From e to eternity

First it revolutionised mathematics, now it's taking on the mysteries of the quantum universe. Raise your glasses to the numbers, says Richard Elwes

WHEN it comes to numbers, some are more enigmatic than others. Just ask Google's chief executive Eric Schmidt. In 2004 the company announced it was aiming to raise $2,718,218,828 from the first sale of its shares. While the exactness of the figure left many perplexed, mathemacians nodded knowingly. They recognised the figure as one of the most important numbers in mathematics, the number known as e expressed in billions of dollars.
A few months later Google was at it again - this time in the hunt for maths-savvy employees. Giant billboard adverts appeared in Silicon Valley and other intellectual hotspots across the US sporting the cryptic message:

"{first 10-digit prime found in consecutive digits of e}.com"

Finding the first 10-digit prime number buried in e's endless stream of figures is no mean feat, not least because the answer, 7427466391,starts at the 101st digit. For those who figured it out. threw up an even more fiendish mathematical puzzle. Crack it and you were invited to submit your Cv to the company's research laboratory.
Google is right to be fascinated by e. From computer science to statistics,this number is everywhere in the mathematical sciences. Along with p,e has transformed our understanding of the very concept of number. Far from being invented by mathematicians, both numbers exist in their own right and crop up throughout the natural world. We have discovered that the number e plays a key role in describing how populations reproduce and grow, and how radioactive decay progresses. But much remains mysterious about this most enigmatic of numbers, and the latest efforts to unearth more of its secrets are having repercussions from mathematical logic to quantum physics.
At approximately 2.718281828, the significance of e and even its definition depends on who you ask, though one simple definition is the limit of (1+ 1/n)n as n tends to infinity. The number first surfaced 1618 through the work of British mathematicians John Napier and William Oughtred on "slide rules", convenient devices for multiplying large numbers in the days before calculators. In 1683 Swiss mathentatician Jacob Bernoulli rediscovered e by studying how bank accounts grow as interest is added year after year.But it was the work of Swiss genius Leonhard Euler in the 18th century that really placed e at the centre of the mathematical universe.
Euler was one of the pioneers of mathematical analysis and it is insights have set the direction of the subject ever since. e features heavily in his work. Most strikingly, in one awe-inspiring equation Euler tied e to the four other fundamental numerical entities 0,1, p and  i, the square root of -1 (see "Eulers identity". ) Since Euler's work, it has been impossible todo mathematics without encountering e at every turn. Even so, agreat deal about it remains unexplained, and for many pure mathematicians today, e is the focus of one of the most perplexing phenomena in the science of numbers.




2007 Shigeru Kondo calculates e to 100 billion decimal places 


2005 Boris Zilber discopvers a function that appears to satisfy Schanuel's conjecture about e and transcendance
1994 Robert Hermiroff and Jerry Barnell calculate  e to 10 million places


1966 Stephen Schanuel formulates his conjecture about e and transcendance
1949 John von Neumann calculates e to 2010 decimal places on one of the first electronic computers


1873 Jacob Bernoulli's work on compound inetrest estimates a value of e for the first time
1853 Charles Hermite proves e is transcendental


1748 William Shanks calculates e to 137 decimal places


1731 Leonard Euler introduces the notion of e
1690 Gottfried Leibniz discovers e as a number in its own right,but calls it
1683 Jacob Bernoulli's work on compound interest estimates a value of e for the first time
1618 e appears indirectly in a mathematics paper by John Napier and William Oughtred

e and p are both examples of transcendental numbers, a type of number whose baffling complexity is the very antithesis of the plain everyday integers 0,1, 2,3,4 and so on. Whereas the integers are comparatively easy for humans to understand, manipulateand program into computers,transcendental numbersare infinitely harder to pin down.
The first proven example of these was discovered in France in 1844 by Joseph Liouville, though the idea has its roots in Euler's work Liouville found a number that cannot be written as a fraction and is entirely unrelated to the integers by any sequence of ordinary arithmetical operations. Starting with Liouville's number, you can multiply it by itself as many times as you wish, combine these powers and divide and multiply by integers in whatever complicated fashion you want, but you will never arrive back in the familiar territory of the integers. This is the definition of a transcendental number.

For centuries, no one even suspected that such strange objects might exist. The ancient Greeks believed that all numbers could be derived from the integers by simple division. According to legend, in around 500 BC when Hippasus of Metapontum proved that some numbers, such as Ö2, couldn't be written as a fraction of integers, his fellow Pythagoreans were so outraged that they had him drowned for heresy.
But even numbers like Ö2 are tame compared with transcendentals. By definition Ö2 x Ö2 = 2,so we get back to the integers after just one step. Objects like Liouville's number had never been imagined, and came as a shock to those who saw the integers as the bedrock of the mathematical world.
If anyone saw Liouville's numbers as nothing more than a curiosity, what happened next would have convinced them beyond doubt of the importance of transcendental numbers. In 1873 another French mathematician, Charles Hermite, proved that e is transcendental too. Coming as it did 100 years after Euler had established the significance of e, this meant that the issue of transcendence was one mathematicians could not afford to ignore

Within 10 years of Hermite's breakthrough, his techniques bad been extended and used to add p  to the list of known transcendental numbers. People then tried to prove that other numbers such as e + p are trarscendental too,but these questions were too difficult and so no further examples emerged.
Hoever while others were struggling to extend this very short list of bizarre numbers,the German logician Georg Cantor delivered a thunderbolt,he showed that far from being a handful of exotic anomalies in fact almost all numbers are transcendental . That is, they infinitely outnumber the non-transcendentals.
The consequences of Cantor's work are profound. It means that the range of numbers that human brains and computers are equipped to handle- essentially those easily derived from the integers - are actually just an infinitesimal sliver of the numerical universe. Swarming around the integers and fractions is an infinitely larger collection of transcendental numbers They are the 'dark matter' of mathematics: they constitute the overwhelming majority of numbers, yet known examples are rare.

Following Cantor's revelation, the challenge was not merely to find more examples, but to account for the whole mass of transcendental numbers known to exist. It quickly became clear that the key to this problem was e and its connection to a mathematical operation called exponentiation.Like addition, subtraction, multiplication and division, exponentiation is a fundamentakl way to combine numbers.

It is easy to define for integers:just as 4x3 is 4 added to itself 3 times (4+4+4),so 43 is 4 multiplied by itself 3 times: 4x4x4.But for non-integer numbers thsi definiion of exponentiation does not work: what might it mean to multiply something by itself p times for example?
The fact that exponentiation can be extended to all numbers is one of the cornerstones of mathematics and it revolves around e. Euler found away to define ex , where x doesn't have to be an integer. He then showed how to write every number ab as ex and provided an easy formula for finding x in terms of a and b.
Exponentiation is the primary obstacle to our understanding of transcendence. For the most part, it is easy to understand what happens to transcendental numbers under additions, subtraction, multiplication and division. You can usually tell if the result will be transcendental or not. But exponentiation is a far tougher problem.

During the 1960s, University of Cambridge mathematician Alan Baker partially addressed this issue. By studying e, be discovered powerful ways to use expontiations to understand whole families of transcendental numbers. In 1970 he won a prestigious Fields medal for the inroads he made into the transcendental wilderness. Since then, however, progress has been slow.
Even after Baker's work, we stilt do not know if e + p. is transcendental. At first sight this may seem to be a question about addition rather than exponenatiation, but the problem is a gap in our understanding of the relationship between e and p and the pivotal issue is actually exponenatiation. The same is true for e x p ex, and reams of other seemingly simple numbers. Most mathematicians believe that all these should be transcendental,but after 130 years of trying, proofs still remain elusive. David Masser, a number theorist at the University of Basel inSwitzerland has described the prospects of a proof as "hopeless", The challenge is to reconcile the numerical landscape we would like to see with the few glimpses we've managed to catch. Even today the main obstacle is still e.

That's not to say there isn't a road map, however. In the early 1960s, number theorist Stephen Schanuel at the University of Buffalo in New York made a huge, sweeping conjecture about e and transcendance. Proving Schanuel's conjecture would immediately settle the matters of e + p ,e x p, ex and many, other numbers. It would subsume Baker's prize-winning work and indeed almost every other known fact about exponentiation and transcendence. It would solve literally hundreds of major open questions in the subject at a stroke, making it the holy grail of transcendental number theory.

The statement of Schanuel's conlecture is technical but it basically says that the interplay between e and transcendence is as simple and streamlined as could posslbly be hoped. We already know some basic conditions that are satisfied when exponentiation and transcendance interact. For instance, in the 1930s the Russian and German mathematicians Alexandr Gelfond and Theodor Schneider independently showed that whenever you have two non-transcendental numbers a and b, then ab is transcendental as long a is not 0 or 1 and b is not fraction of integers. Schanuel's conjecture says that there are no surprises in store, so proving it would quickly allow us to tell if specific numbers such as e + p are transcendental too.

Euler's identity
In the 18th century,the great Swiss mathematician Leonard Euler prove the formula eip +1=0. Subsequently this formula became central to our understanding of number and exponentiation,and is celebrated for the beautiful way it unites the five fundamental constants of mathematics. After demonstrating a proof of this equation in a lecture,the 19th century American mathematician Benjamin Pierce is reputed to have told the audience " absolutely paradoxical; we cannot understand it, and we don't know what it means. But we have proved it, and therefore we know it is the truth". The nobel prizewinning physicist Richard Feynmann has described it as "the most remarkable formula in mathematics".

Surprisingly,Schanuel's conjecture hardly merits a mention in most textbooks about e and transcendental numbers. For a long time it just looked too big and too wildly optimistic a claim.What's more, a proof seemed too distant from our current state of knowledge to be remotely approachable. So many people were sceptical,and even those who believed in Schanuel's conjecture thought that a proof was several generations away,at least. "It is generally regarded as impossibly difficult to prove," says Masser.

Enter Boris Zilber at the University of Oxford. He made a breakthrough in 2004 that has brought the possibility of a solution tantalisingly close. Since then,several mathematicians have made substantial progress towards it,an the race to come up with a comlpete proof is very much on.

Zilber applied techniques from a branch of mathematical logic called model theory to Schanuel's conjecture. In  a ground breaking paper published in 2005,he announced an astonishing discovery: he had found an object in the numerical world that behaves exactly as Schanuel's conjecture predicts exponentiation does (Annals of Pure and Applied Logic,vol 132,p67).

This object is not a number as such,but something more abstract: a function,a rule for generating new numbers from given ones. It looks a lot like ordinary exponentiation and indeed Zilber has named it pseudo-exponentiation. Remarkably,not only has he proved that pseudo-exponentiation exists,and that it satisfies Schanuel's conjecture,he has also shown that it is unique- there is only one pseudo-exponentiation.

It's hard to escape the conclusion that this unique object,which looks like exponentiation and satisfies Schanuel's conjecture, must in fact be exponentiation. That is certainly what Zilber believes. Many mathmaticians agree,and a few have even argued that Zilber's conclusion is so glaring that his work should be considered a proof of Schanuel's conjecture. Any outstanding issues are no more than philosophical niceties.

Certainly the alternative seems doubly unlikely;that Schanuel's conjecture is false and yet there is this hitherto unknown ghostly function that satisfies it.
If Zilber's supposition is correct and pseudo-exponentiation really is about e,then the truth of Schanuel's conjecture and everything it implies follows too. What's more,proof would have consequences well beyond the realm of transcendental numbers. One striking aspect is its application to the area of quantum geometry,the theoretical framework that underpins many attempts to reconcile the disparate worlds of quantum mechanics and Einstein's general theory of relativity into a single theory of quantum gravity.

In the 1980s and 1990s,Fields medallist Alain Connes introduced a range of new geometric objects designed to put quantum physics on a firm mathematical foundation. One of the most important examples is the "quantum torus",an abstract, quantum version of the traditional doughnut-shaped torus.While a classical torus is easy to visualise,it is impossible to picture a quantum torus in the same way. This is because quantum geometry replaces the traditional notions of shape,area,curvature and so on with a more abstract concept of mathematical "space". All the same,the quantum torus is fundamental to ongoing efforts to model the quantum universe.

Connes's insights are deep,but his avant garde mathematics defies all usual geometric intution and is disconcertingly difficult to get to grips with. However,Zilber's work could help demystify Connes's extreme levels of abstraction. From his own insights into e, Zilber has proved that if Schanuel's conjecture is true then the quantum torus is what is known as a "stable structure". Model theorists have been studying stable structures intensively for 30 years and have developed an impressive armoury of methods to analyse them. So the stability of the quantum torus would open up a raft of techniques for understanding Conne's abstract geometry ina  more intuitive way.

It seems that the more we study e,the more important it reveals itself to be. Certainly there is now more than ever hanging on Schanuel's conjecture . A proof would not only herald a new era in our understanding of the numerical universe,but by opening a doorway between the seperate mathematical worlds of logic and quantum geometry,it would also provide much needed insight into some of the hardest questions about our physical universe. All we need is for someone to complete Zilber's attack on Schanuel's conjecture by showing that his pseudo-exponentiation is nothing more than another appearance by that most ubiquitous of numbers e.

Eric Schmidt at Google may be taking note. If mysterious messages about pseudo-exponentiation start appearing on giant billboards near you,you know what to do. Brush up your CV and start memorising e.

Richard Elwes is a mathematician an writer based in Leeds,UK





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