Constructing a number system I have just started working through a book on higher algebra. I'm just at the beginning, where the authors introduce the notation and talk about the various number systems. 
I found this particular paragraph confusing:- "The basic idea in the construction of new set of numbers is to take a set, call it $ S $, consisting of mathematical objects, such as numbers you are already familiar with, partition the set $ S $ into a collection of sets in a suitable way, and then attach names or labels, to each of the subsets. These subsets will be elements of a new number system."
What does the author mean, when he says a "suitable way" here? Does it mean, that I can partition in any way that I find suitable, or are there requirements to be met, for any number system that is constructed by me?
For instance, I'm familiar with the set of natural numbers. So, can I construct $S= ${$1,2,3,4,5,6,7,8,9,10$} and call it a subset of a new number system? Can I go as far as to say that this subset is the only element of my new number system?
 A: Maybe an example would help. We're all familiar with the natural numbers $\mathbb{N} = \{0, 1, 2, \ldots \}$. Then we also have the set of ordered pairs of natural numbers $\mathbb{N} \times \mathbb{N}$. If we partition $\mathbb{N} \times \mathbb{N}$ using the equivalence relation $(a, b) \sim (c, d)$ when $a + d = b + c$, then this equivalence relation partitions $\mathbb{N} \times \mathbb{N}$ out into subsets. You can think of each subset as containing pairs $(a, b), (c, d), \dots$ where $a - b = c - d = \ldots$. So each subset can be thought of as an integer by picking any pair $(a, b)$ out of it and thinking of it as the integer $a - b$.
But this is a bit circular, since the point is that we imagine we didn't formally have the integers yet, but we had an intuition for them, and just formally constructed them from $\mathbb{N}$. We just came up with a new number system, and attached names to the subsets; for example, the subset that contains the pair $(1, 4)$ can be called $-3$.
So suitable means it fulfils some sort of intuition we have about something we think exists in a real-world sense, or at least has some abstract reality. We can come up with silly, meaningless number systems which are technically equally valid, but mean less to human beings. "Suitable" isn't a mathematical term, it's a social and psychological one.
A: The author is attempting to informally convey the notion that many constructions of number systems begin by taking a quotient set of a known number system, i.e. by defining an equivalence relation on the set, which is equivalent to a partition of the set (into equivalence classes). For example, fractions are constructed as pairs of integers quotiented by the equivalence relation $(a,b) \cong (c,d) \stackrel{\rm def}\iff ad = bc,\,$ where the class of $\,(a,b),\ b\ne 0\,$ represents the fraction $\,a/b.\,$  Another well-known example is the ring of integers mod $\,n\,$ whose elements are equivalence classes $\,a + n\,\Bbb Z\,$ of the equivalence relation $\, a\equiv b\pmod n\stackrel{\rm def}\iff n\mid a-b.\,$  
That the equivalence relation (or partition) is constructed in a "suitable way" means that we usually desire addtitionally that the quotient object preserves some (or all) of the algebraic structure of the base number systems, i.e. that the algebraic operations are inherited by the quotient object. Thus we want the integers mod $\,n\,$ to be a quotient ring of $\,\Bbb Z,\,$ not merely a quotient set, which is equivalent to desiring that the equivalence realtion be, additionally, a congruence, i.e. $\,a\equiv a',\ b\equiv b'\,\Rightarrow\, a+b\equiv a'+b'$ and similarly for all other ring operations. 
But before studying the more complex algebraic quotient objects, it is wise to first study the simpler case of quotient sets, because they are the building blocks, or raw materials, out of which the more complex algebraic quotient structures are constructed.
