Intro to Number Sets

There are sev­eral com­mon sets of num­bers that math­e­mat­i­ci­ans use in their stud­ies. In or­der from sim­ple to com­plex, they are:

  1. The nat­u­ral num­bers \(\mathbb{N}\)

  2. The in­te­gers \(\mathbb{Z}\)

  3. The ra­tio­nal num­bers \(\mathbb{Q}\)

  4. The real num­bers \(\mathbb{R}\)

  5. The com­plex num­bers \(\mathbb{C}\)

Each set is con­structed in some way from the pre­vi­ous one, and this path will show you how they are con­structed from the most ba­sic num­bers. You may have come across these terms in a math class that you at­tended, and may have had other defi­ni­tions given to you. In this path, you will ob­tain a firm, com­plete un­der­stand­ing of these sets, how they are con­structed, and what they mean in math­e­mat­ics.

Why are num­ber sets im­por­tant?

Be­fore we go any fur­ther though, it would be nice to know the mo­ti­va­tion be­hind defin­ing the num­ber sets first.

A set is a fancy name for a col­lec­tion of ob­jects. Some col­lec­tions of ob­jects have spe­cial prop­er­ties — such as the set of all blue things, which are spe­cial in that they’re all blue. In math, if a set of ob­jects all have a cer­tain prop­erty, we can make in­fer­ences about them — that is, there are cer­tain things we can say about them that you can de­duce log­i­cally. For ex­am­ple:

In a field, ev­ery nonzero num­ber has a mul­ti­plica­tive in­verse.

You don’t need to know what a field is yet (it’s a spe­cial type of set), but now you can make in­fer­ences about them with­out re­strict­ing your­self to a spe­cific ex­am­ple when talk­ing about them. For ex­am­ple, you know that if a set is a field, then ev­ery num­ber in that set that isn’t zero can di­vide into an­other num­ber in that set (by mul­ti­ply­ing by its “mul­ti­plica­tive in­verse”) and pro­duce yet an­other num­ber in that set.

Con­versely, you can also tell when a set is or isn’t a field based on whether it satis­fies the prop­er­ties a field has. For ex­am­ple, since you can’t di­vide \(3\) by \(2\) (be­cause the re­sult is \(1.5\) which is not a nat­u­ral num­ber), you now know that the nat­u­ral num­bers are not a field.

Now let’s turn to our first set: the nat­u­ral num­bers.


  • Mathematics

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