square.
The instructions
for finding the solution can be expressed
in modern notation
√
√
2 + 0; 35 − (0; 40)/2 =
as
x
=
[(0;
40)/2]
0; 6, 40 + 0; 35 − 0; 20 =
√
0; 41, 40 − 0; 20 = 0; 50 − 0; 20 = 0; 30.
√
• These instructions amount to the use of the formula x = (a/2)2 + b −
a/2 to solve the equation x2 + ax = b !!!
The following points about Babylonian algebra are important to note:
(i) There was no algebraic notation. All problems and solutions were verbal.
(ii) The problems led to equations with numerical coefficients. In particular,
there was no such thing as a general quadratic equation, ax2 +bx+c = 0,
with a, b, and c arbitrary parameters.
(iii) The solutions were prescriptive: do such and such and you will arrive
at the answer and there was no justification of the procedures. But the
accumulation of example after example of the same type of problem
indicates the existence of some form of justification of Babylonian
mathematical procedures.
(iv) The problems were chosen to yield only positive rational numbers as
solutions. Moreover, only one root was given as a solution of a quadratic
equation. Zero, negative numbers, and irrational numbers were not part
of the Babylonian number system.
(v) The problems were often phrased in geometric language, but they were
not problems in geometry. Nor were they of practical use; they were likely
intended for the training of students. Note, for example, the addition of
the area to 2/3 of the side of a square in the above problem.
(vi) They were also able to solve equations that lead to quadratic equations,
for example x + y = a and x2 + y 2 = b, by methods similar to ours.
[Please read p.205-206 of Chapter 5]
• The Chinese (about 200 BC) and the Indians (about 600 BC) advanced
beyond the Babylonians. For example, they allowed negative coefficients in
their equations (though not negative roots), and admitted two roots for a
quadratic equation.
• They also described procedures for manipulating equations, but had no
notation for, nor justification of, their solutions.
• From The Nine Chapters on the Mathematical Art (Jiuzhang Suanshu)
(100 BC-50 AD), we know the Chinese had methods for approximating
roots of polynomial equations of any degree, and solved systems of linear
equations using “matrices” (rectangular arrays of numbers) well before such
techniques were known in Western Europe.
• The solution formula for solving the quadratic equations was mentioned
in the Bakshali Manuscript written in India between 200 BC and 400 AD.
• The mathematics of the ancient Greeks is very strong in geometry and
number theory but rather weak in algebra.
• Elements (around 300 BC) contains several parts that can be interpreted
as algebraic. These are geometric propositions that, if translated into
algebraic language, yield algebraic results: laws of algebra as well as
solutions of quadratic equations. This work is known as geometric algebra.
For example, Proposition II.4 in the Elements states that
“If a straight line be cut at random, the square on the whole is equal
to the square on the two parts and twice the rectangle contained by the
parts.”
• If a and b denote the parts into which the straight line is cut, the
proposition can be stated algebraically as (a + b)2 = a2 + 2ab + b2.
Proposition II.11 states:
“To cut a given straight line so that the rectangle contained by the
whole and one of the segments is equal to the square on the remaining
segment.”
• This asks, in algebraic language, to solve the equation a(a − x) = x2.
• Note that for Greek algebra, homogeneity in algebraic expressions is a
strict requirement; that is, all terms in such expressions must be of the
same degree.
A much more significant Greek algebraic work is Diophantus’
Arithmetica (around 250 AD). Although essentially a book on number
theory, it contains solutions of equations in integers or rational numbers.
Arithmetica introduced a partial algebraic notation - a most important
achievement:
ζ denoted an unknown, Φ negation,1́σ equality, ∆σ the square of the
unknown, K σ its cube, and M the absence of the unknown (what we would
write as x0).
For example, x3 − 2x2 + 10x − 1 = 5 would be written as
K σ αζιΦ∆σ βM α1́σM ε
• Note that numbers were denoted by letters, e.g., α stood for 1 and ε for
5; moreover, there was no notation for addition, thus all terms with positive
coefficients were written first, followed by those with negative coefficients.
Diophantus made other remarkable advances in algebra, namely:
(i) He gave two basic rules for working with algebraic expressions: the
transfer of a term from one side of an equation to the other, and the
elimination of like terms from the two sides of an equation.
(ii) He defined negative powers of an unknown and enunciated the law of
exponents, xmxn = xm+n, for −6 ≤ m, n, m + n ≤ 6.
(iii) He stated several rules for operating with negative coefficients,
for example: “deficiency multiplied by deficiency yields availability”
((−a)(−b) = ab).
(iv) He did away with such staples of the classical Greek tradition as
(a) giving a geometric interpretation of algebraic expressions,
(b) restricting the product of terms to degree at most three, and
(c) requiring homogeneity in the terms of an algebraic expression.
• During the Dark Ages, Diophantus’s work was forgotten and like many
other mathematical treatises from the classical period, Arithmetica survived
through the Arab tradition.
• However, only six of the original thirteen books of which Arithmetica
consisted have survived.
Islamic mathematicians attained important algebraic accomplishments
between 900 AD and 1500 AD. For example, Muhammad ibn-Musa alKhwarizmi (around 780-850 AD) was conisdered by some “the Euclid of
algebra” because he systematized the subject (as it then existed) and made
it into an independent field of study.
Muhammad al-Khwarizmi (around 780-850 AD)
He did this in his book al-jabr w al-muqabalah.
“Al-jabr” (from which stems our word “algebra”) denotes the moving of
a negative term of an equation to the other side so as to make it positive,
and “al-muqabalah” refers to cancelling equal (positive) terms on both sides
of an equation.
Al-Khwarizmi (from whose name the term “algorithm” is derived) applied
these rules to solve quadratic equations.
He also classified quadratic equations into five types: ax2 = bx, ax2 =
b, ax2 + bx = c, ax2 + c = bx, and ax2 = bx + c.
This clasification was necessary as al-Khwarizmi did not admit negative
coefficients or zero. He also had essentially no notation, so that his problems
and solutions were expressed rhetorically.
The following is an example of one of al-Khwarizmi’s problems with
solution:
“What must be the square, which when increased by ten of its roots
amounts to thirty-nine?” (i.e., solve x2 + 10x = 39).
Solution: “You halve the number of roots [the coefficient of x], which in
the present instance yields five. This you multiply by itself; the product is
twenty-five. Add this to thirty nine; the sum is sixty-four. Now take the
root of this, which is eight, and subtract from it half the number of the
roots, which is five; the remainder is three. This is the root of the square
which you sought.”
√
• Symbolically, the prescription is: [(1/2) × 10]2 + 39 − (1/2) × 10.
Al-Khwarizmi gave a geometric justification of his solution procedures:
Construct the gnomon as in Fig.1, and “complete” it to the square in Fig.2
by the addition of the square of side 5.
The resulting square has length x + 5. But it also has length 8, since
x2 + 10x + 52 = 39 + 25 = 64. Hence x = 3.
Fig. 1
Fig. 2
The Babylonians were solving quadratic equations by about 2000 BC,
using essentially the equivalent of the quadratic formula. A natural question
is therefore whether cubic equations could be solved using similar formulas.
At least another three thousand years would pass before the answer would
be known.
Girolamo Cardano (1501-1576)
• Based on the work of Scipione del Ferro and Nicolo Tartaglia, Cardano
published the solution formula for solving the cubic equations in his book
Ars Magna [The Great Art] (1545).
Nicolo Tartaglia actually passed his method to Cardano, who had
promised that he would not publish it, which he promptly did as he
discovered that Scipione del Ferro had actually discovered Tartaglia’s
formula before Tartaglia himself.
What came to be known as Cardano’s formula for the solution of the
cubic x3 = ax + b was given by
√
x=
3
√
√
√
3
2
3
b/2 + (b/2) − (a/3) + b/2 − (b/2)2 − (a/3)3.
• Comments on Cardano’s book:
(i) Cardano used no symbols, so his “formula” was given rhetorically (and
took up close to half a page). Moreover, the equations he solved all had
numerical coefficients.
(ii) He was usually satisfied with finding a single root of a cubic. In fact, if
a proper choice is made of the cube roots involved, then all three roots
of the cubic can be determined from his formula.
(iii) Negative numbers are found occasionally in his work, but he mistrusted
them, calling them “fictitious.” The coefficients and roots of the cubics
he considered were positive numbers (but he admitted irrationals), so
that he viewed (say) x3 = ax+b and x3 +ax = b as distinct, and devoted
a chapter to the solution of each (compare al-Khwarizmi’s classification
of quadratics).
(iv) He gave geometric justifications of his solution procedures for the cubic.
[We shall study it in Section 5.2]
• Lodovico Ferrari, a student of Cardano discovered the solution formula
for the quartic equations in 1540 (published in Ars Magna later).
The formulae for the cubic and quartic are complicated, and the methods
to derive them seem ad hoc and not memorable.
The first attempt to unify solutions to quadratic, cubic and quartic
equations date at least to Lagrange ’s work in Réflexions sur la
résolution algébrique des équations [Reflections on the algebraic solution
of equations](1770/1771).
Lagrange also believed (but could not prove) that the general equation
of degree five was not solvable by radicals, meaning that there could be
no formula for the roots that involved only algebraic operations[addition,
subtraction, multiplication,division, raising to a natural power] on the
coefficients of the equation together with radicals of these coefficients.
Lagrange’s analysis characterized the general solutions of the cubic and
quartic cases in terms of permutations of the roots, laying a foundation for
the independent demonstrations by Abel and Galois of the impossibility of
solutions by radicals for general fifth degree or higher equations.
Before, Abel and Galois, the Italian Paolo Ruffini (1765-1822) published
a lengthy treatise Teoria Generale delle Equazioni [General Theory of
Equations] in 1799 and he claimed to have a proof that the general
equation of degree five was not solvable by radicals.
Due to the less than clear exposition, his arguments were received with
much skepticism (later on a significant gap was found), and were in general
not accepted by the mathematical community at that time.
In 1824, the Norwegian mathematician Niels Henrik Abel (1802-1829)
gave a different proof, published in the first issue of Journal für die Reine
und Angewandte Mathematik in 1826 [1, vol. 1, pp. 66-94]. The paper
contained the following
Abel’s Theorem (1824). The generic algebraic equation of degree five
is not solvable by radicals.
There is some evidence that Abel had similar results for higher-degree
equations of prime degree, as well as results on the roots of equations that
are solvable by radicals.
Possibly only his untimely death at the age of 26 prevented him from
being the one to give a complete solution to the problem of which algebraic
equations are solvable by radicals.
Abel’s Proof
Abel’s idea was that if some finite sequence of rational operations and
root extractions applied to the coefficients produces a root of the equation
x5 + ax4 + bx3 − cx2 + dx + e = 0,
the final result must be expressible in the form
1
m
2
m
x = p + R + p2R + · · · + pm−1R
m−1
m
,
where p, p2, . . . , pm−1, and R are also formed by rational operations
and root extractions applied to the coefficients, m is a prime number,
and R1/m is not expressible as a rational function of the coefficients
a, b, c, d, e, p, p2, . . . , pm−1.
By straightforward reasoning on a system of linear equations for the
coefficients pj , he was able to show that R is a symmetric function of the
roots, and hence that R1/m must assume exactly m different values as the
roots are permuted.
Moreover, since there are 5! = 120 permutations of the roots and m is
a prime, it follows that m = 2 or m = 5, the case m = 3 having been ruled
out by Cauchy.
The hypothesis that m = 5 led to certain equation in which the left-hand
side assumed only five values while the right-hand side assumed 120 values
as the roots were permuted.
Then the hypothesis m = 2 led to a similar equation in which one side
assumed 120 values and the other only 10.
Abel concluded that the hypothesis that there exists a formula for solving
the equation was incorrect.
P. Pesic, Abel’s proof. An essay on the sources and meaning of
mathematical unsolvability. MIT Press, Cambridge, MA, 2003.
C. Houzel, The work of Niels Henrik Abel. The legacy of Niels Henrik
Abel, 21–177, Springer, Berlin, 2004
Geometric proof of Abel’s theorem
V.B. Alekseev, Abel’s theorem in problems and solutions. Based on
the lectures of Professor V. I. Arnold. With a preface and an appendix
by Arnold and an appendix by A. Khovanskii. Kluwer Academic Publishers,
Dordrecht, 2004.
D. Fuchs and S. Tabachnikov, Equation of degree five, Mathematical
omnibus. Thirty lectures on classic mathematics, 79–92, American
Mathematical Society, Providence, RI, 2007.
The Abel Prize
The Niels Henrik Abel Memorial Fund was established on 1 January
2002, to award the Abel Prize for outstanding scientific work in the field of
mathematics.
The prize amount is 6 million NOK (about 750,000 Euro) and was
awarded for the first time on 3 June 2003.
Abel Prize Laureates:
Jean-Pierre Serre (2003), Sir Michael Francis Atiyah and Isadore M.
Singer (2004), Peter D. Lax (2005), Lennart Carleson (2006), Srinivasa S.
R. Varadhan (2007), John G. Thompson and Jacques Tits (2008), Mikhail
L. Gromov (2009), John Tate (2010), John Milnor (2011), Endre Szémeredi
(2012)
Solution of the general quintic by elliptic integrals.
• In 1844, Ferdinand Eisenstein showed that the general quintic equation
could be solved in terms of a function χ(λ) that satisfies the special quintic
equation
(
)5
χ(λ) + χ(λ) = λ.
This function is in a sense an analog of root extraction, since the square
root function φ and the cube root function ψ satisfy the equations
(
φ(λ)
)2
(
= λ, ψ(λ)
)3
= λ.
• In 1858 Hermite and Kronecker showed (independently) that the quintic
equation could be solved by using an elliptic modular function.
R. B. King, Beyond the quartic equation. Birkhäuser Boston, Inc.,
Boston, MA, 1996.
The honor of giving a complete solution to the problem of which algebraic
equations are solvable by radicals was given to the French mathematician
Evariste Galois (1811-1832).
At the tender age of 18, Galois communicated to the Academy of
Sciences in Paris some of his results on the theory of equations, through one
of its members, Augustin-Louis Cauchy (1789–1857). Shortly thereafter,
Galois learned that a number of his results had actually been obtained by
Abel, before him.
Two years later, he submitted a rewritten version, Mémoire sur les
Conditions de Resolubilité des Equations par Radicaux [Memoir on the
Conditions for Solvability of Equations by Radicals], which give a complete
solution of the remained problem of characterizing those equations that are
solvable by radicals.
Galois’s Theorem. A polynomial is solvable by radicals if and only if its
Galois group is solvable.
The importance of this memoir was not recognized properly until 1843,
when Joseph Liouville prepared Galois’s manuscripts for publication and
announced that Galois had indeed solved this age-old problem.
Galois’s work forms one of the beginnings of abstract algebra, namely
the theory of groups and that of fields.