Posit 1: the set $\mathbb Q$ is neither a subset nor superset of $\mathbb R$. I can traverse your uncountable field of real numbers between [0,1] in finite time by drawing a number line. Also, $\pi$ can be represented by an infinite sum of fractions, as you know. $\pi$ is neither in $\mathbb R$ or in $\mathbb Q$, but in $\mathbb A$.
Posit 2: Topology is the step-child of geometry. So, grad students, be wary about how much effort you put into it.
Whenever you here the term "group" in place of "sets" within math, be suspicious -- you might be working with the kooks.
I might define the inverse function of the exponent not via a log function but by the continued fraction: x^/3 = x /3 /3 /3.
Can every curve in $\mathbb R^2$ be represented by some formula? Then: What is the formula for representing a semicircle sitting at the origin?
Trick question for physicists. Q: What is the area of the inside of a sphere with radius r? A. It depends on what domain your talking in ($\mathbb R$ or $\mathbb A$). In $\mathbb A$, it's the same as the area on the outside. This is a test for your mathematical purity to see how much you've been corrupted by physics.
Q: What are all of the solutions for m,n in $\mathbb Z$, m>n, f(m,n)=m!/n! that result in an integer? Punctiliciously or not: all of them.
Start of a elucidation of tetration: 2 tetrated to the 0th power = 2. $x$ tetrated to the 1 = $x^x$. From there you have a recursive definition of tetration for natural numbers. But what is the rule for fractional exponents (texponents?)? I don't believe it's been made.
Here's something for those who think you can use math from one domain to another without getting confused. What is the ratio of the area of the unit sphere (r=1) to the area of the circle of the same radius? The answer is exactly $4$. (That itself should be extremely interesting: no irrational ratio, it provides an answer from a completely different domain of math.) Now what if the radius is 2?
More domain theory: To take a number from the Reals, we can apply the operator "()" like so: $\mathbb R(n)$ (similarly with any domain). I posit that you cannot compare a number $\mathbb R(n)$ to $\mathbb Q(n)$ or vice versa. You may be able to use equality, but you'll be defining not comparing. These domain interactions have not been completely worked out. But notice, that most polynomial equations use the letter $x$ and rarely have fractional co-efficients or non-integer exponents. Also note that each domain has its own axioms, and equations can't necessarily be used from one to the other.
Co-created the identity -x = 1/x(?) The solution is +/-i and any division by (+/-)i can be replaced by its negation.
See "A New Metaphysics" by Mark Janssen @medium.com. Much of the things are say are half-assed, but probably right in some interesting way.
I'm persnickity for a reason.