e (v2)

One of my first blogposts was about e. I hate that post so much that I've decided to write up a new one. 

e: 2.718281828459045...; a number that looks like it flirts with possible rationality, yet ends up being just as transcendental and infinite as π. Sometimes called Euler's number, e was actually discovered by the mathematician Bernoulli in 1685, twenty years before Euler was born. However, Euler did admittedly give e its name, though why it's e, no one really knows, and it happened to catch on. In fact, Leibniz used b to represent the number in 1690...yet Euler's arguably had more staying power, so it's not a shock e's become so universal.

What is e?

In the late 17th century, Jacobo Bernoulli was investigating compound interest, which involves this formula:

Compound interest is, in effect, the resultant interest which occurs after several months of a given interest. Say you get a loan from a bank of £100, with 5% monthly interest on it. Initially, you'd need to pay back £100, but after one month, this rises to £105 = 100 + 0.05x100. After two months, you pay back £110.25 = 100 + 0.05x0.05x100 - and so on. This whole process can be rewritten so that after n months, you'll have to have paid back 100 + 100x0.05ⁿ. The general formula for this process is:

P(1 + r)ⁿ

Where P is the initial amount borrowed, r is the interest rate, and n is the number of months (or time intervals). In my example, P is 100, r is 0.05, and n is, well...anything you want it to be. 

Let's make P = 1, and let r = 1/n. That gives us the limit formula from before, which simply discusses what would happen if the value of n approached infinity. If we do this with various iterations, we'll approach a number. Let's call the output of the limit G.

• n = 1, G = 2
• n = 2, G = 2.25
• n = 5, G = 2.48832
• n = 10, G = 2.59374246
• n = 100, G = 2.704813829
• n = 1000, G = 2.716923932
• n = 10000, G = 2.718145927
• n = 100000, G = 2.718268237

As you can see, the limit is converging to a set value, and that value is e. 

e also happens to have other definitions, such as this one:

Which is simply the summation of all the reciprocals of the factorials from 0 to infinity. 

Another very cool one, and a fundamental property in calculus, is that the function f(x) = e^x is the only function which is exactly equal to its derivative, as so:

Well aside from f(x) = 0, obviously...

In effect, this means that if you graph out f(x) = e^x, at any given point x, the gradient of the function will correspond to e^x.

e^x and ln(x) 

Logarithms are the inverse of exponentiation, and there are various ubiquitous ones you may have come across before. log(x) is common notation for either log₁₀(x) or log_e(x), but I've typically seen it used for the former. With log₁₀(x), where for instance 10³ = 1000, log₁₀(1000) = 3; in general, where a^b = c, log_a(c) = b, and for log₁₀(x), a = 10.

Now, when our a = e (and yes, I'm aware using italics like this is dicey), we can either write log_e(x), or we can use the far cleaner ln(x) - they mean the same thing. 

Since e^x is a function of growth, it only makes sense that we used it to model any form of exponential growth, such as in the case of radioactive decay or the rate of a chemical reaction. The formula

illustrates the rate of nuclear decay, where N is the number of nucleons present in a sample after a given time t, where N₀ is the initial number of nucleons and λ is a constant. I won't get into how this equation works because that's outside the remit of this post, and besides the interesting thing is what happens when we apply ln(x) here:

1. Let N = 1/2 and N₀ = 1. What we have here is an example of half-life, where the radioactive sample has decayed to half the amount that there was before in a given timeframe. Every radioactive isotope has a half-life of sorts - strontium-90 has a half life of about 29 years, which means that if you had 1g of Sr-90, 29 years later you'll only have about 0.5g left - half of it will have decayed. 
2. We end up with 1/2 = e^-λt. Here, we can apply the natural log on both sides.
3. We will get ln(1/2) = -λt. 
4. ln(1/2) = -ln(2). This is due to the log law where log(aⁿ) = n log(a). Read here for more about log laws - admittedly there are several, but they're outside the scope of this post.
5. Therefore, we know that ln(2) = λt.
6. Earlier, I mentioned that λ is a constant; ln(2) is also a constant, it's the solution to the equation: e^x = 2. So if we want to know the half-life t of said radioactive sample, it's just t = ln(2)/λ.

Hopefully that was clear!

Another cool property of ln(x) is that when you differentiate ln(x), you get 1/x. I could prove it, but that would require me to explain the chain rule as well, and this is a post about e and ln(x), not about differentiation.

Euler's formula

Quite often, when you read about e, you'll stumble across

e^iπ = -1

which is also known as Euler's identity, which is called the most beautiful expression in maths. Admittedly that's a very subjective claim, but it is still interesting that e, i (which is √-1), and π all come up together at some point. 

This is all due to Euler's formula, which states that e^ix = cos(x) + isin(x), and when x = π (we're working in radians here), we end up with -1. So overall, I'm not sure if the identity itself is beautiful, or if we've collectively decided a specific case of it should be placed on the altar to the gods. You can find a proof here.

Epilogue

e is certainly an interesting number, one which seems to crop up all over the place, whether it be calculus, trigonometry, or basic finance calculations. And I hope this post was far better than my first one, which was half apology really. I would delete that post, though I don't really see a reason to, so I won't - but it is here.