e: The story of a Number
(appeared in Nov 2015)

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Tucked away in the continuum of numbers, there is one little number that peeps out and links to almost all the ideas of mathematics, says S.Ananthanarayanan.

This number, denoted by the letter, ‘e’, has the unlikely value, and one that can never be exactly determined, of approximately 2.7182818284…., and this number turns out to dominate the world of shapes angles, or growth and rates of change, even the progress of numbers themselves.

Israel born Eli Maor, author, historian and teacher of mathematics, in his book, “e: The Story of a Number”, conducts the non-specialist reader through the fascinating course of discovery and the myriad properties of ‘e’, and along the way, many of the concepts of modern mathematics. In some 200 racy pages, Maor tells the story in simple language, filled with examples and anecdotes to make the reading easy as well as rewarding, to the newcomer and equally to the specialist. A new edition of this book has just been brought out by the Princeton University Press.

The number, e, arises in mathematics as a result of an infinite process, where a number adds to itself other numbers which grow progressively smaller, in such a way that the total tends to reach, but never gets to some final number. We could illustrate infinite processes by looking at an athlete running a course in a certain time, say a minute. Now to run half the course, she takes half a minute, and for the half the remaining half, she takes a quarter of a minute, and for half of what remains, an eighth of a minute, and so on. We can see that here we have a series: ½ + Ό +1/8 + 1/16 + 1/32 + 1/64 +……., which will go on to infinite terms, as we can always divide any number by two, but the total will not grow beyond one minute, which is the time the athlete takes to run the course. In this case, we also know that the final total will be exactly one minute, because of the way we have stated what was happening.

But there could be a different case where a thing only grows by additions that rapidly reduce in size, to infinite terms, and we need to know how large it gets to be after a certain time. In such a case, the final stage, whatever the size be, may not ever be exactly known, as there is always some tiny addition in the tiny bit of time that is left. An example would be a sum of money that is invested at some rate of interest, to double after some given period of time. As the amount doubles in the period, it is clear that it would grow four times in twice the time and only by half in half the time. When a unit sum of money grows by one more unit, we could depict this by the expression: “money grows to (1+1) = 2”, in one spell of the given time. The case of two spells would be shown as : “money grows to (1+1)x((1+1) = 2x2 = 4”. In the same way, the growth in any number of spells of time, say a number denoted by ‘n’, would be: (1+1) x (1+1) x (1+1) x (1+1)….. n times. This amounts to (1+1) raised to the power of n, and is written as : (1+1)n.

Now let us think of another idea, of the interest during the first spell of time being reckoned not at the end of the period, but half way through and again at the end. In such a case, the amount grows to only (1+1/2) in the first half spell, and it is (1+1/2) that grows by half of that amount in the second half spell. The final amount is thus (1+1/2) x (1+1/2), which can be written as (1+1/2)2, and this comes to , or 2.25. Now, we could repeat this process, by adding interest of one fourth the amount every one fourth of the period, which would make the final amount (1+1/4)4, which works out to 2.3707, a little more than when we split the period into two parts. And then by dividing the period into a great many number of parts, say ‘n’ parts, the final number would be (1+1/n)n, which would add a great many, but rapidly reducing amounts to the original total amount of ‘2’. The value of the total, in fact, can be expressed as the series:

1+1/1 +1/2+1/6+1/24+1/120+….,

This number, which can never, even in principle, be exactly evaluated, was found to have so many remarkable properties, that it became a subject of deep study, great advances having been made by the Swiss, Leonhard Euler, who first gave the number the name, ‘e’. The number, in fact, is sometimes referred to as Euler’s number

Logarithms and the calculus

Eli Maor gets the book going with a description, first, of a closely related idea, of logarithms. In the 16th century, the Scotsman, John Napier, noticed that a multiplication, like 4 x 16 = 64, could also be written as 22 x 24 = 26. When we write the numbers in this way, as powers of the number, 2, we notice that the multiplication reduces to simply adding the powers to which the number 2 has been raised, in this case, 2 + 4 = 6. As numbers can also be raised to fractional powers, we could work out a table that shows us the power to which 2 has to be raised to come to any number. With the help of such a table, we could carry out complex multiplications by looking up the power of 2 for each number and then just adding the powers. A reverse table could then tell us the answer, from the total that we get. This is the idea of logarithms, or using tables of powers of some base to carry out multiplication with the lesser labour of addition. Napier’s invention soon stabilised with tables based on powers of the number 10, which is easier in the real world than the number, 2 as the base, and was surely a great help in the progress in science that followed.

Another development of the period was the discovery of the methods of the calculus, independently by Isaac Newton and Gottfried Wilhelm Leibniz. The calculus was first conceived by the Greeks, Archimedes having sown the seeds of making computations by breaking a progression into infinitely small steps. The greater sophistication available in the 17th century allowed more precise definition of relations among variable quantities, like length and breadth or distance and time, and this led the formal and powerful methods of the calculus. One of the concepts in the calculus is the rate at which a value changes and the methods of Newton and Leibniz help work with curved shapes and areas and volumes, forces and motion, growth and decay.

Eli Maor’s easy narrative painlessly introduces the role of the number, e, in all these fields, like the properties of logarithms to the base e, the way the number e behaves when it is raised to powers of other numbers and the remarkable property that the value of e raised to a power is the same as its own rate of change! As logarithms to the base e would involve expressions of e raised to different powers, like 25or 104, this property of an expression like ex, gains great consequence, and is found to play a role widely separated fields like the values of trigonometric (of the study of triangles) ratios, the shapes of suspension bridges or the shapes of seashells, galaxies, in art and even musical scales. Even the distribution of prime numbers, in the sequence of integers, is found to be related to the logarithm, to the base e, of large numbers!

While the treatment of the math is captivating, equally so are the stories about mathematicians. The rivalry of Newton and Leibniz over credit for discovering the calculus, lives of the Bernoullis and the composers Bach, an imaginary meeting of Johann Bernoulli and J S Bach, and the references to different contributors to the growth of the subject present mathematics as a pulsating, living thing.

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