As the title says, how come the value e is special? it has functions and rules just dedicated to it, but what sets it apart from another random number?
In: 397
A lot of people here are giving examples without really explaining what the common thread between them is, why e appears in all of them.
e is kind of in the same situation as pi. There are thousands of formulas that have pi appearing in them despite not being trigonometry. It turns out that while the formulas aren’t about circles, they’re way more often related to angles, or sine and cosine waves, or areas of a circle or volume of a sphere, etc. Even in areas that have nothing to do with circles, we can still find ways to use those trigonometry bits to make it more manageable. As a consequence, pi appears everywhere.
e isn’t to do with circles, but it’s the situation of ‘what if the next state of a thing is based on the current state?’. e is kind of a magic number for that situation, because the line for e^(x) has a slope (rate of change) equal to its value (it’s its own derivative, in technical terms). That makes it the ‘neutral’ case. It turns out a lot of formulas get easier if you put them in terms of the neutral case, rather than some other number. Just like with pi and circles, even in formulas that aren’t explicitly about, say, population growth, putting things in terms of growing functions and derivatives can make things easier, and so e appears.
After a while, you start to realise that it’s not just easier, the underlying mechanisms really are more about growth than you thought!
It just happens to be the base 10 expression of the basic constant that pops out when you describe exponential change.
Kind of like phi is the ratio of two sequentially added numbers which also shows up in natural growth. It expresses an inherent property of the mathematical operation.
One thing that no one else has mentioned is that the indefinite integral of 1/x is the logarithm of x with base e. This relates it to the compound interest formula. But also these are functions and expressions that arise in nature referred to as transcendental numbers. At the end of the day these are numbers or functions that are potentially universal as they only rely on logical methods of discovery as opposed to subjective measures like how humans tend to focus on base 10 and base 12
Ok. Everyone has completely missed why e is important. Other people have already mentioned that the derivative of e^x is e^x. In engineering, we learn that systems are described by differential equations, and all but the most simple differential equations are essentially unsolvable except for an extremely limited set of input functions—essentially functions that differentiate to themselves, which means we can’t tell if our amplifier will have a flat frequency response in our target frequency range or our bridge is going to fall down. What to do? Well euler’s formula tells us that an arbitrary single frequency function can be represented by e^jomega where j is the sqrt of -1 and omega is the frequency, and Fourier tells us that an arbitrary sinusoid can be modeled as an infinite sum of single frequency sinusoids, so (hold on to your hats) we can decompose our arbitrary input sinusoid an infinite sum of single frequency sinusoids, represent those as e^jomega, and since the derivative of e^jomega is e^jomega, our we can now determine important things about our system—specifically whether our bridge or skyscraper is going to fall down, whether our jet is going to fall out of the sky, whether our chemical reaction is going to explode, or whether our amplifier is going to have a flat 20-24KHz frequency respose
Disclaimer : Not every important thing in life is so simple to be explainable to a 5 year old.
It just is. There is no why.
Do you know why things fall? Nobody knows, they know how things fall (i.e. how gravity works), but nobody knows why gravity works.
You might find many answers on what the number is, how it is used etc etc. But nobody knows why.
Think of it like that one kid in school who tops the academics and sports too.
There’s nothing special about the value, *except* those functions based around it. It just happens to be that value.
Saving for later.
e is also the solution to the secretary problem. You want to interview e% of candidates and then select the next best one.
It’s not so much that things are “based around” e. It’s like how one or zero show up a lot because of how basic arithmetic works. Add zero to anything, and the value remains the same, subtract any number from itself and you get zero, multiply anything by one and it stays the same… It’s much more obvious how one and zero are fundamental to the basic functions (+-“/) but e has a similar relationship to exponentials (x^y) and logarithms. It’s baked in and as a result it pops out frequently.
Because it is not just a random number. e is a special number which seems to be an intrinsic property of the universe. Just like pi and c and i, e is related to fundamental properties of the way things work in our universe and in which matter, force, energy and space relate to each other. They are all connected to each other in special ways which are described by the laws of physics.
You can use any number to describe exponential curves, but using e as a base gives you an exponential curve with an advantage: its slope at any point is much easier to calculate than the others.
It’s like assuming 1 for the hypotenuse when you are teaching trigonometry: it simplifies the equations so it’s easier to move on.
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