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1. Completing the cube Let's solve the cubic (1) x3 + px = q.

We'll proceed by analogy with the method of "completing the square":

u-v v

u u v

u-v u-v

v u

In the picture we see that (2) (u - v)3 + 3uv(u - v) = u3 - v 3 .

If we stare at equations (1) and (2) long enough we might be inspired to set x = u-v, which makes equation (2) take the form (3) x3 + 3uvx = u3 - v 3

If we compare equation (3) to equation (1) then we are led to the system (4) p = 3uv, q = u3 - v 3 .

This is a system we have already solved! Altho it looks like a cubic system, it is really a quadratic in the variables u3 and v 3 , once we cube the first equation: (5)

1 u3 v 3 = ( 3 p)3 , u3 - v 3 = q.

Let's remind ourselves how we solved such "product and difference" systems.

Date: 28 July 2008.




2. Linear systems In ancient Iraq even before Hammurabi, and indeed across the ancient world, people had figured out how to determine a pair of numbers given their product and difference. That is, they could solve the system (6) ab = P, a - b = D (a + b)2 = (a - b)2 + 4ab.

for a and b. They did so by first applying the identity (7)

Here is how they pictured this identity.





This identity was closely related to ancient proofs of the "pythagorean" theorem. For the problem at hand they used it in the time-honored fashion: reduce a problem to a simpler one you've already solved. In our case, equation (7) can be used to turn the quadratic system (6) to a linear system. Here's how it works. If we set S = a + b then (8) S= D2 + 4P = 2 ( 1 D)2 + P . 2

(We'll soon see why it is convenient to factor 4 from under the radical.) So now we have the data (9) a + b = S, a - b = D.

1 Everyone knows how to solve this system! The average of a and b is 2 S, which lies halfway between a and b:




Hence (10)



a = 1 S + 1 D, b = 1 S - 1 D. 2 2 2 2

If we apply (10) and (8) to our quadratic system (6) then we obtain an ancient version of the quadratic formula: (11) a=

1 ( 1 D)2 + P + 2 D, b = 2

( 1 D)2 + P - 1 D. 2 2



3. Cardano's formula Let's use our knowledge of quadratic systems to solve the cubic (1). We had used the idea of "completing the cube" to make the substitution x = u - v, which brought us to the cubic system (5). Our quadratic formula (11) tells us that u3 = v3 =

1 ( 1 q)2 + ( 1 p)3 + 2 q 2 3 1 ( 1 q)2 + ( 1 p)3 - 2 q 2 3

Therefore, since x = u - v, we conclude that

1 ( 1 q)2 + ( 1 p)3 + 1 q - ( 1 q)2 + ( 1 p)3 - 2 q. 2 3 2 2 3 It is convenient to use the fact that 3 -1 = -1 to rewrite this slightly:







1 2q


1 ( 1 q)2 + ( 3 p)3 + 2


1 2q


( 1 q)2 + ( 1 p)3 . 2 3

This is Cardano's formula for the solution of the cubic (1). 4. Depressing the cubic We looked at Cardano's formula for one particular cubic (1), but this formula can be used to solve all cubics. You can always reduce to this case, possibly after a simple change of variables. For example, if you have a cubic equation written in the standard way we use today y 3 + ay 2 + by + c = 0 then you can let y = x - 1 a and use the Binomial Theorem to find an equivalent 3 equation in x which has no linear term. Once we solve the resulting equation for x, we can then subtract 1 a from the roots to find the values of y which solve the 3 original equation. Let's take a short digression to remind ourselves about how to expand a binomial such as (x - 1 a)n . Pascal's triangle -- which was in fact known centuries before 3 Pascal, throughout China, India, and the Islamic Empire -- tells us the coefficients we use to collect like powers: 1 1 1 1 1 4 3 6 . . . 2 3 4 1 1 1 1

Each line begins and ends with 1; the other entries are computed by adding the two numbers above it in the previous line. So, the next line of our table would be 1, 5, 10, 10, 5, 1; the one after that would be 1, 6, 15, 21, 15, 6, 1, and so forth. The table tells us, for example, that

1 1 1 (x - 3 a)4 = 1 · x4 - 4 · ( 1 a)1 x3 + 6 · ( 3 a)2 x2 - 4 · ( 3 a)3 x + 1 · ( 1 a)4 . 3 3 1 3a

Note that the powers of x decrease from left to right; the powers of and the signs alternate.




Classically the substitution above, followed by the use of the Binomial Theorem, is called "depressing the cubic". You can then move all of the terms except x3 to the other side of the equation. Altho Cardano would have been appalled at this step, we should have no problem with it, if we understand that the coefficients might be positive, negative, or zero. The algebra of Cardano's formula (12) works just fine in any case. . . 5. The weird case . . . altho there is one weird case -- Cardano called it the casus irreducibilis, but that phrase has no real meaning, so we'll just call it weird. If p < 0 then it may happen that the so-called discriminant (13) ( 1 q)2 + ( 1 p)3 2 3

is also negative. In this case, Cardano's formula (12) expresses the roots in the "imaginary" square-roots of negatives. Does this mean that there are no meaningful solutions? In similar situations this would have been the interpretation, across the globe. On the other hand, it had been known to the medieval Arabic scholars that in this case there is always a positive real root. We can see why this is so by graphing the functions y = x3 and y = -px + q -- the latter has positive slope, since we are assuming that p < 0.

y=x 3 y=-px+q

The first to try and resolve this mystery was the Italian engineer Bombelli. Let's look at his famous example: x3 = 15x + 4. We have p = -15, q = 2, and hence the discriminant is 22 - 53 = -121. Cardano's formula tells us that 3 3 x = 2 + -121 + 2 - -121. What can this mean? How could we possibly get a positive real number out of this? Bombelli said let's just play along with the new rules. Assume that algebra with square roots of negatives makes sense, and obeys the usual rules. Let's these use rules find a cube root of 2 + -121. First of all, we expect that -121 = to 121 -1 = 11 -1. So there is only one new number to worry about, namely -1. Call this number i. What we know about i is that i2 = -1, hence also 3 i = i2 · i = -i and i4 = (-1)2 = 1.



So, we are searching for a number of the form a + bi whose cube is 2 + 11i. If we apply the Binomial Theorem we find that (a + bi)3 = a3 + 3a2 bi - 3ab2 - b3 i = a(a2 - 3b2 ) + b(3a2 - b2 )i. After a few guesses we might hit upon the solution a = 2, b = 1. That is x = 3 2 + 11i + 3 2 - 11i = (2 + i) + (2 - i) = 4. Sure enough, x = 4 does solve our original equation. 6. De Moivre's formula Bombelli gave us real insight: it is often the laws of algebra that are of first importance, not the interpretation of individual numbers. Still, his analysis is unsatisfying, because we have no systematic way to compute cube roots. Bombelli told us that the usual laws of algebra, together with the rule i2 = -1, lead to a simple formula for multiplying complex numbers: (14) (t + iu)(v + iw) = (tv - uw) + i(tw + uv).

Look at the real and imaginary coefficients separately: don't they seem familiar? They are highly suggestive of the addition formulas for sine and cosine: cos( + ) = cos() cos() - sin() sin() sin( + ) = cos() sin() + sin() cos() De Moivre showed that complex numbers could be written in what we now call polar coordinates, using an ancient trick that astronomers had used to simplify multidigit multiplication (before the invention of logarithms). (15) t + iu = r(cos() + i sin()).

The positive factor r is called the modulus of t + iu, and it measures the size of the complex number. The angle is called the argument. If we write the second factor similarly then we can interpret Bombelli's rules in a simple way: To multiply complex numbers multiply the moduli and add the arguments. In symbols: (16) r(cos() + i sin()) × s(cos() + i sin() = rs(cos( + ) + i sin( + )).

Oddly enough, despite this nice representation, even great mathematicians continued long after this to resist the obvious picture of complex numbers in the plane:

iu r t t+iu



Wessel, a Norwegian engineer, was the first to draw this picture and interpret the multiplication geometrically. Mathematicians ignored him. These are now called Argand diagrams, after their rediscoverer. At any rate, we now see clearly how to find r and from t and u, using a bit of simple trig. How does de Moivre's formula (16) help us compute cube roots? Well, cubing a complex number cubes the modulus and triples the argument. To reverse this and find the cube root we should compute the cube root of the modulus and one third of the argument. In other words 3 1 t + iu = 3 r(cos() + i sin()) = 3 r (cos( 3 ) + i sin( 1 )). (17) 3 Notice something fun: since sine and cosine are periodic, there are lots of choices for . In fact we can add or subtract any whole multiple of 2 without changing the value. But changing this way could result in a different cube root, since it amounts to adding or subtracting a multiple of 2/3 to the argument of the root. Hence every nonzero number has exactly 3 cube roots in the complex plane --and 4 fourth roots and 5 fifth roots. . . . For example, since 1 = cos(2n)+i sin(2n), for any integer n, we see that 3 1 could be any of the numbers cos(2n/3)+i sin(2n/3). Altho this seems like there are infinitely many possibilities, they repeat in threes. 7. Quartics Cardano also had a formula to handle quartics, discovered by his student Ferrari. The basic strategy is the same: reduce a given quartic to a simpler problem, by adapting the solution you already know. The specific tool Ferrari used is the method of completing the square, which enabled him to reduce the solution of a quartic to the solution of a cubic. Since del Ferro and Tartaglia had solved the cubic, this is a winning strategy. Suppose we depress our quartic, then shift terms so that it looks like this: (18) x4 + px2 = qx + r. The left-hand side is a quadratic in x2 , and so we will attempt to add terms sx2 and t to both sides so that each is a perfect square: (19) x4 + (p + s)x2 + t = sx2 + qx + (r + t).

We all know that a quadratic is a perfect square precisely when its discriminant is 0. This leads to a system of 2 equations in the new variables s and t: (20)

1 1 ( 2 (p + s))2 = t, ( 2 q)2 = s(r + t).

If we use the first equation to eliminate t from the second, we obtain a cubic equation in s: 1 ( 1 q)2 = s(r + ( 2 (p + s))2 ). 2 We could expand the right-hand side, depress it, then apply Cardano's formula -- but I will leave to you this part of the fun. Let's jump ahead to the next step: whatever we determine s and t to be, both sides of equation (19) become perfect squares: 1 1 (x2 + 2 (p + s))2 = s(x + 2 q/s)2 . If we take square roots of both sides we are left with a pair of ordinary quadratic equations in x: 1 1 x2 + 2 (p + s) = ± s(x + 2 q/s).



8. The Fundamental Theorem After Cardano published the cubic and quartic formulas, people raced furiously to find solutions for all polynomial equations. Everyone understood the rules: use nothing more than ordinary algebra and taking n-th roots to express the roots in terms of the coefficients. It does not matter how nasty the formulas turn out to be. Since de Moivre's formula produces n-th roots in the complex plane, and since all quadratics, cubics, and quartics have their solutions there, it was natural to expect that all solutions to all polynomials can be found in the complex plane. This proved to be true. The great Euler sketched out some ideas for what is now known as the Fundamental Theorem of Algebra, which can be stated rather simply: Any polynomial xn + an-1 xn-1 + · · · of degree n factors into a product of n linear terms (x - r1 ) · · · (x - rn ), where the rj are (not necessarily distinct) complex numbers. In other words, every polynomial has exactly as many roots as its degree, at least if you are willing to look in the complex plane. Euler's ideas were worked out in more detail by Lagrange and de Foncenex. Let's take a quick peek. Euler started with two observations about real polynomials. First, polynomials of odd degree always have at least one root, and hence at least one linear factor. You can see this in the picture:


When x is a large enough positive number -- say a bzillion -- f (x) > 0, because the leading term xn dominates. (If we want to be explicit, we only need to take x bigger than 1 + |an-1 | + |an-2 | + · · · .) But when n is odd, f (x) < 0 when x is a negative bzillion. By the Intermediate Value Property, this says that f (x) = 0 somewhere in between plus bzillion and minus bzillion. Euler's reasoning is sound, but it is a bit tautological. He had only a na¨ ive definition of continuity, which is the essential assumption implicit in his argument. Even his notion of "function" was at first not what we would recognize today. He worked mainly with expressions, not functions. The modern concept of function arose from an intense conversation, lasting much of the 18th century, between leading mathematicians and physicists. Physicists prodded mathematicians to accept the more abstract definition of function in order that it be widely applicable. In fact, the Intermediate Value Property is essentially an axiom about the real line. This axiom is usually expressed a bit more "topologically" by saying that the real line is complete, which is a way of saying that there are no "holes" in the line. In turn, the completeness axiom is equivalent to the statement that every odd-degree polynomial has a real root.



The various logical relationships between such statements was the subject of intense study in the 19th century, and generated heated debates. Two notable contributions were the notion of Cauchy sequence and that of Dedekind cut. Both of these had the goal of beginning with the rationals and completing them -- that is, filling in all the holes. Eventually most everyone was satisfied with the soundness of this notion of completeness. The second thing Euler noted was that the quadratic formula can be used to produce explicit formulas for the roots of any real or complex quadratic. So, Euler's clever idea was to look at the highest power of 2 which divides n, and then induce on that. In brief, Euler proved by a rather intricate argument by induction that if k > 0 and every real polynomial of degree 2k-1 × odd has a complex root then so does every real polynomial of degree 2k × odd. Since every real polynomial of odd degree -- in other words, 20 × odd -- has a real root, and reals are also complex, this says that every real polynomial has a complex root. Euler's implements his clever idea by examining expressions that are symmetric in the roots. This means that any permutation of the roots will leave the value of the function unchanged. This idea was not new. In fact, Newton had proved that every symmetric rational function of the roots is a rational function of the so-called elementary symmetric functions, which are (21) s1 = r1 + · · · + rn , s2 = r1 r2 + r1 r3 + · · · + rn-1 rn , · · · , sn = r1 r2 · · · rn .

These should look familiar, since they are the expression you get when you expand the product (x-r1 ) · · · (x-rn ) and collect terms. More precisely, the j-th coefficient of this polynomial equals (-1)j sj . Newton's theorem means that finding a formula for the roots of a polynomial in terms of the coefficients is the same as finding one which is symmetric in the roots. About a century later, Ruffini, Abel, and especially Galois exploited the rich structure of the collection of all permutations to show that when n > 4 there is no way to express the roots in terms of the coefficients, using only algebraic operations. But the Fundamental Theorem is not about explicit formulas. It merely asserts that the roots all lie in the complex plane. Hence there is no contradiction between the Euler-inspired proof and Galois's much later theory. However, there is a subtle point with Euler's proof, which Euler ignored, but Gauss and others criticized. Euler assumed that all of the roots live together in some domain in which all of the usual rules of algebra are valid. This is true, and not hard to prove, but it required a new perspective on algebra. While France was in the midst of its revolution to overthrow the Bourbon monarchy, which had been restored to power by the British after Napoleon's defeat at Waterloo, Galois was fomenting a revolution in the way we conceive of algebra, whose consequences will almost surely be more long-lasting. 9. Galois's theory Late in the 18th century, on the eve of a series of political and economic revolutions which would sweep the world, Lagrange presented two papers to the Paris Academy which summarized in detail all of the attempts to solve equations of degree 5 and higher. It wasn't a pretty picture. Like Pacioli's Summa of three centuries earlier, which focused attention on cubic equations, Lagrange's summary was comprehensive and seminal. It was also pessimistic.



Pacioli's despair turned out to be misplaced: within a generation two different mathematicians, working independently, had solved cubics, and another had solved quartics. In contrast, Lagrange's mood was spot on: within a generation two different mathematicians, working independently, had shown the impossibility of solving quintics by radicals. Abel's work was the more comprehensive, with both positive and negative results. He had shown, for example, that altho the general quintic could not be solved by radicals, some quintics could. Specifically, he considered all of the permutations of the roots a polynomial equation which play nicely with the rules of algebra. He showed that if all of these happen to commute with one another, then the equation can be solved by radicals. Abel's ideas were not fully understand for a long time, perhaps because he died quite young, before they could be fully elaborated. Galois clarified the situation, and went well beyond Abel's results. He called the group of algebra-preserving permutations the group of the equation. We now call it the Galois group. Galois noted that there is a kind of law of algebra in the group itself, since you can compose two group elements to produce another. Hence there is a kind of "multiplication table" for the group, which is associative and has an identity and inverses. It is not in general commutative, and this is exactly the crux of the problem. When the group is commutative we call it abelian, in honor of Abel. Galois's great theorem asserts that an equation is solvable by radicals if and only if its group can be sliced up (in a very precise sense, which we will skip) into abelian pieces. He borrowed the term solvable to refer to groups which can be sliced up into abelian pieces. The group of an equation of degree less than 5 is always solvable, reflecting the fact that those equations are solvable by radicals. However, in general a quintic has a group which is as far from solvable as possible. This provides an expansive generalization of Abel's original theorems. Altho Galois died very young, and his work lay in obscurity for decades, the theory of groups as abstract structures eventually became a cornerstone of mathematics. In fact, after Galois, algebra really changed directions, heading more towards the realm of abstract structure -- besides groups, there were rings, ideals, fields, homomorphisms, and many other esoteric concepts. For millenia, algebra had been about finding procedures to solve equations. Quite quickly it had become instead the study of abstract structures defined by axiomatic laws of composition. In fact, this new perspective on algebra spread across wide swaths of mathematics, especially geometry and number theory. Much of mathematics became, at least on the surface, the study of axiomatic systems, often with little to anchor them in the real world. Proofs were increasingly formal. Dry definitions, dense jargon, and highly technical lemmas dominated mathematical papers. By the beginning of the 20th century, mathematicians were more likely than not to be narrow specialists. Paradoxically, this was also the era when mathematics became an essential tool in more and more branches of science and engineering. As mathematics became more abstract, it became more useful.



10. Hilbert's program After the invention of calculus, and continuing to this day, the biggest stories in mathematics were driven by functions of continuous variables. The 19th century devoted a large share of its mathematical energy to understanding the subtleties about differentiation and integration, sharpening these tools for more precise work in ever more tricky situations. Much of this work boiled down to the following question. Under what conditions is a function the derivative of its integral? The Fundamental Theorem of Calculus says that continuity suffices. But there are many situations where you don't have continuity at every point. How close to continuous does the function have to be in order that you are still guaranteed the conclusion of the Fundamental Theorem? Eventually this problem was resolved, and the tools of calculus could be applied both to solve many ancient problems -- such as "doubling the cube" -- and also to open up vast new areas of exploration, especially in geometry. Cantor's set theory laid the foundation on which all of this success was built. By the end of the century, David Hilbert claimed that Cantor had created a paradise from which mathematicians would not be driven. Hilbert was a mathematician of towering stature. He was, with Henri Poincar´, one of the two mathematicians e usually regarded as the last who could claim to know all of mathematics. His views on the subject would steer the entire course of mathematics for generations. And yet, there were lots of unhappy folks at Hilbert's party. Some of this was was a conservative reaction, reflecting Gauss' rejection of "completed infinity". Some of this was the usual religious and philosophical carping about how any new discovery in math or science undermines "the faith of our fathers". A large part was simply the rootlessness felt when the discovery of riemannian and lobachevskian geometries knocked euclidean geometry from its hallowed perch. Hilbert had a strong hand in this iconoclasm, when he proved that euclidean and lobachevskian geometries are relatively consistent: if one is "true" then so must be the other. In addition to all of this, early versions of Cantor's set theory were shown to be na¨ at best, and possibly complete nonsense. If you are not careful in building ive your theory, it will contain a version of the "barber's paradox". To wit, in a certain village, the barber shaves everybody who doesn't shave himself. Does the barber shave himself? He does if and only if he doesn't. This hole was patched, but many wondered whether another breach would be discovered . . . and then another . . . and then another. . . To strengthen the redoubt around his cantorian utopia, Hilbert formulated a program to prove that the foundations of mathematics were unshakable. His program reduced mathematics to symbolic formalism. He was saying that mathematics has a syntax, but no semantics. There is no meaning, just formal correctness. The first on Hilbert's famous list of 23 great unsolved problems was a question in set theory (not about paradoxes, but about the so-called "continuum hypothesis"). The second asked for a proof that the axioms of arithmetic are consistent -- that is, contain no fatal paradoxes. Hilbert also wanted to show that the axioms are complete -- that is, every question posed in the formal language of ordinary arithmetic is either true or false. He preached the mantra "we must know, we will know". Every mathematical question is unambiguously either true or false -- and not both!



11. Turing machines On most mathematical issues, Hilbert was invariably right. He was not given to wild speculation. He was careful, methodical. When he made a conjecture he did so after painstaking study. He always marshaled a lot of evidence in support of his guesses. But with the metamathematical question of consistency and completeness, Hilbert was driven by blind faith. There was little hard evidence to support his views. One of his former disciples, the Dutch mathematician Brouwer, formulated an alternative to Hilbert's program. It was a radical departure. To follow Brouwer's "intuitionism" you would have to forswear proof by contradiction, accept the integers as if they were handed down to man from on high, and accept only constructive proofs -- that is, proofs where the answer is not merely proven to exist, but constructed explicitly. Despite the apparent hardship demanded by this regimen, Brouwer attracted many followers and sympathizers. Some of them, such as Poincar´, were quite e influential. Some, such as John von Neumann, were quite close to Hilbert. Hilbert felt betrayed, and pressed his program vigorously, if not always ethically. In the end, most of mathematics rejected Brouwer. The "algebraization" of mathematics was complete. The greatest proponents of this style of mathematics were the "bourbachiques", a group of French mathematicians who met clandestinely to plot the overthrow of the entrenched pedagogical ´lite. They sought to completely rewrite the way that e mathematics is taught. Explicitly drawing inspiration from Hilbert and his star pupil Emmy Noether, they wrote a series of books, under the collective pseudonym "Nicolas Bourbaki". The series was called Elements of Mathematics. The evocation of Euclid was deliebrat. They aimed to build up the foundations of all of mathematics, starting with the axioms of set theory. The scale of the project dwarfed Euclid's monumental accomplishment. At the same time that the bourbachiques were first organizing their assault, the Austrian Kurt G¨del delivered a devastating blow to Hilbert's dream. He proved o that any formal system which is powerful enough to do ordinary arithmetic is either inconsistent or incomplete. Thus, if we believe that ordinary arithmetic is logically consistent, then we must accept that there are "nonstandard" models of it. There is not even one true arithmetic! G¨del's Incompleteness Theorems still left a little wiggle room for the hilbero tians, but eventually Hilbert's entire program would prove to be a false hope. The American Emil Post had anticipated G¨del, and even gone beyond him, to develop o the first general theory of computing. Unfortunately, Post's delicate emotional and mental health could not withstand the strain, and he was institutionalized before he could publish his ideas. Another American, Alonzo Church, and a Briton, Alan Turing, built on G¨del's ideas to show that there is no mechanical computing o process which can successfully handle all possible starting input. Turing's work on this was particularly important, because he showed that there is a "universal" computing process, now known as a Turing machine. This master recipe for computing something said, in effec, "read the recipe and do what it says". Altho his work was completely mathematical and abstract, it led eventually to the construction of the electronic computer. The iPhone is a Turing machine.



Both the study of theoretical computer science -- which is generally much more in need of constructive proofs than existential ones -- and also the availability of real working computers has restored some balance to mathematics. Computers, both abstract and concrete, have returned some of the former glory to the age-old question of "how do I solve this?" Mathematics remains secured to the formalism of set theory and axiomatics, but it is also now more openly experimental. Above all, there is no longer any serious claim that there is one true mathematics.

Univ of Toledo


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