The Origin of Complex Numbers

The Origin of Complex Numbers

Chapter 1 Complex Numbers

Overview

Get ready for a treat. You're about to begin studying some of the most beautiful ideas in mathematics. They are ideas with surprises. They evolved over several centuries, yet they greatly simplify extremely difficult computations, making some as easy as sliding a hot knife through butter. They also have applications in a variety of areas, ranging from fluid flow, to electric circuits, to the mysterious quantum world. Generally, they are described as belonging to the area of mathematics known as complex analysis.

Section 1.1The Origin of Complex Numbers

Complex analysis can roughly be thought of as the subject that applies the theory of calculus to imaginary numbers. But what exactly are imaginary numbers? Usually, students learn about them in high school with introductory remarks from their teachers along the following lines: "We can't take the square root of a negative number. But let's pretend we can and begin by using the symbol ." Rules are then learned for doing arithmetic with these numbers. At some level the rules make sense. If , it stands to reason that . However, it is not uncommon for students to wonder whether they are really doing magic rather than mathematics.

If you ever felt that way, congratulate yourself! You're in the company of some of the great mathematicians from the sixteenth through the nineteenth centuries. They, too, were perplexed by the notion of roots of negative numbers. Our purpose in this section is to highlight some of the episodes in the very colorful history of how thinking about imaginary numbers developed. We intend to show you that, contrary to popular belief, there is really nothing imaginary about "imaginary numbers." They are just as real as "real numbers."

Our story begins in 1545. In that year the Italian mathematician Girolamo Cardano published Ars Magna (The Great Art), a 40-chapter masterpiece in which he gave for the first time an algebraic solution to the general cubic equation

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Cardano did not have at his disposal the power of today's algebraic notation, and he tended to think of cubes or squares as geometric objects rather than algebraic quantities.Essentially, however, his solution began with the substiution .This move transformsinto the cubic equationwithout a squared term, which is called a depressed cubic and can be written as

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You need not worry about the computational details, but the coefficients areand.

Exploration.

To illustrate, begin withand substitute.The equation then becomes, which simplifies to.

Exploration.

If Cardano could get any value of x that solved a depressed cubic, he could easily get a corresponding solution to from the identity . Happily, Cardano knew how to solve a depressed cubic. The technique had been communicated to him by Niccolo Fontana who, unfortunately, came to be known as Tartaglia(the stammerer) due to a speaking disorder. The procedure was also independently discovered some 30 years earlier by Scipione del Ferro of Bologna. Ferro and Tartaglia showed that one of the solutions to the depressed cubic equation is

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Although Cardano would not have reasoned in the following way, today we can take this value for x and use it to factor the depressed cubic into a linear and quadratic term. The remaining roots can then be found with the quadratic formula.

For example, to solve,use the substitutionto get,which is a depressed cubic equation.Next, apply the "Ferro-Tartaglia" formula with and to get.Sinceis a root,must be a factor of.Dividingintogives,which yields the remaining (duplicate) roots of.The solutions toare obtained by recalling, which yields the three rootsand.

Exploration.