(Warning: this post contains a good deal of calculus, some of which was researched for the reason of making it more rigorous.)

The thing is that is much more important than it seems - but the reasons behind it are generally really really deep.

Rigorously, is defined as the number . By this definition, yes it is a "placeholder", just like how is a "placeholder" constant that equals the ratio of the circumference of any circle to its diameter. It just so happens that this mysterious number shows up way more often than its somewhat innocent definition suggests it should.

First, some reasoning on where this definition might come from. In calculus, there's this rule for evaluating integrals that works like the reverse of derivatives. It states that for an arbitrary constant . The problem with this definition is that it doesn't work for (because then otherwise we would divide by which is bad). A natural question to ask is then "what is ?"

A first interesting property to make is that if denotes the integral with bounds and , then . To prove this, note that


This highly suggests that is some sort of logarithm function. But to what base? To find out, let be this unknown base. Then by the definition of the derivative


It turns out that in order for this to equal as desired we must have . (I don't remember exactly how to prove this, but it shouldn't be too hard.) To summarize, the integral of is just - and already we have an interesting idea with this mysterious variable !

Okay, now to the meat of the whole discussion: Taylor Series. When your calculator is asked to compute, say, or , how does it go about doing it? I mean, most trigonometric, exponential, and logarithmic numbers are transcandental, meaning that they aren't the solution of any polynomial equation with rational coefficients. The answer is simple: we can approximate these values by polynomials, called Taylor Series. In a general sense, for any function its taylor series centered at the origin (i.e. ) is given by


where is the derivative of the function. (The idea here is that this polynomial of infinite degree has its first derivative, second derivative, third derivative, etc. all equal to the first, second, third, etc. derivatives of the original function .) Calculators will probably compute "only" up to or so terms, but it's enough to get a pretty darn good approximation.

The hidden beauty of these Taylor Series is that they can be used to prove Euler's formula. Any complex number with magnitude can be written in the form where is the angle between the complex number and the real axis. The claim is that this expression equals . To prove this, one can use the formula shown above to generate the Taylor polynomials


(The last one is interesting, as it implies that !) Now we can say that


This explains tons and tons of properties of complex numbers very easily. For example, we can easily prove that the argument of the product of two complex numbers is equal to the sum of the arguments of the individual complex numbers almost trivially from this new definition, as . Also, watch what happens when we plug in : Boom! The famous Euler's formula is not only beautiful, but it is also the fusion of complex numbers and calculus, two fields of mathematics that at first seem relatively distant from each other.

In conclusion, yes, it's true that in many cases is seen as just a "placeholder", but upon closer inspection there's a lot more going on underneath the surface. In fact, that can be said for any two branches of math, really.