How the elusive imaginary number was first imagined, and how to imagine it yourself
Imagining Numbers (particularly the square root of minus fifteen) is Barry Mazur's invitation to those who take delight in the imaginative work of reading poetry, but may have no background in math, to make a leap of the imagination in mathematics. Imaginary numbers entered into mathematics in sixteenth-century Italy and were used with immediate success, but nevertheless presented an intriguing challenge to the imagination. It took more than two hundred years for mathematicians to discover a satisfactory way of "imagining" these numbers.
With discussions about how we comprehend ideas both in poetry and in mathematics, Mazur reviews some of the writings of the earliest explorers of these elusive figures, such as Rafael Bombelli, an engineer who spent most of his life draining the swamps of Tuscany and who in his spare moments composed his great treatise "L'Algebra". Mazur encourages his readers to share the early bafflement of these Renaissance thinkers. Then he shows us, step by step, how to begin imagining, ourselves, imaginary numbers.
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About the Author
Barry Mazur does his mathematics at Harvard University and lives in Cambridge, Massachussetts, with the writer Grace Dane Mazur.
Read an Excerpt
(Particularly the Square Root of Minus Fifteen)
By Barry Mazur
Farrar, Straus and GirouxCopyright © 2003 Barry Mazur
All rights reserved.
THE IMAGINATION AND SQUARE ROOTS
1. Picture this.
Picture Rodin's Thinker, crouched in mental effort. He has his supporting right elbow propped not on his right thigh, as you or I would have placed our right elbow, but rather on his left thigh, which bolts him into an awkward striving, his muscles tense with thought. But does he, can we, really feel our imaginative faculty at work, striving toward, and then finally achieving, an act of the imagination?
Consider the range of our imaginative experiences. Consider, for example, how immediate is the experience of imagining what we read. Elaine Scarry has remarked that there is no "felt experience" corresponding to this imaginative act. We experience, of course, the effect of what we are reading. Scarry claims that if we read a phrase like
the yellow of the tulip
we form, perhaps, the image of it in our mind's eye and experience whatever emotional effect that image produces within us. But, says Scarry, we have no felt experience of coming to form that image. We will return to this idea later.
Perhaps one should contrast reading with trying to think something up for ourselves. Rainer Maria Rilke's comment on the working of our imagination,
We are the bees of the invisible
paints our imaginative quests as not entirely unfelt experiences (following Scarry), but not contortions (following Rodin) either. Our gathering of the honey of the imaginative world is not immediate; it takes work. But though it requires traveling some distance, merging with something not of our species, communicating by dance to our fellow creatures what we've done and where we've been, and, finally, bringing back that single glistening drop, it is an activity we do without contortion. It is who we bees are.
Thinking about the imagination imagining is made difficult by the general swiftness and efficacy of that faculty. The imagination is a fleet genie at your service. You want an elephant? Why, there it is:
You read "the yellow of the tulip."
And, again, there it is: a calligraphed swath of yellow on your mental movie screen.
More telling, though, are the other moments of thought, when our genie is not so surefooted. Moments composed half of bewilderment and half of expectation; moments, for example, when some new image, or viewpoint, is about to reveal itself to us. But it resists emerging. We are forced to angle for it.
At those times, it is as if the waters of the imagination are roiling; you have cast your fishing line from a somewhat shaky boat, and you feel a tug on that line, but have no clear sense what you have hooked onto. Bluefish, old boot, or some underwater species never before seen? But you definitely feel the tug.
I want to think about the inner articulations of our imaginative life by "re"-experiencing a particular example of such a tug. The example I propose to consider occurs in the history of mathematics. It might be described as a moment of restless anticipation in the face of a slowly emerging act of imagining. Moment, though, is not the right word here, for the period, rather, stretches over three centuries. And anticipation carries too progressivist and perhaps too personal a tone, for this "act" doesn't take place fully in any single mind. There are many "bees of the invisible" in the original story.
If we are successful, we will be reenacting, for ourselves, the imagining of a concept that, for the original thinkers, had never been seen or thought before, and that seemed to lie athwart things seen or thought before. Of course, thinking about things never thought before is the daily activity of thought, certainly in art or science. The cellist Yo-Yo Ma has suggested that the job of the artist is to go to the edge and report back. Here is how Rilke expressed a similar sentiment: "Works of art are indeed always products of having-been-in-danger, or having-gone-to-the-very-end in an experience, to where one can go no further."
In contrast to the instantly imaginable "yellow of the tulip," the square root of negative quantities was a concept in common use in mathematics for over three hundred years before a satisfactory geometric understanding of it was discovered. If you deal exclusively with positive quantities, you have less of a challenge in coming to grips with square roots: the square root of a positive number is just a quantity whose square is that number.
Any positive number has only one (positive) square root. The square root of 4, for example, is 2. What is the square root of 2? We know, at the very least, that its square is 2. Using the equation that asserts this,
([square root of 2])2 = [square root of 2]·[square root of 2] = 2
try your hand at estimating [square root of 2]. Is it smaller than 3/2? Do you see why [square root of 3] · [square root of 5] = [square root of 15]?
Square roots are often encountered geometrically, as lengths of lines. We will see shortly, for example, that [square root of 2] is the length of the diagonal of a square whose sides have length 1.
Also, if we have a square figure whose area is known to be A square feet, then the length of each of its sides, as in the diagram below, is [square root of A] feet.
The square root as "side"
Suppose that each box in this diagram has an area equal to I square foot. There are a hundred boxes, so A = 100, and the dimensions of the large square are [square root of A] by [square root of A] — that is, 10 by 10.
In Plato's Meno, Socrates asks Meno's young slave to construct a square whose area is twice the area of a given square. Here is the diagram that Socrates finally draws to help his interlocutor answer the question:
The profile of this diagram is a 2 × 2 square (whose area is therefore 4) built out of four 1 × 1 squares (each of area 1). But in its midst, we can pick out a catercorner square (standing, as it seems, on one of its corners). By rearranging the triangular pieces that make up the diagram, can you see, as Socrates' young friend in the Meno did, that the catercorner square has area 2, and therefore each of its sides has length [square root of 2]?
The sides of the catercorner square play a double role: they are also the diagonals of the small (1 × 1) squares. So, as promised a few paragraphs earlier, we see [square root of 2] as the length of the diagonal drawn in a square whose sides are of length 1.
The early mathematicians thought of the square root as a "side"; the sixteenth-century Italians would at times simply refer to the square root of a number as its lato, its "side." Thus, at first glance, negative numbers don't have square roots, for (as I discuss later) the square of any numerical quantity (positive or negative) is positive. In fact, a second and third glance will tend to confirm the suspicion that negative numbers are not entitled to have square roots.
If we think of square roots in the geometric manner, as we have just done, to ask for the square root of a negative quantity is like asking: "What is the length of the side of a square whose area isless than zero?" This has more the ring of a Zen koan than of a question amenable to a quantitative answer. Nevertheless, these seemingly nonexistent square roots were, early on, seen to be useful. But the first users of square roots of negative numbers were queasy about the practice of invoking such airy objects. These strange square roots were called imaginary numbers, meaning they were difficult to place among real mathematical objects.
And then, an astonishingly satisfying image of these square roots emerged. A way ofimagining these otherwise unpicturable "numbers" was found independently, and almost simultaneously, by two, or possibly three (or more), people. What a dramatic act: to find a home in our imagination for such an otherwise troublesome concept!
This "way of imagining" has become our common intellectual property. It and the numbers it helped us imagine have found thorough and ubiquitous use, not only by mathematicians but by every engineer who works with the calculus, by every physicist.
The aim of this book is not to give a historical account. Rather, it is to re- create, in ourselves, the shift of mathematical thought that makes it possible to imagine these numbers.
Poetry, to be sure, has "shifts of thought" at its core, the "turn" of the poem, in both its viewpoint and its typography, being celebrated in the word verse. Poetry demands our paying attention to these turns. For people who pay such attention while reading poetry but who have never done anything similar with mathematics, I hope the style of presentation I have adopted — which passes back and forth between reflections on the imaginative work of thinking about poetry and thinking about mathematics — will be helpful.
In proceeding with our mathematical theme, we want first to feel the uncomfortableness of the early mathematicians who encounter imaginary numbers; then to sense the possibility that some shift, some new viewpoint in thinking about number, may help to tame the concept of imaginary number; then to be conscious of the emergence of this viewpoint within ourselves. Finally, we will see that our new attitude toward number unifies otherwise disparate intuitions and helps us interpret an amazing formula that perplexed sixteenth-century mathematicians.
As for prerequisites, the less mathematics you know, the better prepared you are for the task ahead. To follow the mathematics presented here, you will only need to have the skill to perform certain simple multiplications and substitutions when the text requests this, and to allow with equanimity the occasional appearance of simple algebraic equations of the type encountered in the first weeks of high school algebra. If you can do, or follow, the sample exercises in this endnote, you are ready for the math in this book.
Let us start by considering that imaginative construct, the faculty of imagination itself.
A certain mathematical article opens with the invitation:
Imagine ... an infinite completely symmetrical array of points.
In the prologue to Shakespeare's Henry V, the Chorus asks that you, the audience, let the actors,
... ciphers to this great accompt,
On your imaginary forces work.
Paul Scott's The Raj Quartet begins with a request of the reader:
IMAGINE, then, a flat landscape, dark for the moment, but even so conveying to a girl running in the still deeper shadow cast by the wall of the Bibighar Gardens an idea of immensity.
What a problematic instruction: to be told to imagine! What are we doing, and do we have the language to say what we are doing, when we fulfill that instruction?
Our English word imagination has a direct antecedent in Latin, but the earlier Latin word, which connoted "object of the imagination" (at least as a side meaning), is visio, whose standard meaning is "sight." For a discussion about this (and for a comprehensive history and commentary on what has been said about the imagination), see Eva Brann's majestic The World of the Imagination — Sum and Substance. Here is Quintilian explaining the Greek origin of the Latin term visio:
What the Greeks call "phantasies" we rightly term "sights" through which the images of absent things are so represented in the mind that we seem to discern and have them present.
Quintilian's definition of sights as meaning "objects of the imagination" is a serviceable definition, as far as it goes. It includes things we have seen before but which happen to be absent. Its reach, however, does not encompass the unicorns and sphynxes that tinkers and joiners of the imagination have thrown together for us.
One might try to extend Quintilian's definition, following the lead of Jeremy Bentham, by claiming that the imagination is a faculty by which "a number of abstracted ideas are compounded into one image." Bentham's definition goes a bit further than Quintilian's, but not much, for surely there are objects of thought that cannot be parsed in terms of the algebra of simple, previously known images.
And Bentham's definition, which has the imaginative faculty playing the menial role of editing table for videotapes of the mind's eye, would hardly satisfy Wordsworth, who would prefer a loftier function of the imagination: the function of connecting mere fact with "that infinity without which there is no poetry." For Wordsworth, the imaginative faculty is the transcendental alchemist that turns, for example, the "mere" gold band of a wedding ring into a symbol of eternal unity.
Quintilian, Bentham, Wordsworth, et al., notwithstanding, there are those who simply shrug off "imagination" as an "'onomatoid,' that is, a namelike word which in fact designates nothing because it signifies too broadly." Is it one thing, deserving of the pronoun it? Coleridge makes a distinction in Biographia Literaria between what he calls the imagination and its less daring sibling fancy, which "is indeed no other than a mode of memory emancipated from the order of time and space." In some circles, the concept of the imaginative faculty (or, at least, the idea that you can say anything about it) raises philosophical suspicion; in other circles, its very mention raises religious fears. For example, a recent review of high school history textbooks reports that, to satisfy the religious right, the word imagine is largely banished from textbooks. An editor at McGraw-Hill is quoted as saying, "We were told to try to avoid using the word 'imagine' because people in Texas felt it was too close to the word 'magic' and therefore might be considered anti-Christian."
Nevertheless, there are certain experiences of the intellect that cannot be discussed at all without grappling with the issue of the imagination.
3. Imagining what we read.
When we look at a page of writing, our mind's eye sees something quite different than the white page, the black ink. John Ashbery, in his prose poem "Whatever It Is, Wherever You Are," writes of reading:
[T]he yellow of the tulip, for instance — will flash for a moment in such a way that after it has been withdrawn we can be sure there was no imagining, no auto-suggestion here, but at the same time it becomes as useless as all subtracted memories.
He muses about the inventors of writing:
To what purpose did they cross-hatch so effectively, so that the luminous surface that was underneath is transformed into another, also luminous but so shifting and so alive with suggestiveness that it is like quicksand, to take a step there would be to fall through the fragile net of uncertainties into the bog of certainty ...
and suggests that the images conjured by reading flash onto our mental screen and convey "certainty without heat or light." For Scarry, the "vivacity" of the yellow Hash of the tulip compels conviction, and the suddenness of its appearance in our mind precludes our having any "felt experience of image-making." She says that
the imagination consists exclusively of its objects, that it is only knowable through its objects, that it is remarkable among intentional states for not being easily separable into the double structure of state and object.
Excerpted from Imagining Numbers by Barry Mazur. Copyright © 2003 Barry Mazur. Excerpted by permission of Farrar, Straus and Giroux.
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Table of Contents
1 - THE IMAGINATION AND SQUARE ROOTS,
2 - SQUARE ROOTS AND THE IMAGINATION,
3 - LOOKING AT NUMBERS,
4 - PERMISSION AND LAWS,
5 - ECONOMY OF EXPRESSION,
6 - JUSTIFYING LAWS,
7 - BOMBELLI'S PUZZLE,
8 - STRETCHING THE IMAGE,
9 - PUTTING GEOMETRY INTO NUMBERS,
10 - SEEING THE GEOMETRY IN THE NUMBERS,
11 - THE LITERATURE OF DISCOVERY OF GEOMETRY IN NUMBERS,
12 - UNDERSTANDING ALGEBRA VIA GEOMETRY,
APPENDIX: - THE QUADRATIC FORMULA,
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