Problem 18.4 Algebraic Solution of Cubics
A little more history: Most historical accounts assert correctly that Khayyam did not find the negative roots of cubics. However, they are misleading in that they all fail to mention that his methods are fully sufficient to find the negative roots, as we have seen above. This is in contrast to the common assertion (see, for example, [SE: Davis & Hersch]) that Girolamo Cardano (15011578, Italian) was the first to publish the general solution of cubic equations. In fact, as we shall see, Cardano himself admitted that his methods are insufficient to find the real roots of many cubics.
Cardano published his algebraic solutions in his book, Artis Magnae (The Great Art), which was published in 1545. For a readable English translation and historical summary, see [AT: Cardano]. Cardano used only positive coefficients and thus divided the cubic equations into the same 13 types (excluding x3 = c and equations reducible to quadratics) used earlier by Khayyam. Cardano also used geometry to prove his solutions for each type. As we did above, we can make a substitution to reduce these to the same types as above:
If we allow ourselves the convenience of using negative numbers and lengths, then we can reduce these to one type: x3 + ax + b = 0, where now we allow a and b to be either negative or positive.
The main "trick" that Cardano used was to assume that there is a solution of x3 + ax + b = 0 of the form x = t1/3 + u1/3. Plugging this into the cubic we get
If you expand and simplify this, you get to
(Cardano did this expansion and simplification geometrically by imagining a cube with sides t1/3 + u1/3.) Thus x = t1/3 + u1/3 is a root if
Solving, we find that t and u are the roots of the quadratic equation
which Cardano solved geometrically (and so can you, Problem 18.1) to get
Thus the cubic has roots
This is Cardano's cubic formula. But, a strange thing happened. Cardano noticed that the cubic x3 = 15x + 4 has a positive real root 4 but, for this equation, a = -15 and b = -4, and if we put these values into his cubic formula, we get that the roots of x3 = 15x + 4 are
But these are the sum of two complex numbers even though you have shown in Problem 18.3 that all three roots are real. How can this expression yield 4?
In Cardano's time there was no theory of complex numbers and so he reasonably concluded that his method would not work for this equation, even though he did investigate expressions such as . Cardano writes ([AT: Cardano, page 103])
It is not clear what book he is referring to, but the "aliza problem" presumably refers to the mathematician known as al'Hazen, Abu Ali al'Hasan ibu al'Haitam (9651039), who was born in Persia and worked in Egypt and whose works were known in Europe in Cardano's time. Al'Hazen had used intersecting conics to solve specific cubic equations and the problem of describing the image seen in a spherical mirror this latter problem in some books is called "Alhazen's problem."
In addition, we know today that each complex number has three cube roots and so the formula
is ambiguous. In fact, some choices for the two cube roots give roots of the cubic and some do not. (Experiment with x3 = 15x + 4.) Faced with Cardano's Formula and equations such as x3 = 15x + 4, Cardano and other mathematicians of the time started exploring the possible meanings of these complex numbers and thus started the theory of complex numbers.
a. Solve the cubic x3 = 15x + 4 using Cardano's Formula and your knowledge of complex numbers.
Remember that on the previous page we showed that x = t1/3 + u1/3 is a root of the equation if t + u = - b and t1/3 u1/3 = -(a/3).
b. Solve x3 = 15x + 4 by dividing through by x - 4 and then solving the resulting quadratic.
c. Compare your answers and methods of solution from Problems 18.3c, 18.4a, and 18.4b.
So What Does This All Point To?
So what does the experience of this chapter point to? It points to different things for each of us. I conclude that it is worthwhile paying attention to the meaning in mathematics. Often in our haste to get to the modern, powerful analytic tools we ignore and trod upon the meanings and images that are there. Sometimes it is hard even to get a glimpse that some meaning is missing. One way to get this glimpse and find meaning is to listen to and follow questions of "What does it mean?" that come up in ourselves, in our friends, and in our students. We must listen creatively because we and others often do not know how to express precisely what is bothering us.
Another way to find meaning is to read the mathematics of old and keep asking, "Why did they do that?" or "Why didn't they do this?" Why did the early algebraists (up until at least 1600 and much later, I think) insist on geometric proofs? I have suggested some reasons above. Today, we normally pass over geometric proofs in favor of analytic ones based on the 150-year-old notion of Cauchy sequences and the Axiom of Completeness. However, for most students and, I think, most mathematicians, our intuitive understanding of the real numbers is based on the geometric real line. As an example, think about multiplication: What does a ´ b mean? Compare the geometric images of a ´ b with the multiplication of two infinite, non-repeating, decimal fractions. What is?
There is another reason why a geometric solution may be more meaningful: Sometimes we actually desire a geometric result instead of a numerical one. For example, a friend and I were building a small house using wood. The roof of the house consisted of 12 isosceles triangles which together formed a 12-sided cone (or pyramid). It was necessary for us to determine the angle between two adjacent triangles in the roof so we could appropriately cut the log rafters. I immediately started to calculate the angle using (numerical) trigonometry and algebra. But then I ran into a problem. I had only a slide rule with three-place accuracy for finding square roots and values of trigonometric functions. At one point in the calculation I had to subtract two numbers that differed only in the third place (for example, 5.68 - 5.65); thus my result had little accuracy. As I started to figure out a different computational procedure that would avoid the subtraction, I suddenly realized I didn't want a number, I wanted a physical angle. In fact, a numerical angle would be essentially useless imagine taking two rough boards and putting them at a given numerical angle apart using only an ordinary protractor! What I needed was the physical angle, full-size. So I constructed the angle on the floor of the house using a rope as a compass. This geometric solution had the following advantages over a numerical solution: