[fredilo]

The real numbers have some very unreal properties. Especially, their uncountable infinite cardinality is mind boggling.

A person can have a finite number of thoughts in his live. The number of persons that have and will ever live is countably infinite, as they can be arranged in a family tree (graph). This means that the total thoughts that all of mankind ever had and will have is countably infinite. For nearly all real numbers, humankind will never have thought of them.

You can do a similar argument with the subset of real numbers than can be described in any way. With description, I do not just mean writing down digits. Sentences of the form "the limit of sequence X", "the number fulfilling equation Y", etc are also descriptions. There are a countably infinite descriptions, as at the end every description is text, yet there are uncountably many real numbers. This means that nearly no real number can even be described.

I find it hard to consider something "real" when it is not possible to describe most of it. I find equally hard when nearly no real number has been used (thought of) by humankind.

The complex extension of the rational numbers, on the other hand, feel very natural to me when I look at them as vectors in a plane.

I think the main thing people stumble over when grasping complex numbers is the term "number". Colloquially, numbers are used to order stuff. The primary function of the natural numbers is counting after all. We think of numbers as advanced counting, i.e., ordering. The complex "numbers" are not ordered though (in the sense of an ordered field). I really think that calling them "numbers" is therefore a misnomer. Numbers are for counting. Complex "numbers" cannot count, and are thus no numbers. However, they make darn good vectors.

[seanhunter]

For people who read this parent comment and are tempted to say “well of course complex numbers can be ordered, I could just define an ordering like if I have two complex numbers z_1 and z_2 I just sort them by their modulus[1].”

The problem is that it’s not a strict total order so doesn’t order them “enough”. For a field F to be ordered it has to obey the “trichotomy” property, which is that if you have a and b in F, then exactly one of three things must be true: 1)a>b 2)b>a or 3)a = b.

If you define the ordering by modulus, then if you take, say z_1 = 1 and z_2 = i then |z_1| = |z_2| but none of the three statements in the trichotomy property are true.

[1] For a complex number z=a + b i, the modulus |z|= sqrt(a^2 + b^2). So it’s basically the distance from the origin in the complex plane.

[dxdm]

> The number of persons that have and will ever live is countably infinite

I don't think you can say that their number is infinite. Countable, yes. But there is no rule that new people will keep spawning.

[fredilo]

If you say that humanity will end at some point, then yes, there is only a finite number of people. This, however, does not go against my overall argument. In that case, you end up with a finite number of thoughts. That's even smaller than countable infinite and makes the real numbers even weirder.

[RugnirViking]

im not very good at all this, having just a basic engineers education in maths. But the sentence

> There are a countably infinite descriptions, as at the end every description is text

seems to hide some nuance I can't follow here. Can't a textual description be infinitely long? contain a numerical amount of operations/characters? or am I just tripping over the real/whole numbers distinction

[fredilo]

> Can't a textual description be infinitely long?

That's a good question. The usual answer is no.

The idea is that every book/description that has ever been written can be seen as string of finite length over a finite alphabet. For example, the PDF of the book is a file, i.e., a string of finite length over the byte-alphabet.

Another way to think about it, that every book/description has to have been written. Writing started some time in the past. Since then a finite amount of time has passed. Assuming one writes one character per second at most, one obtains an upper length on the number of characters in the book. This implies that it is finite in length.

That being said, one way to define the real numbers is to start with infinite sequences of rational numbers. Next, one defines when they converge against the same number. A real number x is then defined as the classes of sequences that converges against x. The set of infinite sequences of rational numbers is uncountable infinite. That's where the cardinality of the real numbers at the end of the day comes from.

The reason I bring this up is because one can view an "infinite sequence of rational numbers" as "infinitely long textual description". So your question really scratches the core of the problem.