In a Venn diagram, where are other number sets located? I remember of this image I've learned at school:

I've heard about other number (which I'm not really sure if they belong to a new set) such as quaternions, p-adic numbers. Then I got three questions: 


*

*Are these numbers on a new set?

*If yes, where are these sets located in the Venn diagram?

*Is there a master Venn diagram where I can visualize all sets known until today?


Note: I wasn't sure on how to tag it.
 A: This Venn diagram is quite misleading actually.
For example, the irrationals and the rationals are disjoint and their union is the entire real numbers. The diagram makes it plausible that there are real numbers which are neither rational nor irrational. One could also talk about algebraic numbers, which is a subfield of $\mathbb C$, which meets the irrationals as well. 
As for other number systems, let us overview a couple of the common ones:


*

*Ordinals, extend the natural numbers but they completely avoid $\mathbb{Z,Q,R,C}$ otherwise. 

*$p$-adic numbers extend the rationals, in some sense we can think of them as subset of the complex numbers, but that is a deep understanding in field theory. Even if we let them be on their own accord, there are some irrational numbers (real numbers) which have $p$-adic representation, but that depends on your $p$.

*You can extend the complex numbers to the Quaternions (and you can even extend those a little bit).

*You could talk about hyperreal numbers, but that construction does not have a canonical model, so one cannot really point out where it "sits" because it has many faces and forms.

*And ultimately, there are the surreal numbers. Those numbers extend the ordinals, but they also include $\mathbb R$.


Now, note that this diagram is not very... formal. It is clear it did not appear in any respectable mathematical journal. It is a reasonable diagram for high-school students, who learned about rationals and irrationals, and complex numbers.
I would never burden [generic] high-school kids with talks about those number systems above.
A: Strictly speaking $\mathbb{R}$ is not a subset of $\mathbb{C}$, rather it is isomorphic to a subfield of $\mathbb{C}$. Same for $\mathbb{Q}\subseteq\mathbb{R}$. Now $\mathbb{Z}$ is also isomorphic to a subring of $\mathbb{Q}$, not a proper subset. Whenever you have two algebraic structures $A$ and $B$ with respect to same binary operations, it may be possible to 'identify' $A$ with some subset of  of $B$, that means to show an isomorphism between $A$ and a subset of $B$ with respect to the defined opereations; it this case you can write $A\subseteq B$, in some loose sense.
