The exponential function (y=ex) is a definable amount of incremental change – a very useful function for defining the “unit step-size of incremental change”.
“e” itself is a specific value of the above exponential function; 2.718281828…… is the value of y when x=1.
[If you hate any form of maths, what follows is not for you. But if you can track some basic algebra and understand a little mathematical terminology, what follows is worth the read; because as it turns out, the concept of “incremental step-size” is quite an important idea if you want to understand how to build a universe…]
A Quick Intro
It is often, considered amazing that “ex” appears to crop up virtually everywhere in science, (from physics to biology, from economics to climate change), but in truth it is not amazing at all! The one constant in this universe, is “change”; and the reason “ex” appears everywhere is that “ex” is all about change!
Mathematically, “y = ex” is an exponential function. An exponential function is any function of the form
y = f(x) = ax
For practical purposes we can think of exponential functions as a mathematical tool that allows us to define a multiplicative step-size of change (i.e. they allows us to define a macroscopic step-size of change using a base microscopic unit.). Thus y = 2x is an amount of change define using a base step-size of “2”, with “x” being the required number of steps.
Part 1 – A Few Histories
A Quick History of “e”
In the late 1600s Calculus was jointly but independently developed by Isaac Newton and Gottfried Leibniz.
In the years following the work of these great men, many mathematicians applied calculus to compute the derivative (i.e. the rate of change) of many different types of functions. They struggled however with the exponential function (f(x) = ax).
Calculus shows that the derivative of a function can be calculated from the graph of the function. Calculus shows that the slope (i.e. the tangent; the instantaneous rate of change) at any point on a curve is the limit of a decreasing quantity delta-x (Δx).
Imagine we are trying to figure out the derivative of y=2x. Calculus tells us that the slope at any point on the curve is the
Limit ( as Δx → 0 ) of ((2x+Δx – 2x) / Δx)
[Note: Going forward we will simply use “Limit” as shorthand for “Limit as Δx decreases towards zero”.]
A little bit of algebra converts the above into
Limit of (( (2x)(2Δx) – (2x) ) / Δx)
A little bit more converts this into
Limit of ( (2x)(2Δx – 1) / Δx)
And finally since (2x) does not have a Δx term we can take it out of the limit, which means the derivative of y=2X becomes
(2x)(Limit of ((2Δx – 1) / Δx))
So the derivative (dy/dx) of the function y=2x is in fact just the function itself (i.e. 2x ) multiplied by a number — which is simply the limit (as Δx decreases towards zero) of ((2Δx – 1) / Δx). This seems a nice and simple derivative; all we have to do now is calculate this limit to find the number.
Calculating the limit however turned out to be surprising difficult to do. A simple guess using a very small Δx (say Δx = .00001) will yield an approximate result
(2.00001 – 1) / (.00001) = 0.6931
But approximate results were considered far from ideal; mathematicians of the day wanted to find a consistent methodology. It was Swiss mathematician Leonhard Euler (1707-1783) who eventually solved the problem sometime around 1730. Euler reasoned that if
Limit of ((2Δx – 1) / Δx ) ≅ (2.00001 – 1) / (.00001) = 0.6931
Limit of (10Δx – 1) / Δx) ≅ (10.00001 – 1) / (.00001) = 2.3026
then surely there must be a number in between – to which Euler assigned the symbol “e” – such that
Limit of ((eΔx – 1) / Δx) ≅ (e.00001 – 1) / (.00001) = 1.0000
It turns out that that number is approximately equal to 2.718281828459 – a number, curiously enough, previously stumbled upon by Jacob Bernoulli when studying compound interest some 50 years earlier…
So the Limit of ((2.71828…Δx – 1) / Δx) as Δx deceases towards zero is in fact the simplest number of them all: the number 1.
And thus computing the derivative of this particular function could not be easier. The derivative of ex is (1)(ex) and, by use of the so-called chain rule, the derivative of eax is (a)(eax ). This, of course, made computing the derivative of any function involving “e” remarkable easy. But what was really so great about this discovery was that it simplified computing the derivatives of ALL exponential functions!
So for example, to differentiate y=2x, mathematicians would now first convert the number 2 into a function of “e”. To do so they firstly needed to answer the question, what power do we raise “e” to, in order to make it equal to “2”?
Fortunately this question, it turns out, was actually a simple logarithm question; and logarithms had been around for over 100 years by that time…
Using “e” as the base for the logarithm, the log of “2” to the base “e” (written as ln(2)) gives us the power to which to raise “e”, in order to get to “2”; and it turns out that that power is approximately 0.6931. So
y = 2x = (e0.6931)x = e(0.6931)(x)
⇒ dy/dx = (0.6931)( e(0.6931)(x)) = ( ln(2) )(2x)
y = 10x = (e2.3026)x = e(2.3026)(x)
⇒ dy/dx = (2.3026)( e(2.3026)(x)) = ( ln(10) )(10x)
So the derivative of any exponential function y = ax is the function itself (ax ) multiplied by ln(a). In general form we write dy/dx = ln(a)(ax). Thus “ln” became known as the “natural logarithm” of all exponential functions, and “e” became known as the “base of the natural logarithm”.
A Quick History of Logarithms
In the latter part of the 16th century science was really just beginning to take off, but the doing of science however involved a lot of data and an enormous amount of mathematical calculations. These calculations, more often than not involved multiplication and division of some very big numbers, which, using only pen and paper, would have been very tedious, very time consuming, and very easy to get wrong.
Scottish Mathematician John Napier (1550-1617) set out to simplify these calculations by reducing much of the laborious multiplication and division to simple addition and subtraction.
Napier’s idea was really rather simple. It had long been known that multiplying any two numbers in a geometric progression is the same as adding their powers.
A geometric progression is a progression by a common ratio, it is a base number raised to successive powers. The successive powers however are just a simple arithmetic progression of 1. Napier’s idea was to relate in table form each number in a geometric progression with its associated power in an arithmetic progression. For example, below is a simple table which relates the geometric “doubling” progression with its successive powers in an arithmetic progression.
So for example if we want to multiply 8 by 64, this is the same as adding the powers 3 and 6 and then looking up what number in the table that corresponds with the power of 9.
8 * 64 = 23 * 26 = 29 = 512
Using this methodology, is great if one is looking to multiply numbers that come in steps of 2, but of no use for the numbers in-between, or, for that matter, numbers less than 1. To solve the problem of gaps Napier decided make the ratio of the progression much smaller.
Napier choose a ratio of 1 over 10 million (or 0.0000001). He constructed a new table, which started off at 10,000,000 and worked its way backwards. [Why Napier choose this backwards progression is a longer story, but long story short it is likely he did so because he did not want to deal with then new-ish concept of decimal fractions.] The construction of Napier’s table took almost two decades, but eventually in 1614 John Napier published his work in a book titled “Description of the Wonderful Rule of Logarithms”.
[Napier coined the word logarithm from the fusion of the Greek words “logos” (meaning ratio) and “arithmos” (meaning number). So logarithms, as Napier conceived them, are basically the “ratio of numbers”.]
A year after publication Henry Briggs, a younger contemporary of Napier suggested some modifications to Napier’s work. With Napier’s agreement, Briggs set to work changing the base ratio to the new decimal system (i.e. changing the base ratio from 0.0000001 to 10 such that log10(10) = 1), and changing the starting point from 10,000,000 to 0 such that log10(1) = 0 and with these changes in place he then worked his logarithms forwards. Briggs Published his work in 1624 and, in doing so, log to the base 10 was born – and as a result by the end of the 17th century the use of the decimal system and logarithms was widespread…
Napier very nearly found “e”
It is a pity Napier concocted the term “logarithm”, as it gives the impression of something quite complicated*, when in reality the idea behind logarithms was something really quite simple.
[* As with so many things in maths and physics, naming really has let the side down. Badly named things – like Chaos Theory, Imaginary Numbers, and Logarithms – have only ever served to confuse]
What Napier effectively did – although he might not have thought about it this way himself – was to build a simple relationship between “step-size” and “number of steps”. A geometric progression progresses by the base ratio, and so this base ratio is the step-size. While the exponents/powers progress in a simple counting fashion, so these exponents are the number of steps.
Simple though this idea was, the real juice came in Napier’s execution of the idea; Napier understood that to create a practical and usable table, the step-size needed to be very small.
So although Napier is famous for the introduction of logarithmic tables, history might better see his major contribution to mathematics as being the introduction of the infinitesimally small step-size. In effect Napier had sought to reduce the coarseness of the geometric progression, by reducing the step-size. Had he been willing to deal with decimal fractions and worked his way forwards, rather than backwards, he could conceivably have easily done something quite like this
This progression fills in the gaps, as Napier had wanted, by making the progression very fine-grained. But the interesting thing about starting at 1 and using this tiny fine-grained step-size, is that if we take exactly 10 million of these steps (each of size 1/10million), the corresponding number in the underlying geometric progression is 2.718281828459045……..
So we see that a very small growth rate applied an exact number of times leads to “e”….
Continuous Compound Interest
“e” is often thought of as related to “continuous” compound interest but this is misleading! In reality “e” could more accurately be thought of as simply an infinitesimally small growth rate applied an infinite number of times; and this process can happen continuously or non-continuously. The only reason that it is generally perceived that the compounding needs to happen continuously is because of the way it was originally formulated.
In the late 1600s the Swiss Mathematician Jacob Bernoulli discovered “e” when he was examining the difference between a simple interest payment after a fixed term, and the continuous payment and compounding of accumulated interest into the principal amount (on which future interest is paid), during the entirety of the fixed term.
Bernoulli found that, an interest rate of 100% per annum compounded annually grows $100 into $200 in one year, but if it is compounded every 6 months then the $100 grows into $225 after one year, and compounding every 3 months grows the $100 into $244.14. This process of reducing the compounding period and increasing the number of periods can be taken to infinity (so that we are compounding every micro second), but Bernoulli found that the number doesn’t grow and grow; instead it converges to $271.83. Bernoulli found that continuous compound interest follows the formula
A = P( 1 + r/n )n
Where “A”= the dollar amount after n periods. “P” is the original principle amount (in this case $100). “r” is the original rate of interest. And “n” is the number of compounding periods.
If n is taken to infinity the number 2.71828459045…. emerges; and thus this is where “e” became associated with “continuous” compound interest. But this is misleading, because it would be equally valid to say that…
This non-continuous compounding, yields the same final result, the same total return. So it doesn’t matter whether the interest payments are compounded continuously or non-continuously – in fact it doesn’t even matter if all the steps are clustered together or equally spaced out. All that matters is that all the incremental-steps are taken for a given step-size; and when this happens we always get the same final result, the same total return!
Thus we see that “e” has nothing explicitly to do with continuous time! But once again we see that an infinitesimally small growth rate applied the exact inverse number of times leads to “e”… But why $271.83 rather than any other amount?
Part 2 – Transcendental “e”
2.718281828459… Messy Macro (but very Neat Micro)
Any large number of compounding steps applied over a set period of time will yield a final total return. This total return is effectively the same as if we had taken one single large geometric step.
Conversely, any single large geometric step can be broken down into a larger number of smaller equally sized steps. For instance a total return of 100% is a simple doubling of the principal amount, and consequently is a single geometric step of size 2 (i.e. multiply the original principal amount by 2); but this same doubling can be carried out by compounding a number of smaller step-sizes. So therefore we can say that each of the following compounding methods is equivalent…
For each case above we can write the single “macro-step” as a function of 1000 “micro-steps”
Of each of the macro steps above however, only “e” has a micro step-size (1/1000) which is the exact inverse of the number of steps (1000).
In other words, if we divide the whole unit (i.e. 1) into a certain number of very small sub-units (i.e. 1/n) and then multiply all the n sub-units together (i.e.(1+1/n)n ); the answer we get will always be some approximation (dependent on the size of n) of “e”….
Step-Size vs. No. of Steps
To arrive at the macro-number “e” the number of steps must be the exact inverse of the micro step-size; if any other number of steps is applied to the micro step-size (1/1000) will brings us to a different macro number. So 693 steps of the micro step-size (1/1000) brings us to 2, and 2304 steps will bring us to 10…
In truth these macro-numbers are only correct to two decimal places. If we want to increase the precision, we need to reduce the step-size and increase the number of steps by a proportional amount.
As we take n to infinity, we tend towards infinite precision of the macro-number 2. The same of course holds true for the macro-number “e”…
In the everyday speak we could say that “e” is a number we converge to when we take an infinitesimally small growth rate and applies it an infinite number of times. Mathematically however “e” is defined as a limit:
e = Limit as n → ∞ (1 + 1/n)n = 2.718281828459045……
For practical purposes, “e” can be difficult to pin down because the question arises as to what size “n” do we want to use to calculate “e”?
If, for Instance, we set n=1000 we find that the exact value of (1+ (1/1000))1000 is 2.71692393223552. So even with only 1000 steps of 1.001, e is correct to 2 decimal places, with 10,000 steps of 1.0001 it is correct to 3 decimal places, and with 100,000 steps of 1.00001 “e” is correct to 4 decimal places, and so on to infinity…
Making the number of steps smaller and smaller, will give more trailing digits of “e”, but since there is no limit to how much a number can be divided, there is therefore no limit to the number of trailing digits we can add to “e”. This why “e” is transcendental…
That “e” is transcendental is simply a consequence of the fact that “e” is a manufactured quantity, defined using limits to infinity, and infinity is an idea not a number. However although the concept of infinity might be vague, “the concept of “e””, on the other hand, is actually all about very finely-grained precision…
Part 3 – “e” without limits
The concept and formula for “e” is neat and very precise because the number of steps must be the exact inverse of the tiny step-size.
In order to discuss the “preciseness” of “e”, it is better, for the purposes of this discussion, to limit the influence of infinity and define “e” with very precise numbers.
To do so, we will use both a very large number and a very small number which are exact inverses of each other. We will use the symbol Ħ for the very large number, and we will use the symbol ħ for the very small number.
Ħ = 10,000,000,000,000,000,000,000,000,000,000,000 = 10+34
ħ = 0.000,000,000,000,000,000,000,000,000,000,000,1 = 10-34
Since ħ = 1/ Ħ we can write “a precision equation” for “e” as
e = ( 1 + ħ )Ħ
Thus the macro-number “e” or “Macro-e” is simply ( 1 + ħ ) applied Ħ number of times. Using Ħ we can also define “Micro-e”
e1 = ( 1 + ħ )Ħ
e1/Ħ = ( 1 + ħ )
eħ = ( 1 + ħ )
“d” — a “Fundamental Unit”
So here, at last, we finally get to what is so important about “e”.
Micro-e is what’s important about the concept of “e” . Micro-e is what makes the exponential function (ex) so useful to “the calculus of infinitesimals” (as calculus was historically known).
In early calculus the symbol “d” was used to represent an infinitesimally small unit of change. So in keeping with history let us assign “d” to be equal to micro-e
d = eħ = ( 1 + ħ )
In this incarnation “d” (or “micro-e”) becomes is a unit of fine-grained precision, the smallest possible linear step-size; we could think of “d” as a Fundamental Unit of Linear Change.
Defining the “Step-Size of Change”
So (1+ħ) is a “Unit Step-Size” of linear change; a finely-grained unit of mathematical change. We will use this very fine-grained step-size to define other, more coarsely-grained mathematical steps!
( 1 + ħ )xĦ = dxĦ = (eħ)xĦ = (eħĦ)x = ex
Thus (1+ħ) applied xĦ number of times yields the exponential function (ex).
So (1+ħ) allows us to define, with precision, any step-size (ex) in terms of this smallest, almost infinitesimal, unit of change. Using “ex” we can easily jump back and forth between various macro step-sizes; in effect we can define any step-size we like from infinitesimally small to infinitely large using an appropriate power (positive for growth or negative for decay) of “ex”. Thus “ex” is a definable step-size – it is a defined amount of change that occurs in a defined amount of time.
So now that we have 1) used the precision unit “d” to define the minimum step-size per unit time, and 2) used the precision step “ex” to define a desired step-size in terms of the precision unit “d”, we are finally in a position to define a function to apply the desired step-size (ex) over time (t).
Part 4 – “e” over Time
Step-Size per Unit Time
An exponential function is any function of the form.
y = f(x) = ax
We can think of all exponential functions as step functions, with the base being the step-size, and the power being the number of steps. Exponential functions are particularly useful for dealing with growth and decay over time. Thus more often than not we find that exponential functions are in fact functions of time.
y = f(t) = at
Let’s say a bacterium population doubles in size every 6 minutes. This is a simple doubling function and can be written as y=2t [where y is the population, 2 is the multiplicative step-size, and t is the time period in units of 6 minutes]. Using this function, we can say that after 1 hour (i.e. after 60 minutes or 10 time periods of 6 minutes each) the population is y = 210 = 1024.
We can, of course, generalize this idea to any exponential growth (and decay) over time by expressing the step-size in terms of ex and thus the doubling function becomes
y = 2t = (e0.693)t = e0.693t
So by combining step-size expressed in term of “ex” with the number of steps expressed in term of “t”, we arrive at a generalized expression for incremental change over time…
y = f(x,t) = ext
Part 5 – Conclusion
When it comes to using mathematics in the real world, “ex ” stands head and shoulders above every other function. “e to the power of x” is the mathematical function that allows us to define the coarseness of incremental change — by defining with precision the size of a unit step of incremental change. Doing so allows us to use “e to the power of xt ” to examine all types of incremental behaviour over time.
So forget about the misleading connection with “continuous” compound interest; continuous change is just one end of a spectrum of incremental change, which can range from very coarse to very smooth (so smooth it is virtually continuous).
“e” is a mathematical tool to define, with precision, an incremental step-size of change. And this ability to mathematically adjust the incremental step-size of change, allows us to easily jump back and forth between coarse-grained and fine-grained behaviour over time.
Furthermore (and this is the real prize in truly understanding e), just as “e to the power of x ” is the mathematical tool that allows us to define the incremental behaviour of growth (or decline), “e to the power of ix” is the equivalent mathematical tool that gives us the ability to define the incremental behaviour of rotations and oscillations; and this little mathematical gem means that being able to define an infinitesimally small unit of “angular change” will result in what is generally perceived to be the most beautiful equation in all of mathematics