Comment by fredilo
1 day ago
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.
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.
> 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.
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