The concerned reader might wonder how it is possible to assert that there is _one_ correct definition of an extension of the factorial function to the real/complex numbers. Why is the gamma function better than any other extension?
The answer is that the gamma function is the unique logarithmically convex extension of the factorial function.
The story of what happens if we defined, for example, 1/0 = 0 is a little more complicated. I will include it because it is sufficiently interesting.
Equality of fractions is defined differently when zero divisors are allowed in the denominator. A zero divisor is a non-zero number that can multiply with another non-zero number to get zero. For example, if we work mod 12 then 3 is a zero divisor because 3·4 = 0.
If we want to allow zero or zero divisors in our denominators then we say that a/b = c/d if and only if there is some value s such that s·(a·d - b·c) = 0 where s is anything allowable in the denominator. If we are working with the integers, including this s term does nothing because s has to be something that can be a denominator and we only allow non-zero denominators.
So, even if we define 1/0 = 0 then literally every fraction would be equal to every other fraction.
These conventions can be broken (like, for example, addition of floating point numbers is not associative as pointed out by other comments) but it is definitely not "natural". In other fields of mathematics, like measure theory, it is possible to define things like "zero times infinity is zero" which is traditionally undefined but is a convenient shortcut and does not break anything that people working in measure theory care about.
You could look for a math history book (such as "An Imaginary Tale" by Nahin) or something like the Princeton Companion to Mathematics or frankly any other popular math book written by established mathematicians.
There are two very good ways of understanding Euler's formula and one is the "circular motion" explanation given by another comment. Both are very similar.
The other is that "multiplication of complex numbers is rotation" (which can be demonstrated purely by algebraic manipulation) and that "exponentiation is repeated multiplication". If we know what e^(ix) is then we also know what e^(2ix) is. It is the same "vector" as e^(ix) but the length of the vector will be squared and the angle it makes with the real axis will be doubled.
It is trivial to differentiate exponents like a^(x) and we get that the derivative is simply a constant multiple of itself (depending only on "a"). We choose "e" to be the choice of real number that makes the constant 1. (We can also rigorously justify that such a choice of real number exists.)
Now, what is the value of e^(ix) for very small positive values of x? It is approximately the value of e^(ix) at zero plus x times the value of the derivative at zero. (This is just the Taylor series.) In other words, for small x, e^(ix) is essentially 1 + ix except we know our answer should have magnitude 1 so we interpret e^(ix) as having magnitude 1 and angle x for small x. The properties of exponentiation as repeated multiplication and multiplication of complex numbers being rotation justifies interpreting e^(ix) as having magnitude 1 and angle x for all x.
This is not very rigorous but it is the gist of the matter. Many tools in modern analysis were created to make arguments like this rigorous so this could definitely be considered a good way to understand complex exponentiation.
This is obviously not a completely serious question but it is definitely looks like a question someone might ask when learning about complex numbers for the first time.
The answer is completely historical in nature. Imaginary numbers began as being interpreted as the square root of -1 for the purposes of solving polynomial equations (hence the name.) Later, their field structure and their interpretation as vectors-with-multiplication became their primary use but the name remained.
Mathematicians don't really use "vectors" in the traditional sense like in physics but deal with abstract vector spaces where a "vector" is simply a member of a "vector space" which is "a set of things with addition and scalar multiplication and a few other nice properties".
However, if something needs to be done with vectors in a plane, complex numbers are extremely useful because scaling and rotation can be represented as multiplication. Therefore natural operations in the complex numbers often correspond to natural operations in whatever you are trying to study with complex numbers.
For maximum safety you should assume your adversary knows which character set your password comes from. Frankly, there are few reasons not to assume this.
I imagine that a lot of people responding to this do not realize how abstract modern pure mathematics is.
The number theory that makes up the basis of cryptography was established in the 1700s. For example, Euler's theorem is the basis of RSA and was proven in 1763. The theorem is a small generalization of Fermat's little theorem which was known (but not proven) in 1640. These theorems are really just simple facts about groups and other cryptosystems, such as elliptic curve cryptosystems, are essentially the same facts except the multiplicative group of integers is replaced with an elliptic curve group.
These concepts could be taught to advanced high school students with no formal pure mathematical training. The "hot" areas in modern mathematics require not only an additional 4 years of undergraduate mathematics but usually ~2 years of a PhD program to begin to understand the current papers.
This is extremely different from other fields such as theoretical computer science which seems to have applications almost immediately. Even professional mathematicians likely do not research in hopes of applications hundreds of years later.
I will not claim that modern mathematics cannot possibly have applications. I will, however, claim that pure mathematics is an extremely poor way to allocate funds if you are simply looking for a return on investment in terms of "useful theorems proved per dollar". Mathematics research should be justified by stating that people trained in pure mathematics can be useful in industry, other applied fields or to teach mathematics.
Its pretty standard in the US (at least in the fields I am familiar with) that if you quit a PhD program you can still leave with a Master's if you've been in the program for long enough.
Of course this quote also suggests that maybe people who go in to PhD programs have higher rates of mental health issues. I would definitely argue that PhD programs are both bad for mental health and also attract people with poor mental health in the first place.
I have known plenty of people who didn't feel like they had the skills (both technical and/or social) to go in to industry so they applied to grad school.
Isn't he also trying to say that perhaps the bias that some women have against male doctors might also contribute to the statistic? In other words, women heart attack patients might be more stressed or worried with male doctors leading to higher mortality.
You yourself admit to being biased against male doctors.
Does anyone else think technology has a lot to do with this?
Recall the experiment where a rat is placed in a cage with a "pleasure button" that stimulates reward centers in its brain. The rat proceeds to never stop pressing the button.
I believe that things like social media act in the same way. Not to mention smartphones, "clickbait", video games. Machine learning algorithms that increase a service's abilities as a pleasure button.
The philosophy behind HN is that popular community websites have lower quality content. In my view, this is simply a "regression towards the mean" of people wanting a dopamine rush.
The answer is that the gamma function is the unique logarithmically convex extension of the factorial function.