For any e-curious person, some of the basic questions that one might have are
Who came up with this constant ?
Where did it first appear ?
Why is this number so important ?
Why should one make a function out of this constant ( exp(x) ) ?
What is its relation to complex numbers ?
What is its connection with hyperbola , as it appears in coshx , sinhx etc ?
This book is a collection of stories about various developments around e. However there is a common thread which runs across all the stories. Each of the story has in its essence, a development which lead to the world we are living in, where we take e for granted in most of the applications we deal with.
It all began with Napier’s 20 year effort. John Napier worked on a single idea for 20 years. What was his idea ? Multiplication and Division of large numbers are difficult operations than addition and subtraction. If numbers could be represented as an exponent of some number, then multiplication will turn in to addition and division would turn in to subtraction. Napier thought of a base and created elaborate tables where by you can multiply and divide by looking up relevant numbers in the tables and performing addition and subtraction instead. Henry Briggs, a geometry professor from Gresham college suggested a few improvements like having a base of 10. Thus the concept of logarithm was born.
What’s logarithm got to do with e ?
Finding area under the curve y = 1/x was a problem which racked quite a few mathematicians. Fermat had developed a way to find out the area under the curve y = x^n , but for n=-1 , he said it did not work. The crucial breakthrough took place when Anton de Sarasa, a Jesuit mathematician noted that area under the curve y = 1/x behaved like logarithms. For lengths which grow geometrically on the x axis, the area grew arithmetically. Thus he was able to connect Napier logarithms with the behavior of area under 1/x and thus was born, logarithmic function.
Subsequent to this, there were developments in the area of calculus that brought e closer to discovery. One of the fundamental theorems of calculus states that rate of change of area function with respect to t is equal , at every point x=t, to the value of the original function at that point.
Simply put, for y = 1/x kind of situation , d(A)/d(t) = -1/t …It was already established that A was log(t) though base was something which was not yet fixed. base of the logarithm could have been anything. If we let the base be some number b, then we see that the equation turns in to a problem where we need to find the base such that derivative of a function is the function itself. Thus the base of logarithm e was born. ln(x) and exp(x) thus become known to be inverse functions of each other.
So, as one can see that exp(x) is closely related to hyperbola ( y = 1/x) and this spawned a new set of trigonometric entities called sinhx, coshx, tanhx etc. These are the analogs of sinx , cosx, tanx relevant to the circle. Thus e became to Hyperbola what pi became to a circle.
The author subsequently takes the reader through multiple facets of e, like the use of logarithm/e in representation on the music scale, its presence in the eulers formula , its presence in the nature ( sunflower spiral, nautilus shell ) . its presence in ancient and modern architecture etc
e was first noticed in continuous compounding of money. The expression (1+1/n)^n as n tends to infinity was known to converge to a number 2.7182.. But it took a century for the math to develop that so called number 2.7182 in to a tractable exponential function, e^x and finally stamping the property of transcendental on e.
Today , one cannot think of doing math/fin with out exp(x) function , for this makes a whole set of problems analytically tractable. This book makes one pause and look at the massive efforts of various mathematicians / scientists who contributed to the understanding of e.