Integer

An in­te­ger is a num­ber that can be rep­re­sented as ei­ther a nat­u­ral num­ber or its ad­di­tive in­verse. −4, 0, and 1,003 are ex­am­ples in­te­gers. 499.99 is not an in­te­ger. In­te­gers are real num­bers; they are ra­tio­nal num­bers; and they are not frac­tions or dec­i­mals.

A Mathier Definition

In­stead of de­scribing the prop­er­ties of an in­te­ger we’ll de­scribe the mem­ber­ship rules for the set \(\mathbb{Z}\). After we’re done, any­thing that’s been al­lowed into \(\mathbb{Z}\) counts as an in­te­ger.

Start by putting \(0\) and \(1\) into \(\mathbb{Z}\). Now, pick an el­e­ment of \(\mathbb{Z}\), pick an­other el­e­ment of \(\mathbb{Z}\), and add them to­gether (you can pick the same el­e­ment twice). Is that num­ber in \(\mathbb{Z}\) yet? No?! Well let’s put it in there fast. We can do the same thing as be­fore ex­cept in­stead of adding, we sub­tract, and if the differ­ence isn’t in \(\mathbb{Z}\) yet, we put it in there. Any­thing that could be let into \(\mathbb{Z}\) with these pro­ce­dures is an in­te­ger.

This is not an effi­cient al­gorithm for build­ing out \(\mathbb{Z}\), but it does show the pri­mary mo­ti­va­tion for hav­ing in­te­gers in the first place. Nat­u­ral num­bers (pos­i­tive in­te­gers) are closed un­der ad­di­tion, mean­ing that if you add any two el­e­ments in the set, the sum will be in the set, but nat­u­ral num­bers are not closed un­der sub­trac­tion. In­te­gers are what you get when you ex­pand nat­u­ral num­bers to make a set that is closed un­der sub­trac­tion as well.

For­mal construction

Given ac­cess to the set \(\mathbb{N}\) of nat­u­ral num­bers, we may con­struct \(\mathbb{Z}\) as fol­lows. Take the col­lec­tion of all pairs \((a, b)\) of nat­u­ral num­bers, and take the quo­tient by the equiv­alence re­la­tion \(\sim\) such that \((a,b) \sim (c,d)\) if and only if \(a+d = b+c\). (The in­tu­ition is that the pair \((a,b)\) stands for the in­te­ger \(a-b\), and we take the quo­tient so that any given in­te­ger has just one rep­re­sen­ta­tive.)

Writ­ing \([a,b]\) for the equiv­alence class of the pair \((a,b)\), we define the or­dered ring struc­ture as:

  • \([a,b] + [c,d] = [a+c,b+d]\)

  • \([a, b] \times [c, d] = [ac+bd, bc+ad]\)

  • \([a,b] \leq [c,d]\) if and only if \(a+d \leq b+c\).

This does define the struc­ture of a to­tally or­dered ring (proof).

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  • Mathematics

    Math­e­mat­ics is the study of num­bers and other ideal ob­jects that can be de­scribed by ax­ioms.