Why Do We Really Use Euler's Number For Growth? An image that asks "Why e^(rt) and not 2^(rt)?

Euler’s number is synonymous with natural growth and decay. Whenever we talk about phenomena such as population growth or investment returns, the associated mathematical formulations involve ‘e’ — Euler’s number.

Back when I first encountered this phenomenon in high school, I must admit that I just took it for granted. But over the years, if there’s one thing that I have learned, it is to question things. With the usage of Euler’s number, a simple series of questions started popping up in my head:

“WHY do we use ‘e’?

Why does it HAVE to be a constant that is equal to ‘2.718281828459045…’?

Why can’t it be just ANY other constant such as ‘2’ or ‘3’ or ‘10’?”

Well, it turns out that there is a wealth of significant reasons that justify our usage of Euler’s number. These reasons not just justify its usage, but go on to show why ‘e’ is one of the most significant constants in all of mathematics.

In this essay, I start with a brief historical account of ‘e’ and then go on to explore the reasons why we use it for expressing growth and decay. In the process, I also cover certain unique properties that this constant possesses.

If you are a math enthusiast, who, like me, has always been fascinated by Euler’s number, then this essay is right up your alley! Broadly speaking, it is an attempt to explore ‘why’ we do things a certain way in mathematics. Without any further delay, let us get started.

This essay is supported by Generatebg

The Historical Journey of Euler’s Number

The Elusive Hunt

Although this might be surprising to note, Euler’s number (e) is a relative newcomer to mathematics when compared with other constants such as π. The first reference to the constant appears in an appendix to John Napier’s work on logarithms in 1618. This appendix is believed to have been written by William Oughtred. Nonetheless, there is no explicit recognition of this constant.

Several years later, in 1624, Henry Briggs seems to have given a numerical approximation of the base 10 Logarithm of ‘e’ but did not explicitly mention or recognise the constant.

In 1647, Saint-Vincent had computed the area under a rectangular hyperbola but had not recognised its connection with logarithms, and consequently, did not identify ‘e’ either. Much later, we discovered that the number ‘e’ is such that the area under a rectangular hyperbola from 1 to e is equal to 1.

Why Do We Really Use Euler’s Number For Growth? —Area under the rectangular hyperbola leading to ‘e’ (Image from Wikimedia Commons)
Area under the rectangular hyperbola leading to ‘e’ (Image from Wikimedia Commons)

In 1661, Christiaan Huygens had discovered a curve that he called “logarithmic”. In modern terms, we refer to this as the “exponential curve”, with the form: y = k*(a)^x. Huygens also managed to recognise the connection between the rectangular hyperbola and logarithms. Yet, the number ‘e’ went unrecognised.

In 1668, Nicolaus Mercator published Logarithmotechnica which contains the series expansion of log(1+x). Mercator was the first person to use the term “natural logarithm” in this work when he calculated logarithms to the base ‘e’. However, the number ‘e’ did not explicitly appear in his work.

The Unexpected Discovery

In 1683, Jacob Bernoulli had been studying the problem of compound interest and ended up showing that the value of [1+(1/n)]^n had to lie between 2 and 3 as n tends to infinity. This is arguably the first approximation of the constant ‘e’.

Why Do We Really Use Euler’s Number For Growth? An image showing Bernoulli’s approximation of e
Image created by the author

Having traced e through its history so far, one would expect the value of ‘e’ to be discovered via its connection to logarithms. But it turned out to be the case that its first approximation came from a growth calculation (compound interest). This is what makes its discovery somewhat ‘unexpected’ for me.

In 1731, ‘e’ in its present form appeared for the first time in a letter written by Leonhard Euler to Christian Goldbach. In 1748, Euler published Introductio in Analysin infinitorum. It was in this work that he gave the full treatment of his ideas involving ‘e’. 

He showed infinitely recurring expressions for ‘e’ involving the factorial function and continued fractions. He also calculated the value of ‘e’ correctly up to 18 decimal places (although he never mentioned how).

Why Do We Really Use Euler’s Number For Growth? An image showing Euler’s work in relation to e
Image created by the author

Many of our modern mathematical notations actually come from Euler. So, rather unsurprisingly, Euler’s notation for ‘e’ just stuck. However, what is surprising is the fact that we do not know why he chose ‘e’ to represent the constant.

One unpopular opinion is that he chose ‘e’ to represent his surname. Another line of thought suggests that since he had already used ‘a’ in his work for something else, the next available vowel turned out to be ‘e’. Nonetheless, it is one of those mysteries that we might never uncover.

Now that we have completed our brief tour through the historical journey of ‘e’, let us turn our attention to what differentiates it from other conventional mathematical constants.

Why Is Euler’s Number Unique?

Most of the fundamentally profound constants in mathematics such as π and √2 have been around for a long time. This is because these constants were discovered due to their geometrical significance.

Why Do We Really Use Euler’s Number For Growth? — Left — π’s relationship with a unit circle (Image from Wikimedia Commons) and Right — √2’s relationship with a unit isoceles right triangle (Image from Wikimedia Commons)
Left — π’s relationship with a unit circle (Image from Wikimedia Commons) and Right —  √2’s relationship with a unit isoceles right triangle (Image from Wikimedia Commons)

Unlike these constants, Euler’s number does not arise out of geometrical significance. It arises because of its significance in the study of growth/decay/change. As human beings, we started taking interest in the study of change relatively recently as compared to our interest in geometry.

So, it is no wonder that ‘e’ is a relative newcomer in the field of mathematics. It is also one of the reasons why Euler’s number plays a significant role in calculus (more on that in a bit).

Now that the stage is all set, let us directly attack the main question that this essay is trying to answer.

What Happens When You Consider ‘2’ Instead of Euler’s Number (e)?

The original question set that I posed in the introduction is as follows:

“WHY do we use ‘e’?
Why does it HAVE to be a constant that is equal to ‘2.718281828459045…’?
Why can’t it be just ANY other constant such as ‘2’ or ‘3’ or ‘10’?”

To answer this question, I plan to follow the method of contradiction. I am going to consider ‘2’ as a growth constant and see what happens first.

Let us start by plotting 2^x:

Why Do We Really Use Euler’s Number For Growth? — Aplot with x on the X-axis and 2^x on the Y-axis. The range of X-axis is (-6,6), and the range of Y-axis is (0,34)
X vs 2^(x) — Plot created by the author

On visual examination, this does seem to represent exponential growth. Let us assume this plot represents some sort of population growth with x representing time. If we consider the positive (non-zero) realm only, it appears that the final population is twice the initial population on each particular day.

Why Do We Really Use Euler’s Number For Growth? — A growth chart that shows that the population doubles each day when the function considered is 2^x (from day 1 onwards).
Math illustrated by the author

In other words, for unit frequency (one day, in this case), the population appears to be doubling. I use the word, ‘appear’ because we are treating the problem in the discrete domain.

Growth in nature and real life is a continuous phenomenon, and for this reason, it is mathematically (and empirically) correct to treat it as such. If we do this, we quickly end up in the domain of calculus, and get the following outcome:

Why Do We Really Use Euler’s Number For Growth? — Aplot with x on the X-axis and 2^x on the Y-axis. The range of X-axis is (-6,6), and the range of Y-axis is (0,34). The instantaneous rate of change at any point on the function curve is given by: dy/dy = d(2^x)/dx = [2^(x+dx) — 2^x] = {(2^x)*[2^dx)] — 2^x}/dx = (2^x)*{[(2^dx)]-1}/dx
X vs 2^(x) — Plot and illustration created by the author

From the final result, we see clearly that the rate of change of the function 2^x depends on itself whilst being multiplied by a factor. As it turns out, this factor leads to a constant value as the limit of dx tends to zero.

Why Do We Really Use Euler’s Number For Growth? — lim[{(2^dx)-1}/dx] = 0.6931…
Math illustrated by the author

Generalising the Result

We could generalise what we have just found as follows:

Why Do We Really Use Euler’s Number For Growth? — For any function y = a^x, dy/dy = (a^x)*{[(a^dx)]-1}/dx = (a^x)*c
Math illustrated by the author

In fact, there is nothing special about 2. When we consider different numbers, we get different constants (factors):

Why Do We Really Use Euler’s Number For Growth? — When y = 3^x, dy/dy = 3^x * c, where c = 1.0986…; When y = 4^x, dy/dy = 4^x * c, where c = 1.3863…; When y = 5^x, dy/dy = 5^x * c, where c = 1.6094…
Math illustrated by the author

Arriving at Euler’s Number

When we require the constant factor to be equal 1 (that is, c = 1), Euler’s number emerges naturally (!):

Why Do We Really Use Euler’s Number For Growth? — When y = a^x, dy/dx = a^x * c, For c = 1, a = e
Math illustrated by the author

Consequently, the constant factors for the various numbers we saw previously turn out to be the natural logarithms of the respective numbers:

Why Do We Really Use Euler’s Number For Growth? — When y = 3^x, dy/dy = 3^x * c, where c = 1.0986… = ln(3); When y = 4^x, dy/dy = 4^x * c, where c = 1.3863… = ln(4); When y = 5^x, dy/dy = 5^x * c, where c = 1.6094… = ln(5)
Math illustrated by the author

This, in turn, leads us to further interesting properties of Euler’s number.

Significant Properties of Euler’s Number

When ‘e’ is used as the base for an exponential function (that is, y = e^x), we arrive at a situation where the derivative of the function at any point is itself. It is the ONLY function to possess this property:

Why Do We Really Use Euler’s Number For Growth? — A plot of x (on the X-axis) vs. e^x (on the Y-axis) illustrating that when y = e^x, dy/dx = (e^x)dx. That is, the rate of change is equal to the function itself.
X vs e^(x) — Plot and illustration created by the author

Furthermore, this also leads to a situation where the area under the curve for the function e^x at any right-hand-side limit of x is equal to itself as well. This property is significant and beneficial in integral calculus.

Why Do We Really Use Euler’s Number For Growth? — A plot of x (on the X-axis) vs. e^x (on the Y-axis) illustrating that the area under the cuve for the function e^x at any right-hand-side limit of x is equal to itself as well.
X vs e^(x) — Plot and illustration created by the author

These properties make ‘e’ the ideal base to study growth (and decay) in a variety of fields ranging from physics to biology.

Final Thoughts

In layman’s terms, ‘e’ is the maximum possible value for a growth function that continuously compounds at a 100% growth rate after a unit time (period). 

In even simpler terms, Euler’s number is arguably the identity measure for growth and decay in nature.

We did not invent ‘e’. It shows up in nature as far as growth and decay are concerned. This is the reason why we ‘choose’ to express growth equations using ‘e’ as the base.

Growth (G) = G_0 * e^(rt), where G is the final value, G_0 is the initial value, r is the growth rate, and t is the number of (time) periods.
Math illustrated by the author

It is possible to use any other constant as the base to express growth (or decay) equations. However, the mathematics turns out to be convoluted and unnecessarily complicated.

Euler’s number brings elegance and efficiency with it when it is employed mathematically. Since we are creatures of nature ourselves, it is perhaps no wonder why we gravitate towards such natural elegance and beauty!

Credit and References: J. J. O’Connor and E. F. Robertson and Grant Sanderson.

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Further reading that might interest you: The Thrilling Story Of Calculus and How To Really Solve The Three 3s Problem?

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