Dr. Euler's Fabulous Formula

Cures Many Mathematical Ills

Paul Nahin

• Publisher: Princeton University Press
• Year: 2011
• ISBN:
  9781400838479
• Topic: Matematica
• Subtopic: Storia e filosofia della matematica

In the mid-eighteenth century, Swiss-born mathematician Leonhard Euler developed a formula so innovative and complex that it continues to inspire research, discussion, and even the occasional limerick. Dr. Euler's Fabulous Formula shares the fascinating story of this groundbreaking formula—long regarded as the gold standard for mathematical beauty—and shows why it still lies at the heart of complex number theory. In some ways a sequel to Nahin's An Imaginary Tale, this book examines the many applications of complex numbers alongside intriguing stories from the history of mathematics. Dr. Euler's Fabulous Formula is accessible to any reader familiar with calculus and differential equations, and promises to inspire mathematicians for years to come.

Chapter 1

Complex Numbers

1.1 The “mystery” of

.

Many years ago a distinguished mathematician wrote the following words, words that may strike some readers as somewhat surprising:

I met a man recently who told me that, so far from believing in the square root of minus one, he did not even believe in minus one. This is at any rate a consistent attitude. There are certainly many people who regard

as something perfectly obvious, but jib at

. This is because they think they can visualize the former as something in physical space, but not the latter. Actually

is a much simpler concept.¹

I say these words are “somewhat surprising” because I spent a fair amount of space in An Imaginary Tale documenting the confusion about

that was common among many very intelligent thinkers from past centuries.

It isn’t hard to appreciate what bothered the pioneer thinkers on the question of

. In the realm of the ordinary real numbers, every positive number has two real square roots (and zero has one). A negative real number, however, has no real square roots. To have a solution for the equation x² + 1 = 0, for example, we have to “go outside” the realm of the real numbers and into the expanded realm of the complex numbers. It was the need for this expansion that was the intellectual roadblock, for so long, to understanding what it means to say i =

“solves” x² + 1 = 0. We can completely sidestep this expansion,² however, if we approach the problem from an entirely new (indeed, an unobvious) direction.

Figure 1.1.1. A rotated vector

A branch of mathematics called matrix theory, developed since 1850, formally illustrates (I think) what the above writer may have had in mind. In figure 1.1.1 we see the vector of the complex number x + iy, which makes angle α with the positive real axis, rotated counterclockwise through the additional angle of β to give the vector of the complex number x′ + iy′. Both vectors have the same length r, of course, and so

. From the figure we can immediately write x = r cos(α) and y = r sin(α), and so, using the addition formulas for the sine and cosine

Now, focus on the x′, y′ equations and replace r cos(α) and r sin(α) with x and y, respectively. Then,

Writing this pair of equations in what is called column vector/matrix notation, we have

where R(β) is the so-called two-dimensional matrix rotation operator (we’ll encounter a different sort of operator—the differentiation operator—in chapter 3 when we prove the irrationality of π²). That is, the column vector

, when operated on (i.e., when multiplied³) by R (β), is rotated counterclockwise through the angle β into the column vector

.

Since β = 90° is the CCW rotation that results from multiplying x + iy by i, this would seem to say that i =

can be associated with the 2 × 2 matrix R (90°)

Does this mean that we might, with merit, call this the imaginary matrix? To see that this actually makes sense, indeed that it makes a lot of sense, recall the 2 × 2 identity matrix

which has the property that, if A is any 2 × 2 matrix, then AI = IA = A. That is, I plays the same role in matrix arithmetic as does 1 in the arithmetic of the realm of the ordinary real numbers. In that realm, of course, i² = − 1, and the “mystery” of

is that it itself is not (as mentioned earlier) in the realm of the ordinary real numbers. In the realm of 2 × 2 matrices, however, there is no such “mystery” because the square of the “imaginary matrix” (a perfectly respectable 2 × 2 matrix) is

That is, unlike the ordinary real numbers, the realm of 2 × 2 matrices does have a member whose square is equal to the negative of the 2 × 2 matrix that plays the role of unity.

To carry the analogy with the ordinary real numbers just a bit further, the zero 2 × 2 matrix is

, since any 2 × 2 matrix multiplied by 0 gives 0. In addition, just as (1/a) ·...