# The number e is arbitrary

## The child prodigy of the world of numbers

Three hundred years ago one of the greatest scientists in Switzerland was born: the mathematician Leonhard Euler. A special number is named after him: e or 2.718 281 828 459 045 235 360 287 471 352 662 497 757 247 093 699 959 574 966 967 627 724 076 630 353 547 594 571 ...

From Roland Fischer

Can you tell the story of a number? The story of a mathematician, the story of a proposition and the struggle for its proof: there would be nothing in it. The story of the numbers can also be told. But the story of a number? Wasn't it already there as soon as the number line was opened, which completely covers everything that romps between minus and plus infinite?

There are, however, numbers that suddenly appear on the map like a new continent and that dramatically change geography in the realm of numbers. The zero and the number pi are part of it. Until they are discovered, they are hidden like an unrecorded island, not white spots waiting to be explored, but lonely elevations in the flood of numbers, of whose existence nobody suspects anything. Sometimes the mathematician comes across a piece of such land during his intellectual expeditions, and sometimes the coastal stretches of supposedly small islands turn out to be parts of a large continent.

To stay with the analogy, one could designate Pi as the old world and e as the new (although there is not a whole Atlantic between the two, they are rather closely spaced: is a little larger than 3, e a little smaller). e is the America of mathematics, and Leonhard Euler from Basel made sure that his discovery marked a turning point: He helped to measure the new continent.

To properly appreciate the discovery of numbers in such a format, some number theory is needed. The number line is staked out from the whole numbers. Between these pegs there are also all fractions, the rational numbers (from the Latin ratio: ratio). Since there are an infinite number of integers, there are of course an infinite number of rational numbers. And between two rational numbers that are arbitrarily close to each other (such as a ninety-ninth and a hundredth) there is room for an infinite number of others. The rational numbers are "close" to the number line, as the mathematician puts it, and the Greeks already knew that. That is why they believed that there was nothing else left on this straight line, i.e. that every number had to be able to be represented rationally by a fraction. There were some among them who claimed to have found irrational numbers, but the Greeks were never quite ready to accept that cracks were beginning to appear in one of the pillars of their harmonious world. So they simply helped themselves by seeing the root of 2, for example, as a geometrical fact that could be inferred constructively, but not as a number.

Today we know that the rational numbers still leave any number of holes on the number line for irrational numbers. E was also hidden in a hole until the number suddenly rolled onto the desk of a bank clerk in perhaps Venice in the seventeenth century, it can no longer be traced that precisely. It first appeared in a thoroughly prosaic field, namely that of calculating compound interest. There is the question of whether it is fair to the investor to keep the interest to be paid under lock and key for the whole year and only take it out for the next round. You could also put him on the account monthly (then he would work for you for the rest of the year) or daily, or even every second, or, in the mathematical borderline case, continuously. When the theoretically versed banker expressed this borderline case mathematically, he came across the famous formula that would later serve as the definition for e. And at the same time, this origin also provides the most handy, albeit still awkward, illustration for Euler's number: If a capital of one franc is invested at an interest rate of one hundred percent and the interest is continuously offset, then one receives one at the end of the year Capital of e francs (instead of two francs as with annual interest).

Such expressions, which tend towards a limit value, are often good for surprises. Mathematically speaking, e is equal to the limit value of (1 + 1 / n) to the power of n, when n approaches infinity. One might be tempted to put the brackets equal to 1 for an infinitely large n, because the limit of 1 / n in this case tends to 0, so that the whole expression 1 to the power of n would be equal to 1. But one could just as well assume that there is always a value greater than 1 in brackets and that every number that is greater than 1 grows to infinity if it is further potentiated forever. But you can also take the calculator to hand and put it to the test, and you will find that the truth lies between the extremes; namely, the result tends slowly, very slowly from 2 (for n = 1) to the rather awkward value of 2.71828 ...

But it was not financial reasons that gave e her exceptional position in mathematics. The compound interest problem is little more than a historical marginal note. It became much more important in a field that was born at the end of the 17th century and was to revolutionize mathematics: differential calculus. In particular, the mathematical formulation of physics would be impossible without differential concepts. Describing quantities is just as important in physics as describing the changes to which these quantities are subject. You don't just want to know how far it is from Bern to Zurich, you also want to know how quickly the change of location takes place and what the speed results from. And the acceleration in turn describes the change in speed.

There are all sorts of examples in physics where a quantity and its change are directly linked. The radioactive decay (the more the material emits, the more it decays, i.e. the faster the amount changes) and the population development (the more people, the faster their number grows, at least if the birth and death rates remain the same) are known. . All these phenomena, in which the quantity is closely linked to its change, are described by exponential equations, and the mother of all exponential equations is the exponential function e to the power of x. The exponential function is the only function that matches its derivative.

And that is the reason why the e jumps towards you in almost every physical problem: e-functions form basic solutions for the mathematical description of all possible natural processes. Everything that somehow vibrates, rocks or otherwise acts back on itself is a case for e to the power of x.

To a certain extent, e is the neuter of the differential world; The 0 for addition and 1 for multiplication play a similar role. But evidently there is much more to e than just the most natural, pure basis of an exponential function. Accordingly, Pi has many other meanings than the original geometric one (namely the ratio of circumference and diameter). e dances at all kinds of weddings. A nice example is the appearance of e in an expression that describes the so-called prime number density (the number of prime numbers in a certain interval). The more numbers you look at, the less prime numbers appear. But if you put the prime number density up to the number N in relation to the natural logarithm (with the base e), then this value tends to the limit value 1 with increasing N. In any case, patient counting has shown that no one has yet derived this limit value exactly. Mathematicians are amazed that this value also contains the number e, because prime numbers actually belong to the whole-numbered realm, and e is one of the half-numbered numbers that exist: Mathematicians call numbers of this genus their properties that are difficult to grasp with classical means because of «transcendent».

Leonhard Euler was a mathematical giant, in many places he is considered the greatest mathematician of all time - one author calls him the "Mozart of mathematics". In this respect, it is quite understandable that e now bears his name. However, if you look up the keyword "Euler" in a mathematical encyclopedia, then it is almost lost in a myriad of entries: there are Euler's formulas, Euler's theorems, Euler's equations, and there are also Euler's numbers (some with the magistral e have nothing to do). All of these contributions clearly go back to Euler and make its discoverer immortal in a similar way to the entomologist who lives on in the name of a new species. With e, however, things are more complicated: Euler by no means discovered the number, nor was he the only one who recognized its meaning. The only thing that is undisputed is that the designation "e" goes back to Euler, as does the "i" for the imaginary unit or the Greek "_" for the name of a sum.

And Euler ennobled e in another way as well: with the number he created what is probably the most beautiful mathematical equation: e to the power of iPi + 1 = 0. You don't need to go into the practical meaning of the equation in detail (which actually denotes differential calculus complex space and thus opened up a host of new applications) to get an idea of the enthusiasm of mathematicians for this formula. It connects the five most important constants of mathematics (the three units 0, 1 and i [the square root of -1], plus pi and e) in the simplest way, as well as the three most important mathematical operations: addition, multiplication and power . And she puts all of that together with a simple equal sign.

Much has been written about why Euler, after whom e is still called Euler's number, probably chose the letter e. It could hardly have been vanity, the reference to the exponential function is much more likely. But who has done Euler the honor of erecting a mathematical monument by naming him e has not yet been clarified by the mathematical historians. There is no doubt that the person, whoever it was, rightly chose the name.

#### 300 years of Euler

Leonhard Euler is considered to be one of the greatest scientists that Switzerland has produced. April 15th marks the 300th anniversary of his birthday. The anniversary is celebrated extensively in Basel, his birthplace (there are also events in St. Petersburg and Berlin, where Euler worked). In the university library there is an exhibition on “Leonhard Euler and the delights of science”, plus events throughout the city, from the highly endowed symposium to city tours in Euler's footsteps to the film series on science and ingenuity.

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