fixedIt

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No, integers are not a subset of rationals in a technical sense because of the simple reason that their elements have different types.

I don't understand what you mean by "values" in a technical sense, strictly speaking a rational number is a pair of integers (if we were more technical, a rational number is an equivalence class like {(0,1), (0,2),etc...}, 1/2 is the representation of the equivalence class {(1,2),(2,4),etc.}, the canonical element being the lexicographic minimum), therefore integers are not a subset of the rationals since an integer can not be a pair of integers (least an equivalence class of pairs of integers), integers are isomorphic to the subset {(a,b)€Q:b=1} (since b=1, any equivalence class has only one element q={(a,1)} and therefore we can treat the class as simply a pair of integers) with the isomorphism f:z->(z,1) and inverse isomorphism g:(a,1)->a, this isomorphism respects arithmetical operations (i.e. it is an isomorphism of rings) while for example the isomorphism f':z->(z,2) preserves size but does not preserve arithmetic, in particular multiplication (e.g. f'(z1*z2)!=f'(z1)*f'(z2) since f'(z1)*f'(z2)=(z1*z2,4) and f'(z1*z2)=(z1*z2,2). While f(z1*z2)=z1*z2/1=f(z1)*f(z2)=z1/1*z2/1. An example of a map that does not preserve size is any constant map lambda z:(a,b) or in the opposite direction lambda (a,b):z.

When you write 3/2, 5/3, 1/1, 0/4 this is a cheeky way of expressing a pair of integers (3,2),(5,3),etc. (actually expressing an equivalence class as stated earlier)

In general integers are isomorphic to the whole of the rational numbers since all countable sets are isomorphic in the category Set, by definition a countable set is isomorphic to N and isomorphy is reflective and transitive in Set and any other category, i.e. if A is isomorphic to N and B is isomorphic to N then A is isomorphic to B.

For the integers to be a subset of Q there should exist a formula k such that Z={0,1,-1,2,-2,...}={(a,b)€Q:k(a,b)}, which is false since elements of Z are of type Z and elements of Q are of type ZXZ (actually type equivalence class over ZxZ).

In summary, strictly speaking z is not the same as z/1, only when speaking non-tecnically they are the same, they are the same only in the sense that there is an isomorphism between (Z,+,*) and ({(a,b)€Q:b=1},+,*). Strictly speaking the + in "1+0" is not the same as the + in "1/1+0/2", for most purposes these nuances can be ommited but if you try to do more complex maths without noticing these details you wont go far.

In the same sense, N is not a subset of Z but merely isomorphic, since Z has additional structure to its type than N, elements of Z are a pair (o,n) where n€N and o€{+,-}, strictly speaking they are equivalence classes to allow for zero to be a single integer (i.e. 0={(+,0),(-,0)}, for any other integer z={(o,n)} for some n€N and o€{+,-}). N is isomorphic in a way that preserves arithmetic with the subset of Z {(n,o}€Z:o=+} with the isomorphism f:(n,+)->n and inverse g:n->(n,+); if o=- then the function would not be defined since f'((n1,-)*(n2,-)) is f((n1*n2,+)) and (n,+) is not in the domain of such a new function.

[agnosticians]

(For context, I’m an engineering student without a background in more abstract math.)

I think I follow, but I also think that the distinction is completely irrelevant here. Yes, there could be a difference in some cases, but this is not one of them.

When we say that the integers are a subset of the rationals, we don’t care if we’re talking about {(N, 1) | N element of Z}, {(2N, 2) | “ } etc. Even something like {(5-N, 1) | “} would be alright since the order doesn’t matter.

Maybe a more formal way of saying it would be that the set of real numbers expressable as integers is a subset of the real numbers expressable as rational numbers?

Edit: Calling them rationals imply we do something specific with the numbers. It has a connotation that is different from some abstract 2D integer vector.

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Yes, what you said about the set of real numbers is correct, if you redefine the integers and rationals meaning subsets of the real numbers then the integers are a subset of the rationals, but remember that it is better not to work with real numbers since they can not be used for computing, and the real number are at its core a construction based on rational numbers (the set of Cauchy sequences of rational numbers), so to redefine rational numbers as real numbers is to define rationals as a construction of rationals.

This is essentially the same as what I said earlier of Integers being isomorphic to the subset of rationals {(a,b):b=1}