My (unofficial) definition

e describes a "zillionth" added a "zillion" times, compoundly, within a given time frame.

Leonhard Euler

Swiss mathematician, and the most prolific mathematician in history. He published over 8 thousand books and papers, on every aspect of pure and implied mathematics, physics, and astronomy. He introduced many new functions and created the calculus of variations. His notations such as e and π are still used today.

The letter e hints to his name.

Having a conceptual understanding of e is fundamental to most science disciplines. In physics, e is present in any physical situation where forces accumulate. In biology, e is the fundamental unit used in formulas that describe growth (and decay) in cultures, microbiological growth etc..

This paper is divided into 2 major sections. The first section involves constructing a standard formula to describe growth. Only basic mathematics is needed to understand the flow of thought. As an added bonus, it gives us some insight in the process of mathematical modelling.

The second section uses the formula constructed in the first section to determine e

e is a unit of growth, just like the meter is a unit of distance. To understand e, it is best to use an example in biological growth. Indeed, e is the most fundamental unit in biological mathematics...

In the following example we envisage a colony of bacteria that forms a "slime mould". The slime mould is a good example since it is a bunch of bacteria that behave as a single organism. Work through the following logic.

The colony could be said as having a principal (initial) amount of slime mould. To symbolise the fact that we start with a whole of something (one colony), we set this amount to 1.

Say it grows by half every hour: So at the end of the first hour we have ...

And at the end of the second hour, we have ...

And so on. The logic is simple: After every hour, we take whatever amount we have from the last calculation and add half of that to it. Each result represents how much this mould has grown by the end of that hour.

Note in particular that we are talking about compound growth. Meaning that each successive operation is based on the lastly attained result, not the initial result.

Instead of using pictures, we can explain this mathematically:

Start with a unity (1), or colony of a slime mould...

• 1 unit

After the first hour, the bacteria will have increased to ...

• 1 + 1/2 units

After the second hour, the bacteria will have increased to ...

• (1 + 1/2) + (1 + 1/2)/2 = 1 + 1/2 + 1/2 + 1/4 = 2 + 1/4

After the third hour, we have ...

• (2 + 1/4) + (2 + 1/4)/2 = 2 + 1/4 + 2/2 + 1/8 = 3 + 3/8

After the fourth hour, we have ...

• (3 + 3/8) + (3 + 3/8)/2 = 3 + 3/8 + 3/2 + 3/16 = 5 + 1/16

And so on ... These results may not be very revealing, but they will be when you check the following table. Each result from the above calculation can be reworked in such a way that a pattern becomes visible:

• (1 + 1/2)

• 1 + 1/2= (1 + 1/2)¹

• 2 + 1/4=(1 + 1/2)²

• 3 + 3/8= (1 + 1/2)³

• 5 + 1/16 = (1 + 1/2)⁴

Having established the pattern it becomes very easy to predict what will happen in the future, e.g. we can predict that in the above example after the 14^th hour, we will have (1 + 1/2)¹⁴ units of bacteria.

Putting it more abstractly, we can say that if the slime mould grows with a factor of 1/k^thevery hour, after n hours the colony will have increased to ...

For example, if something grows by 1/5^th each minute, after 3 minutes, it will have grown to

• (1 + 1/5) * (1 + 1/5) * (1 + 1/5) = (1 + 1/5)³

There's only one problem with this formula, it represents growth in spurts, like, if we want to visualise the above formula of (1 + 1/5)³, we have the following happening in time:

Note that our formula did not represent what happens in between times 0 to 1, 1 to 2, and 2 to 3. Obviously, the colony would still be growing between these times. However our formula only captures this growth at particular time intervals.

The number of time slices are represented by the power factor n in the formula (1+1/k)ⁿ

If the formula was to represent growth more accurately, the time intervals would have to be much smaller (and therefore more numerous) and consequently the factors it grows by (1/k) would be much smaller for each of these time slices.

We will futher explore this, in particular the idea that the growth factor changes as the number of time slices change.

In order to arrive at a more general or universal description of growth, we need to take our logic into the abstract ...

Idealise particular values

Abstract-ness requires that there are no particular values. Whereas before, n could stand for amount of years, or minutes, or whatever, now n stands for a number of time-slices.

Represent continuity

At the same time, the formula will have to represent the continuity that exists in growth.

That is, given a time span of any length. No matter how long the span is, in order to represent continuity, the span will need to be divided into an infinite number of time slices.

Idealise relationships

Any relationship that exists in the formula will be reduced to its simplest form.

There exists a relationship between the number of time slices (n) and the ratio of growth (1/k) at each of these intervals. That is, the more time slices we have, the smaller the growth spurts will be for each of these time slices.
So as the number of intervals increase (each one of shorter duration), the growth during each of these intervals diminishes by an equal amount. In short, the simplest way to idealise this relationship is to say that n = k

For example, if we divide a given time span into, say, 100 intervals, we will reduce the growth factor at each of these intervals with 1/100^th . Mathematically, if n=100, then 1/k = 1/100

In order to determine e we will start with the simplest description of growth and gradually work your way up in the logic, making sure the above conditions are satisfied.

To do this, we take the most simplistic view of growth, and gradually reduce the time intervals. So in the next example, the same situation is portrayed each time, that is, every formula happens over the same time span but we gradually make the slices smaller (so we need more of them).

From what we observed before, this will result in the growth factor becoming smaller.

Remember that to idealise the relationship between growth factor and number of time-intervals, we will equal the growth factor change to the times-interval change.

Imagine a unit (1) that grows by its own size (doubles) over a given time span

time interval is 1, growth factor is 1

To represent growth more accurately, we gradually increase the time intervals and proportionally diminish the growth factors:

time interval is 2, growth factor is 1/2

time interval is 3, growth factor is 1/3

time interval is 4, growth factor is 1/4

time interval is 5, growth factor is 1/5

and so on ...

The curve becomes exponential (as is characteristically the case where powers are involved).

As you can see, to represent continuous growth, the time intervals will become infinitely small which results in an infinite amount of time intervals. Also, the growth factor will become infinitely small at each time interval.

Mathematically we can represent this as

e is the limit reached when n gets infinitely large in the idealised formula of growth.

Note that this limit can never be actually reached, the infinite series of numbers that e represents stands testimony to this.

e is irrational, because it can't be expressed as a ratio of whole numbers.

e is transcendental because it is can't be produced using the standard algebraic computations (it transcends them).

The larger n, the more accurate growth is portrayed and the more accurate the number e will be. For example, for n=1000, e = 2.716923... which is accurate to 2 decimal digits.

So whenever growth needs to be represented in a formula, the best mathematicians can do is to make that entity interact with e which automatically indicates that some continuity is present in this entity.

Just as 5m gives a concrete measure of length to the number 5, e⁵gives a measure of growth to the number 5 (more on that later).

This continuity can only be approximated using our current mathematical tools by using the concept of infinity. We slice up the entity into an infinite amount of slices.

Although we have used time as an example of continuity, when you think about it, virtually everything in life has continuity. It is easy to see why some mathematicians, such as Newton and Leibniz, developed a thorough study of the fundamental concept that continuity depends on, change, in the Integral and Differential Calculus

We will see that by making e interact with itself a particular amount of times, we can describe a more particular situation:

We will see that starting with a principal (initial) amount P, we can describe its growth by

where A is the derived amount.

Following examples show some applications of this formula. Note that the "time" factor doesn't necessarily need to be time, it can be anything that has continuity (such as the continually changing filtering as light passes through a substance in the second example). Note that if x is negative, we have the opposite of growth: decay.

Example 1

In a cohort of Foxglove plants, the number y surviving at time t, measured in months, conforms to the equation ...

y= 100 e^-0.231t

Example 2

As light passes through glass or water, its intensity I at depth x decreases exponentially according to the equation ...

I = I₀ e^-kx

where I₀= intensity before entering and k= filtering factor (e.g. k=0.5/cm means that each cm of filter thickness removes half the light reaching it).

Example 3

The atmospheric pressure p decreases exponentially with the height h miles above the earth:

p = 29.92 e^-h/5