What’s Natural About e?

In the opening post of this blog, I had introduced Euler’s Identity, which states

The identity combines five numbers – 0, 1, e, i, and π – and three mathematical operators – addition, multiplication, and exponentiations – and the equality. In other words, this identity captures many diverse parts of mathematics and links them, thereby demonstrating that what we call ‘mathematics’ is a unified field in which one area neatly dovetails into the next. For a few more links I suggest you read the earlier post. 

In this post, however, I wish to focus on the number e. I will be devoting three posts to it, including this one. If this now seems excessive, I hope that, after you have read the three posts, you will have had a change of heart and mind. Indeed, my hope is that you would wish for a fourth. And a fifth! I could, of course, include it all in one post. However, I have realized that the last few posts have been considerably longer than I had planned for this blog. Granted that each post did deal with a unified theme, the fact still remains that they were quite long. Hence, in the interest of not squelching all the curiosity of the reader, I feel it is best, where possible, to publish shorter posts.

In this post I wish to deal with the definition of e and its relation to a concept of mathematics that most students learn in the 9^th or 10^th grades. I also wish to address the significance of e that arises from the definition. 

The second post on e will deal with some common bounds we can place on its value. The first post will have given us some indication of these bounds. However, in the second post I will take a more formal approach to this. This will involve looking at a few infinite series that mathematicians have derived as ways of calculating the value of e. The third post of e will deal with the issue of e being an irrational number. Along the way, in both future posts, we will learn a few more mathematical tricks to keep in our quiver should we ever need them.

When I was introduced to e, my professor at the Guru Nanak Khalsa College in Bombay (now Mumbai) just told us that it was the base of the natural logarithm (log_ex) and the base of the exponential function (e^x). I asked him a few questions like:

1. Why was it this number and not some other number that was the base of both the logarithm and exponential functions?
2. What was the significance of the number e?

Unfortunately, all my professor could tell me was that the approximate value of e was 2.718. When I pressed him for more information, he summarily asked me to leave his class. Perhaps my mother will now understand why I hated attending classes there. I mean, if even the mathematics class was going to be transformed into one mind-numbing exercise of rote learning, the other subjects didn’t have a prayer!

I have dealt with one mathematical issue that causes me trauma elsewhere. The trauma that this professor caused me remains to this day and surfaces when I hear students confidently tell me that the value of e is 2.718281828. (Yes, they can use their calculators now to get more digits than my professor had memorized!) When I hear something like that I have a strong urge to tug at my hair, which, fortunately, is somewhat difficult for me!

Anyway, let me proceed with the definition of e and then hopefully address the two questions above.

Consider the function

Students who have learned about compound interest will recognize the similarity the above expression has to the formula for compound interest given by

where P is the principal invested, R is the interest rate as a percentage per compounding cycle, N is the number of compounding cycles, and A is the amount upon maturation of the investment. If we divided both sides of the equation by P and express the interest rate as a number rather than a percentage, the formula gets transformed to

where G is the ‘growth’, that is the ratio of the maturation amount to the principal invested, r is the interest rate per compounding cycle, and n is the number of compounding cycles. 

Before proceeding, let’s consider an example so we understand how the formula works. Suppose we invest ₹1,000 at 10% interest per annum compounded annually for 3 years. Then, P = 1000, r = 0.1 (corresponding to R = 10%), and n = 3. Hence,

This gives A = ₹1331 or G = 1.331.

With the same numbers, but assuming that interest is compounded every 6 months, the value of R and r will be halved and the value of N and n will get doubled. This is because 10% per annum is the same as 5% semiannually. And in 3 years, there are actually 6 periods of 6 months each. Hence, R = 5%, r = 0.05, N = n = 6. This gives

Hence, A = ₹1340.10 and G = 1.340096. [Note: I have rounded A to 2 decimal places as is the convention for currency.]

Suppose now, that the interest rate is 100%. Then R = 100% and r = 1. Now, if we invest some amount for, say, 3 years, we will get:

But suppose, we invest only for 1 year. Then we will have

Suppose, now we keep reducing the duration of the compounding cycles. If we have 2 compounding cycles in a year, each lasting 6 months, we will have

If we change this to compounding every 4 months, we will have 3 compounding cycles, giving us

We can, of course, continue increasing the number of compounding cycles.

For the sake of the discussion, I will rename G as f(x) and n as x, yielding the following table:

The third column gives the change in the value of f(x) from the previous row. What we can observe is that the values of f(x) keep increasing from one row to the next. Also, the value of Δf(x) keeps decreasing from one row to the next. In fact, if we plotted the graph of the function, shown below, this is what we would expect.

Since the graph becomes almost horizontal, it seems that the rate at which the function increases its value keeps decreasing. This is indeed the case as can be seen from the table below.

What we can see here is that x is increasing by orders of magnitude, while the corresponding values of Δf(x) keep getting smaller and smaller, while remaining positive. 

Now there are 31,536,000 seconds in a year. If we put this as the value of x we will get f(x) = 2.71828177847, which represents an increase of 0.00001354128 from the value when x = 100,000.

At this stage, let us take a short detour. Suppose we have a sample of bacteria in a petri dish with enough nutrients for the bacteria to grow and undergo mitosis unhindered. Assuming no mutations occur, there will be no way of distinguishing any particular bacterium from another. All the bacteria in the sample are, in other words, identical. All are consuming nutrients and all will reach the next stage of mitosis simultaneously. Hence, 100% of the sample has the potential to undergo mitosis. But how frequently does mitosis occur? 

Some bacteria need about 24 hours of feeding on the nutrients before they undergo mitosis. So here we have a doubling every day. But suppose we once again restricted ourselves to 1 day but somehow sped up the process of mitosis. What would happen? The tables above tell us exactly what would happen. If we have 100,000 cycles of mitosis in our day with only 1/100,000 of the sample undergoing mitosis each time, we will end up having 2.71826823719… times the number of bacteria with which we started.

We can see that, as the number of cycles increases indefinitely, with the fraction of bacteria undergoing mitosis each time correspondingly decreasing, the growth will be given by the limiting value of the function f(x) as x gets infinitely large.

Coming back to the issue of compound interest, every unit of currency we invest is identical to every other. Since we proposed a 100% interest rate, every currency unit is subject to growth at all times. However, if we reduce the compounding period indefinitely and correspondingly decrease the fraction of the currency units that actually multiply, at the end of the year we will have a growth equal to the same limiting value of f(x).

Now currency is an artificial human construct. However, bacteria belong to the natural world. Many other things grow in the natural world. Similarly, there are things that decay, like radioactive nuclei. All these natural phenomena are, like our sample of bacteria or the invested money, continuously growing. Continuous growth, subject to sufficiently large environments and resources to expand into, and continuous decay, subject to sufficiently large numbers of species to undergo decay, are ubiquitous natural phenomena.

The limiting value I have referred to is the number denoted by e. And we can see that we have answered both the questions I had posed. The significance of e is that it represents the limiting value of growth (i.e. a multiplicand) or decay (i.e. a divisor) when the growth or decay is continuous. And it is the base of the natural logarithm, which shows up when we know the final population and need to solve for time, and the base of the exponential function, which shows up when we need to solve for the population after a period of growth, because it represents the behavior of all natural systems. 

Now, was that too hard for my professor to tell me? I do not think so. But, unfortunately, I have to face the dismal possibility that he had no clue about any of this, having resigned himself to learning by rote rather than learning by inquisitiveness.