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 mathssavvy
employees. Giant billboard adverts appeared in Silicon Valley and other
intellectual hotspots across the US sporting the cryptic message:
"{first 10digit prime found in consecutive digits of e}.com"
Finding the first 10digit 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.
7427466391.com 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 aweinspiring 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.
A HISTORY OF e 

YEAR  HISTORY/CALCULATIONS BY 
DECIMAL PLACES 
2007  Shigeru Kondo calculates e to 100 billion decimal places  100,00,000,000 
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  10,000,000 
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  2010 
1873  Jacob Bernoulli's work on compound inetrest estimates a value of e for the first time  
1853  Charles Hermite proves e is transcendental  137 
1748  William Shanks calculates e to 137 decimal places  18 
1731  Leonard Euler introduces the notion of e  
1690  Gottfried Leibniz discovers e as a number in its own right,but calls it b  
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 nontranscendentals.
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 4^{3 }is 4 multiplied by itself 3 times: 4x4x4.But for
noninteger 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 e^{x} , where x doesn't have to be an
integer. He then showed how to write every number a^{b }as
e^{x} 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
e^{x}, 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, e^{x} and many, other numbers. It would subsume Baker's prizewinning 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 nontranscendental numbers a and b, then a^{b} 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 e^{ip} +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 "Gentleman...is 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 pseudoexponentiation. Remarkably,not only has he proved that pseudoexponentiation exists,and that it satisfies Schanuel's conjecture,he has also shown that it is unique there is only one pseudoexponentiation.
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 pseudoexponentiation 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 doughnutshaped 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 pseudoexponentiation 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 pseudoexponentiation 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 www.newscientist.com/channel/fundamentals
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