Understanding e And The Exponential Function
2.71828 18284 59045 23536 02874 71352 66249 77572…
The mathematical constant “e” also known as Euler’s number is arguably one of the most important constants in mathematics. It comes up everywhere: from natural logs, Euler’s identity to wave equations and radioactivity. But where does this value actually come from? What does it mean? Recently, I watched Grant Sanderson’s 3Blue1Brown lockdown series (which I highly recommend you to watch) where he briefly explains where e comes from. Intrigued, I decided to explore this topic further…
THE ORIGINATION OF EULER’S NUMBER…
Leonhard Euler, born on April 15th 1707 in Switzerland, discovered the constant e in the early 18th century when he was trying to solve a problem first proposed by another mathematician, Jacob Bernoulli, half a century earlier.
Bernoulli’s problem was related to compound interest:
“An account starts with $1.00 and pays 100 percent interest per year. If the interest is credited once, at the end of the year, the value of the account at year-end will be $2.00. What happens if the interest is computed and credited more frequently during the year?”
When you compound annually there is a 100% increase on the $1 so you end up with $2. Compounding semi annually, the interest is credited twice in the year, the interest rate for each 6 months will be 50%, so the initial $1 is multiplied by 1.5 twice, yielding $1.00 × 1.5² = $2.25 at the end of the year… and so on…
This formula can be generalised, for n compounding intervals with a starting value of 1 and a growth rate of 100%:
The question Bernoulli was trying to answer was what happens to the value of m as n gets bigger and bigger ( in other words, tends to infinity):
As you compound more frequently, the rate of growth increases because the investment will generate earnings from both its initial amount and the accumulated earnings from preceding periods, making growth relative to the current size.
The limit as n grows large is the number that came to be known as e. That is, with continuous compounding, the account value will eventually reach $2.7182818… The number e is, therefore, the maximum possible result when compounding 100% growth for one time period.
This can be written for any starting amount, growth rate and time period. e represents the idea that all. . All continually growing systems are scaled versions of e. (similar concept to how every number can be considered a scaled version of 1). So, although we picked 100%, we could have chosen any unit of growth — there would just be a different exponent on e.
We can also conduct binomial expansion for the formula above to form a series expansion for e, as n tends to infinity:
In this series, as n gets really large, (the series approaches e). In that limit, n(n-1) / n² tends to 1, and so does n(n−1)(n−2)/n³ and so on… So, for large enough values of n the dependence of n can be ignored, for these early terms in the series. The larger the value of n more accurately the terms of the binomial expansion can be simplified. Thus, we are left with:
e can, therefore, be written as the sum of an infinite series:
THE EXPONENTIAL FUNCTION: eˣ
Building on from the formula we derived above: as e is the infinite n limit of (1+1/n)ⁿ, eˣ would be the infinite n limit of {1+1/n}^nx. If we let y=nx, eˣ becomes the infinite y limit of (1+x/y)ʸ.
This expression can be expanded using binomial expansion to get a power series for eˣ. For simple binomial expansion, the exponent has to be a whole number however, as x is continuous — for general values of x, y=nx will not be whole number. However, if x can be written as a rational number a/b, where a and b are integers, because we are looking at the limit as n tends to infinity, for n any multiple of b, y will be an integer and the function (1 + x/y)ʸ is continuous in y. If x is an irrational number it can be approximated using a sequence of rational numbers which will produce the same result. To show that y is an integer we can write y=m.
Therefore, you are left with the following series. This is also the Maclaurin series expansion for eˣ (a special case of the Taylor series, centred at 0), which can be derived using the expansion function for the series.
eˣ IS THE DERIVATIVE OF ITSELF:
One notable property of the exponential function eˣ is that it is the only function that is the derivative of itself. This property makes this function very important in the field of calculus. We can show this by differentiating the series expansion above:
It is clear that, this pattern will keep repeating and thus the derivative of eˣ is just itself. Replacing eˣ with eᵃˣ gives a more general formula which when differentiated using first principles shows that the derivative of eᵃˣ is aeᵃˣ ( a constant (a) times the function value)
Relating it back to the compound interest interpretation given earlier: if a bank account is growing under compound interest, with an interest rate of R per unit time, that means that at any instant in time the growth rate is R times the current balance.
One of the most interesting and widely used applications of e is in complex numbers, which is a whole other topic. It involves Euler’s identity e^iπ = -1 which emerges from a more general formula: e^ix = cos(x) + isin(x) (Euler’s relation). I look at the intuition behind this formula in more detail in another article.
With numerous applications in mathematics and many branches of science, Euler’s number is one of the most important constants to be discovered. It characterises this idea of continuous, smooth growth — which is how things change in nature.
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