The sum of these terms, Euler noted, is the irrational number 2.718281828459 ..., which ‘for the sake of brevity’ he will represent as e. This number is the base of natural logarithms and one of the most important mathematical constants. Euler then noted that if we use e as the base of our exponents, then the function ex can be calculated for any x by an infinite series:
This is known as the exponential function, an example of what is called the Taylor series.9
In Chapter VIII, Euler turned to trigonometric functions. He began by reviewing the fact that if the diameter of a circle is 1 its circumference is an irrational number, 3.14159265...which ‘for the sake of brevity’ he says he will call π. He then described properties of the trigonometric functions, which associate to the measure of an angle in a right-angled triangle the numbers created by various ratios of the sides. The sine function, for instance, associates to the measure of one of the acute angles in a right-angled triangle the ratio of the length of the side opposite that angle to the length of the hypotenuse. The sine function can be generalized from acute angles to arbitrary angles as follows: Draw a right-angled triangle ABC, with hypotenuse BC of length 1, in the (x, y)-plane so that vertex B lies at the origin (0,0), vertex A lies on the positive x-axis, and vertex C lies above the x-axis. Let a be the measure of the angle ∠ABC, measured counterclockwise from the positive x-axis. Then sin a is the ratio of the lengths AC/BC, but since BC = 1, sin a = length AC = the y-coordinate of the point C. If we take ‘y-coordinate of C’ as the definition for sin a (a is the measure of angle ∠ABC), then we have a definition that works for any angle: rotate BC through an angle a (starting from the positive x-axis and measured counterclockwise) and record the y-coordinate of C. Then the sine goes from 0 to 1 (at 90 degrees) back to 0 (180 degrees), thence to –1 (270 degrees), again to 0 (360 degrees), and repeats that pattern through successive cycles of 360 degrees, producing a pattern called the ‘sine wave’ familiar from oscilloscopes. The general cosine function is defined in the same way, except taking the x-coordinate of C as the value. As the angle varies, the cosine goes from 1 to 0 to –1 to 0 to 1, repeating just as does the sine function, making the same pattern as the sine function, but out of phase.
Euler then runs through various more or less obvious properties of sines and cosines, including the fact that, from a simple application of the Pythagorean theorem, (sin x)2 + (cos x)2 = 1.
Continuing to summarize things that Newton and other predecessors knew, Euler next showed how trigonometric functions involving sines and cosines could also be expressed in terms of infinite series. For instance, the function sin x can be expressed as the following infinite sum of terms, which we’ll put in this font so that we can simply and easily follow the terms:
while cos x, whose pieces we’ll put in this font, is:
And Euler then used these functions to show how all the other trigonometric functions likewise could be expressed as infinite series.
Euler’s fluency at calculating now enabled him to arrange these trigonometric functions so that they added up to something identical to the exponential function with base e. He did so with the aid of the imaginary number which he would later – years after he wrote the Introductio – symbolize as i. Although i is not a ‘real’ number – a number with a place on a number line – it is used in real mathematical operations and allows mathematicians to solve otherwise insoluble equations. If, for instance, you insert it in the exponent of ex it shows up in each term of the infinite series associated with it:
But i2 is −1, and therefore i3 = −i, i4 = 1, i5 = i, etc. So the series becomes:
Euler observed that if you group together the multiples of i, you obtain:
Or, as he wrote toward the end of Chapter VIII of the Introductio (using i where he used as does the English translation),10
eix = cos x + isin x
This equation establishes the deep connection between exponential and trigonometric functions. When the great Indian mathematician Srinivasa Ramanujan (1887–1920) discovered this connection on his own while in high school, he wrote it down excitedly – and was so crestfallen to discover that he was not the first that he hid all his calculations in the roof of his house.11
This equation is magic enough, but there’s more. Suppose x is π. The sine of π is 0, and the cosine of π is −1. Then eiπ = −1, or eiπ + 1 = 0.
Another way to show the truth of this equation graphically is the following. Suppose we insert π for x. Then the formula a few paragraphs above becomes:
Mathematicians can add such a sequence as a series of vectors, with each one beginning at the tail end of the one preceding it, and with the imaginary number i rotating a vector 90 degrees counterclockwise.12 If we start at 0, the first term (1) is a vector that takes us 1 unit out on the x-axis, to coordinate (1, 0). The second term (iπ) takes the form of a vector that starts at (1, 0) and, rotated counterclockwise with respect to the first, goes straight up π units, ending at coordinate (1, π). The third term, (iπ2/2!), takes the form of a vector that starts at (1, π) and – rotated another 90 degrees from the previous one – runs in the opposite direction from the first, going across the y = 0 line to the point (− (π2/ 2 − 1), π). The fourth term is a vector that runs downward, ending up below the x-axis, and so forth. Because the vectors keep rotating 90 degrees counterclockwise, and keep getting shorter because the denominator increases much faster than the numerator, the result is a polygonal spiral that converges on the point (−1, 0) (see diagram on next page).
Euler’s simple formula (according to some definitions, it is not an equation in this form, for it contains no variables) contains five of the most fundamental concepts of mathematics – zero, one, the base of the natural logarithms e, the imaginary number i, and π – as well as four operators – addition, multiplication, exponentiation, and equality – and each exactly once. It states that an irrational number multiplied by itself an imaginary number times an irrational number of times – plus one – equals exactly zero. The numbers πe, 2π, and eπ are all thought to be irrational. But eiπ picks out that special place in the architecture of numbers where rational, irrational, and imaginary numbers mix in a way that spookily ‘balances out’ to exactly zero. It has been said that all analysis is centreed here in this equation.13 Among other things, Euler’s result here demonstrated that imaginary numbers, despite Descartes’ scorn, were not on the margins, but at the very centre of mathematics. They would play a greater and greater role in mathematics – and then, with the advent of quantum mechanics in the twentieth century, in physics and engineering and any field that deals with cyclical phenomena such as waves that can be represented by complex numbers. For a complex number allows you to represent two processes such as phase and wavelength simultaneously – and a complex exponential allows you to map a straight line onto a circle in a complex plane.
Polygonal spiral, showing how the infinite series converges to −1.
It may be true, then, that Euler’s formula e iπ + 1 = 0 is but one implication, one step, in his exploration of functions. It is ‘only’ an equation, a single one of the thousands of steps in the ongoing process of scientific inquiry, a mere implication in Euler’s extended exploration of functions. And yet, some steps in an inquiry acquire, and deserve, special status. Certain expressions serve as landmarks in the vital and bustling metropolis of science, a city that is continually undergoing construction and renovation. They preserve the work of the past, orient the present, and point to the future. Theories, equipment, and people may change, but formulas and equations remain pretty much the same. They are guides for getting things done, tools for letting us design new instruments, and repositories for specialists to report and describe new discoveries. They summarize and store, anticipate and open up.
Euler’s equation, too, emblematized the way that its author had recast mathematics. Mathematics, like other sciences, does not develop along a predetermined track, but follows a historically contingent path in which each generation of
scientists inherits assumptions, techniques, and concepts from the generation before, transforms them, and passes them on in turn. Thanks to this process, we perceive the field as structured in a particular way, as having a certain ontology, with different phenomena assigned to distinct domains. Every equation implicitly refers to this inherited structure. But Euler rearranged this ontology, reorganizing it so that analysis was at the centre, with geometry and algebra as neighbourhoods. Looking backward, mathematicians may take the latest organization as self-evident – which, no doubt, is why the mathematician Carl Friedrich Gauss is said to have remarked that anyone to whom eiπ + 1 = 0 is not obvious is not a mathematician. When you are fully literate, nothing comes as a surprise. But mathematicians are made not born; in infancy they are not yet mathematicians, and have to learn it – and in such learning often experience extensive transformations and reorganizations of mathematical knowledge that they have only partially acquired. The brief formula eiπ + 1 = 0 is the most succinct expression of this process.14
There is yet one more, still deeper reason why this formula has become an icon. As Devlin once wrote of Euler’s equation, ‘Euler’s equation reaches down into the very depths of existence. It brings together mental abstractions having their origins in very different aspects of our lives, reminding us once again that things that connect and bind together are ultimately more important, more valuable, and more beautiful than things that separate.’
Devlin’s remark suggests the chief reason why an equation such as Euler’s attracts value and interest beyond the particular scientific inquiries that gave birth to it. It serves as a clear and concise example of what an equation and formula does: it shows how what seemed to be disparate and even incompatible elements (rational, irrational, and imaginary numbers) are implicated in a unity, and does so concisely, with few moving parts, so to speak. It simultaneously simplifies, organizes, and unifies. It brings what equations do out into the open. It is an equation that shows what it is to be an Equation.
Interlude
EQUATIONS AS ICONS
Journalist: Do the Russians have anything like GISMO?
Scientist (Rod Taylor): No, I’m sure they’d like it, though.
Journalist: Can you give us the equation?
Scientist: No, I’m sure they’d like that even better.
– The Glass Bottom Boat (1966)
Equations have a subtle influence on the fabric of our language and our thought far beyond science. Cloaking thoughts in mathematical dress seems to make them more authoritative, certain, precise, and eternal. Jokes, maxims, political bumper stickers, and uplifting self-help slogans are often revamped as equations: ‘Knowledge = Power’, ‘War = Killing People’, ‘Preparation + Patience = Success.’
Equations are written about humor, as the following diagnosis of the 2007 episode in which CBS Radio fired talk show host Don Imus for a racist remark: ‘White guy plus black slang equals comedy. But there’s where the equation breaks down. White guy plus black slang minus common sense equals tragedy.’1
Or consider George Orwell’s famous equations from his novel 1984, which, though obviously overtly false, point to a different kind of truth:
War = peace
Ignorance = strength
Freedom = slavery
While these have a superficial resemblance to equations of maths and science, they are really just metaphors in disguise. The ‘=’ sign in them does not mean the mathematical notion of ‘equal to’ or even equivalence. In mathematics, ‘=’ is quantitative, and means ‘is exactly the same as’, referring to the number of items in a set, or to a specific measurable amount. This is the foundation stone of the discipline. The way that knowledge is power, to take one example, is qualitative and much different, and must be addressed by broaching the philosophical complexities of the meanings of the seemingly self-evident words ‘sameness’, ‘equality’, and ‘is.’
Still, these fanciful equations are intriguing, for they exhibit the dangerous hope that other kinds of knowledge can be couched in equational terms, with neat packaging, balanced amounts, and simple units. Equations, that is, can seduce us into thinking that this is the way to think, and that other ways are inferior or even defective. A correspondent to a science magazine, after receiving an email from someone at a breakfast cereal company asking him to produce an equation for the best time to add the milk, made light of the way the public seems obsessed with finding equations for even the most trivial of actions. The letter unleashed comments from others – who had seen requests for equations for making sandwiches, parking cars, and ‘the perfect sitcom’ – warning that the practice had a dark side, not only because it was bad science, but because it encouraged irresponsible behaviour among scientists and mistaken views about the nature of science among the public.2
Specific equations, too, can have a wide range of symbolic meanings. Take 2 + 2 = 4, the slightly elder sibling of 1 + 1 = 2. In fiction and reality, it has been used to symbolize the superiority of the irrational over the rational, the rational over the irrational, and the divine over both the rational and the irrational.3 In Dostoyevsky’s novel Notes from Underground, for instance, the narrator describes it as ‘insufferable’, as a ‘piece of insolence’, as sterile and rational, as something dead and beneath bare consciousness, which the narrator finds is ‘infinitely superior to two times two makes four.’ In George Orwell’s novel 1984, on the other hand, Winston, the protagonist, uses 2 + 2 = 4 as a self-evident truth, the touchstone of sanity and rationality, available to thinking at any and every moment, the one shining light to grasp that objective reality, which assures and even guarantees for him that objective reality is there; for the Party, 2 + 2 = 4 is the final resistance that must be defeated in the way of the success of doublethink and the Party’s rule, the one outside standard that must be eradicated. Orwell, in turn, was only quoting a genuine slogan by the leaders of the Soviet Union, which used 2 + 2 = 5, written on billboards and in electric lights, as a symbol of optimism, of the ability of labour to triumph over nature, of the fact that ‘miracles could be worked through the sorcery of naked force.’4 The correct equation – dry, rational, and stale – was false, it seems, because it did not capture human creative ability, while the incorrect one was true because it symbolized the way human creative ability can overcome limitations of nature. Meanwhile, the architect and inventor Buckminster Fuller liked to define synergy with the motto ‘1 + 1 = 4’, meaning that efficient and imaginative use of parts produces more than is possible with conventional methods. Finally, the eminent Oxford theologian Marilyn McCord Adams, arguing that ‘Human nature is not created to function independently, but in omnipresent partnership with its Maker’, speaks of the ‘self-effacing Spirit’ who ‘is ever the midwife of creative insight, subtly nudging, suggesting, directing, directing our attention until we leap to the discovery that 2 + 2 = 5.’5
Novelists who have used equations in bizarre ways include Italo Calvino, whose book Cosmicomics features Einstein’s general relativity in one story. Another is Mark Leyner, whose book Et Tu, Babe includes a character who claims to have tattooed on his penis Max Planck’s energy formula E = hv, hence, something associated with radiation and power – and is humiliated to have to confess in front of a judge that it is actually d = 16t2 – Galileo’s law of falling bodies.
If equations have a dark side, it is that they can also tempt us to think that knowledge resides in the equation itself, rather than in the ongoing construction and renovation of the city of science (what Plato called more questioning). They can promote the erroneous view that science consists of a set of facts or beliefs to be memorized, rather than a quest for greater understanding to be achieved by moving beyond the facts or beliefs we already have.
5
The Scientific Equivalent of Shakespeare:
THE SECOND LAW OF THERMODYNAMICS
S’ – S ≥ 0
DESCRIPTION: The entropy of the world strives toward a maximum.
DISCOVERER: An in
ternational cast of characters
DATE: 1840s–1850s
A good many times I have been present at gatherings of people who, by the standards of the traditional culture, are thought highly educated and who have with considerable gusto been expressing their incredulity at the illiteracy of scientists. Once or twice I have been provoked and have asked the company how many of them could describe the Second Law of Thermodynamics. The response was cold: it was also negative. Yet I was asking something which is about the scientific equivalent of: Have you read a work of Shakespeare’s?
– C. P. Snow, The Two Cultures
The first two laws of thermodynamics are easy to state. Rudolf Clausius, who formulated the second one and who coined the word ‘entropy’ as a name for a measure of disorder, expressed them this way: ‘The energy of the world is constant; the entropy of the world strives toward a maximum.’ A popular formulation in simple language is: ‘You can’t win. You can’t break even, either.’ Max Planck’s symbolic formulation of the second law, with S the entropy at an earlier time and S’ the entropy at a later time, is given above. This law is essential to the activities of the world around us. If you do not understand this, you can have little understanding of how the world works. This was surely C. P. Snow’s motivation in saying that asking if someone can describe the second law of thermodynamics is like asking, ‘Have you read a work of Shakespeare’s?’ It should be equally shameful for people who think themselves cultured to have to answer no to either question.
A Brief Guide to the Great Equations Page 10
