[Maths is interesting series] What is e?
What is e? Isn’t it just the fifth alphabet of the English language? True, but today I would like to talk about the letter (constant) e in the other ‘language’ — mathematics, which is known as the Euler’s number*.
*Not to confuse with Euler-Mascheroni constant γ, the other mathematical constant which is also named after the famous Swiss mathematician and physicist Leonhard Euler
If you are old enough, you have probably come across this ‘mathematical letter’ e when you study mathematics at high school. But how much do you really know about this constant apart from the simple fact that it is an irrational number that has an infinite number of digits to the right of the decimal point and of value approximately 2.7182…? For the case of π, you probably know it has geometrical interpretations and is directly related to circles. But how about e? For what reasons mathematicians introduce e to represent such irrational number in the first place? I will try answer these questions and explain more in this article.
A brief history of e
To begin with, let me first talk about the ‘biography’ of e. Though named after Euler, e this mathematical constant was actually first ‘mentioned’ by the other mathematician John Napier. In the appendix of his work published in 1618, there appeared to be a table giving the natural logarithms of various numbers. Why I said “mentioned”? Because the work did not contain or mention the constant itself and people thought of logarithms very differently at that time, i.e. the modern concept that logarithms are the exponents to which another fixed number, namely the base, that one must raise to get the required number was not yet developed. Moreover, it is assumed that this table was written by another mathematician William Oughtred.
Although there were other references and occurrences of e in various mathematics literature in the next few decades, the discovery of the constant e itself is usually credited to the Swiss mathematician Jacob Bernoulli in 1683. At that year, he accidentally discovered the constant by trying to work out how much money was left in his bank saving account (LOL). Of course this is a just a joke that I made up, but Bernoulli did discover the number e by looking at the problem of continuous compound interest and was the first one to define the number by a limiting process. In today’s mathematics, Euler’s number e is usually thought of to be related to limits and infinity and is defined as
This expression was first written down also by Bernoulli. He did not realise there is any connection between his work and the work on logarithms in the beginning though.
Then in the 17th century, Euler did a lot of work on e and made various discoveries regarding this irrational number, including expressing e in terms of an infinite series
representing complex number using e and the famous, most beautiful Euler’s equation, also known as the Euler’s identity (see Fig. below).
For convenience, Euler started to the letter e for this constant and was the first one to do so$. The first appearance of e in a publication was in Euler’s Mechanica (1736). From that point onwards, the use of letter e to represent the constant became more and more common and eventually became standard. Because of Euler’s work and his choice of the symbol e for the constant, e is often called the Euler’s number in the honour of this great mathematician of the 18th century. (though it seems not many honour and appreciate Euler’s work in today modern world, see below {LOL, JUST A JOKE}). You can check out [1] if you would like to find out more about the history of e.
$ The first time the number e appears in its own right is actually around 1690, in correspondence from Gottfried Leibniz to Christiaan Huygens. The letter b was used there instead though.
So what is e and what is it useful for?
Now let’s talk about the maths of e. Unlike some other ‘mathematical letters’ such as π or the golden ratio φ, Euler’s number e is actually quite special in the sense that it does not have direct and trivial geometrical interpretations#. This number is usually thought of and defined in connection to some rate of change processes, in particular the classical compound interest problem as mentioned above.
# There are some not so trivial geometrical interpretations if you wish, see [2] for example.
You should have probably come across the concept of interest rate and have an idea of what is compound interest at schools. In very simple words, compound interest just means “interest of interest”, i.e. adding the interest to the principal sum of a loan or deposit. For example, assuming a bank offers an annual interest rate of 100% (very unrealistically generous bank), then in the case of simple interest, you will get a total of $2 (principal sum + interest) after 1 year if you save $1 in the bank deposit account. However, in the case of compound interest, suppose the same principal sum ($1) is now compounded every half year, then after 1 year you will get (1+1.0/2)2=$2.25, which is a little bit more.
Wait, so how compound interest is related to the Euler’s number e? Well, if you are smart enough, you may already figure it out by looking at the definition of e mentioned in the previous section. Let’s imagine the extreme compounding case where the compounding period becomes smaller and smaller, from half year to one month or one day, or even becoming infinitesimally small. In this limiting case, which is known as continuous compounding, what is the total sum you will get given the same overall time period, say 1 year?
Infinity? No, the answer is in fact given in terms of e. Though the compounding frequency going to infinity in continuous compounding, the interest rate over such infinitesimal compounding period is becoming infinitesimally small at the same time. The latter cancels out the former to some extent and therefore the resulting final interest you compute is actually finite in the end, see the Fig. below (for the annual interest rate of 100% case).
In general, the total amount after some time t of continuous compounding is given by
where P_0 is the initial amount and r is the annual interest rate.
You can check out the following video by Numberphile for a more detailed discussion of e.
Euler’s number is indeed an very important number in mathematics. Apart from calculating compound interest, e also has many other applications and appears in quite a lot of different fields in mathematics, such as in statistics and calculus. For instance, signal processing in engineering, calculating the radioactivity of some unstable nuclei in physics and estimating the number of patients arriving in an emergency room between some time period all involves e. So e is indeed very usual and appears everywhere in everyday life!
Finally, let me end this article in a very geeky way by the “sound” of Euler’s number as follows XD
Reference and further readings:
[1] http://www-history.mcs.st-and.ac.uk/HistTopics/e.html (article about the history of e)
[2] “e in geometry” by Waqar Ahmad, https://geomathry.wordpress.com/2017/01/29/e-in-geometry/
[3] “What is the Number e?” by Math Centre, https://www.youtube.com/watch?v=R0oUeLQIbIk
[4] “The number e is everywhere” by Randell Heyman, https://www.youtube.com/watch?v=b-MZumdfbt8
[5] https://en.wikipedia.org/wiki/E_(mathematical_constant) (wiki)
The original version of the article in Chinese:
https://medium.com/@godfrey.leung.cosmo/有趣數學系列-甚麼是e-8e8ca4831743
