Complex number

A com­plex num­ber is a num­ber of the form \(z = a + b\textrm{i}\), where \(\textrm{i}\) is the imag­i­nary unit defined as \(\textrm{i}=\sqrt{-1}\).

Doesn’t make sense? Let’s back­track a lit­tle.

Mo­ti­va­tion—ex­pand­ing the defi­ni­tion of numbers

Say we start with a child’s un­der­stand­ing of num­bers as count­ing num­bers, or nat­u­ral num­bers. Us­ing only nat­u­ral num­bers we can add and mul­ti­ply whichever num­bers we want, but there are some re­stric­tions on sub­trac­tion: ask a child to sub­tract 5 from 3 and they’ll say “you can’t do that!”.

In or­der to al­low the com­pu­ta­tion of num­bers like \(5-3\), we need to define a new kind of num­ber: nega­tive num­bers. The set of nat­u­ral num­bers, nega­tive num­bers and \(0\) is called whole num­bers, or in­te­gers. Un­like nat­u­ral num­bers, with in­te­gers we can add and sub­tract what­ever we want. (In math terms, we say the set of in­te­gers is closed to ad­di­tion.)

We still have a prob­lem, though: what about di­vi­sion? Go­ing back to our child anal­ogy, say we’re talk­ing to a third grader. If you ask them to di­vide 8 by 2, no prob­lem. But what if you as them to di­vide 2 by 8? “You can’t do that!”

Just like with nega­tive num­bers, we need to define a new kind of num­ber: frac­tions, like \(\frac{1}{2}, \frac{5}{3}\) or \(-\frac{6}{7}\). By adding frac­tions to the in­te­gers, we get a more gen­eral set of num­bers—ra­tio­nal num­bers. With ra­tio­nal num­bers we can mul­ti­ply and di­vide how­ever we want—the set of ra­tio­nal num­bers is closed to mul­ti­pli­ca­tion. Well, ex­cept di­vid­ing by zero. No one has figured out how to do that with­out break­ing math­e­mat­ics, so that’s awk­ward.

Now say our child is a sev­enth grader, so they know about square roots. They can ap­ply the square root op­er­a­tion to any perfect square, no prob­lem - \(\sqrt{9}=3\). But what if you ask them to find the square root of a non-perfect square, like \(\sqrt{2}\)? “You can’t do that!”

Once again, we need to ex­pand our set of num­bers to in­clude a new kind of num­ber: ir­ra­tional num­bers. Un­like we dis­cussed with nega­tive num­bers or frac­tions, ir­ra­tionals can do a whole lot more that just define square roots. In fact, ir­ra­tionals are so cool that ac­cord­ing to some sources the Pythagore­ans couldn’t deal with their ex­is­tence and ended up kil­ling the guy who in­vented them.

Ad­ding ir­ra­tional num­bers to our set of ra­tio­nal num­bers, we get the real num­bers. Math­e­mat­i­cally, we say that the re­als are a com­plete or­dered field, which is ac­tu­ally one of the ways of defin­ing them.

How­ever, there’s still one prob­lem: the real num­bers are not an alge­braically closed field. In a prac­ti­cal sense, this means we still don’t have clo­sure un­der the square root op­er­a­tion \(\sqrt{}\), be­cause we can’t define the square roots of nega­tive num­bers.

By now you’ve prob­a­bly no­ticed the pat­tern, though: any time some­one says “you can’t do that!”, math­e­mat­i­ci­ans in­vent a new kind of num­ber to prove them wrong.

In­tro­duc­ing: imag­i­nary numbers

In or­der to al­low us­ing the square root op­er­a­tion on nega­tive num­bers, we once again have to define a new kind of num­ber: imag­i­nary num­bers.

Ac­tu­ally, we have to define just one num­ber—the imag­i­nary unit, \(\textrm{i}\). \(\textrm{i}\) is defined as a solu­tion to the quadratic equa­tion \(x^2+1=0\). In other words, we can define \(\textrm{i}\) as equal­ing \(\sqrt{-1}\).

All the other imag­i­nary num­bers (square roots of nega­tives) fol­low di­rectly from this defi­ni­tion of \(\textrm{i}\): for any nega­tive num­ber \(-a\), we can define \(\sqrt{-a}=\textrm{i}\sqrt{a}\).

To be con­tinued…

This ar­ti­cle is un­finished, and will later in­clude an ex­pla­na­tion of the com­plex plane as well as the alge­braic prop­er­ties of com­plex num­bers.

Finish ar­ti­cle by adding ex­pla­na­tions of the com­plex plane and the alge­braic prop­er­ties of com­plex num­bers.


  • Number

    An ab­stract ob­ject that ex­presses quan­tity or value of some sort.