You are watching: Are integers a subset of rational numbers

Why did they pick to speak $piersonforcongress.combbZ subset piersonforcongress.combbR$ in piersonforcongress.com? In various other words, why space 2 and also 2.0 considered the same?

When you room working in $piersonforcongress.combbR$, does the make any kind of difference whether some elements, eg. 2.0, additionally belong to $piersonforcongress.combbZ$ or not?

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edited Dec 8 "14 at 19:37

cHao

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inquiry Dec 7 "14 at 10:30

AivarAivar

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If you desire to do things really formally, the integers room in reality

**not**a subset that the reals: they room entirely different constructs (which ns guess is more or much less what you room saying in your question). However, the reals do contain the set$$\,ldots,,-2.0,,-1.0,,0.0,,1.0,,ldots,$$which "looks just like" the integers. The usual terminology is the the sets space isomorphic. If girlfriend take any kind of true statement in the arithmetic of integers, and replace each integer by the corresponding real number, the an outcome will be a true statement about the real numbers. For example, the statement$$2+3=5$$corresponds to$$2.0+3.0=5.0 .$$

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answered Dec 7 "14 at 10:48

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Formally speaking, this counts on the context.

Sometimes it"s nice come have everything as a subset. It"s straightforward to have that an creature is a real number, since it allows us come talk about the subspace topology, and that the work coincide v what we intend them come be.

Sometimes it"s not as nice, in which situation we like to talk about "embedding", namely there is an embedding (which is distinct up to some properties preserved) which identify a subset that the real numbers through the integers.

If you desire to construct things native the ground up, then you"re right. You develop the natural numbers, climate you build the integers and then the rationals and finally the real numbers (and you can continue). Each step comes with some canonical embedding, i beg your pardon we can then extend and have a canonical means to identify the natural numbers v a subset of the real numbers, and so on.

But sometimes it"s nicer come say "Okay, now that we have $Bbb R$ and all those canonical subsets i m sorry behave prefer $BbbN,Z,Q$ and also so on, let"s redefine them together these subsets." now we can talk around subspaces directly and subrings and also subfields and so on and so forth.

Similarly you can want to have actually $Bbb R$ as a subset the $Bbb C$, and sometimes together a subspace (and thus a subset) that $Bbb R^3$; and sometimes friend will want to have actually these objects separated as "different types", and keep track on the embeddings.

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If you want to emphasis on the example of $2$, then keep in mind that $Bbb Z$ and $Bbb Q$ and also $Bbb R$ are all rings (whatever the means) which has actually a unit, and $1$ is that unit, for this reason each has a version of $1$ and $2$ is identified as a shorthand for $1+1$. Walk it matter where you carry out this addition? No, it doesn"t, because the term $1+1$ is syntactical in the language of rings, and will have similar properties in $Bbb Q$ and $Bbb R$, and also the exact same properties in $Bbb Z$. Because it doesn"t matter for the an easy properties that $2$ in which of this rings we take into consideration it, we have the right to just think about those ring coincide on these numbers.