2 Complex numbers

God cre­ated the nat­ural num­bers, all the rest is the work of man, the math­e­mati­cian Leopold Kro­necker is reported to have said. So let’s take it from where God left off. (The nat­ural num­bers are the ones we use for counting: 1, 2, 3, 4, and so on.)

By sub­tracting nat­ural num­bers from nat­ural num­bers we can create inte­gers (or whole num­bers) that are not nat­ural num­bers. For example, 4 − 5 = −1.

By dividing inte­gers by inte­gers we can create rational num­bers (frac­tions) that are not inte­gers. For example, 23.

By taking the square root of a pos­i­tive rational number (as well as by var­ious other means) we can create real num­bers that are not rational. For example, √2, the square root of 2. Every rational number can be written as a repeating dec­imal. For example, 13 = 0.3333333…, 3227555 = 5.8144144144…, 21 = 2.0000000…. A real number that is not rational (in other words, an irra­tional number) cannot be written in this way.

Finally, by taking the square roots of neg­a­tive real num­bers (as well as by var­ious other means) we can create imag­i­nary num­bers that are not real. For example, √(−1), the square root of –1. For this we use the letter i.

Do not be misled by the con­ven­tional labels “real” and “imag­i­nary.” No number is real in the sense in which, say, apple pie is real. Both the real num­bers and the imag­i­nary num­bers are cre­ations of the human mind. The former are no less imag­i­nary (in the ordi­nary sense of “imag­i­nary”) than the latter. So far you may not have had much use for imag­i­nary num­bers, but this is going to change.

To be able to count, we need nat­ural num­bers. For accounting we need rational num­bers. For mea­suring we need real numbers—though this is arguable, since no mea­sure­ment can be made with infi­nite pre­ci­sion. But for doing quantum physics we unar­guably need imag­i­nary num­bers. To be pre­cise, we need com­plex num­bers.

Think of a com­plex number c as an arrow in a plane. It has a mag­ni­tude, which is a real number c = |c|, and it has a direc­tion, spec­i­fied by an angle γ, which is called its phase. Pos­i­tive real num­bers are com­plex num­bers whose phase equals 0° — they point to the right or “east­ward.” Pos­i­tive imag­i­nary num­bers are com­plex num­bers whose phase equals 90° — they point to upward or “to the north.” Neg­a­tive real num­bers are com­plex num­bers whose phase equals 180° — they point to the left or “west­ward.” And neg­a­tive imag­i­nary num­bers are com­plex num­bers whose phase equals 270° — they point down­ward or “to the south.”

To add two com­plex num­bers (con­sid­ered as arrows), move the second arrow (without changing its mag­ni­tude or direc­tion) so that its tail coin­cides with the tip of the first. Now draw (or imagine) an arrow that extends from the tail of the first arrow to the tip of the second. This is the sum of the two com­plex num­bers. (It doesn’t matter which arrow is taken to be the “first” and which the “second.”)

The fol­lowing figure illus­trates how the sum of two com­plex num­bers depends on the angle between the cor­re­sponding arrows (or, what comes to the same, on the dif­fer­ence between their phases). Observe that in the right dia­gram the mag­ni­tude of the sum is smaller than the mag­ni­tude of one the com­plex num­bers being added.


vector addition

Figure 2.3.1 The sum (black) of two com­plex num­bers (gray, each shown twice) depends on the angle between them. The com­plex num­bers being added in the left dia­gram have the same mag­ni­tudes as those being added in the right dia­gram, yet their sums are quite different.


To mul­tiply two com­plex num­bers, all we have to do is to mul­tiply their mag­ni­tudes and add their phases. The square of the com­plex number c thus has the mag­ni­tude |c|2 and the phase 2γ. For the same reason the root of c has the mag­ni­tude √|c| and the phase γ/​2.

So what, according to these rules, is the square root of −1? Since −1 is neg­a­tive, its phase equals 180°, so the phase of i = √(−1) equals 90°. And since the mag­ni­tude of −1 equals 1, that of i also equals 1. Thus the square root of −1 is an arrow of unit mag­ni­tude pointing upward.

Let us write a com­plex number c in the form [c:γ]. In this nota­tion, c is short for the mag­ni­tude |c| of c, γ is the phase of c, and the product of two com­plex num­bers [a:α] and [b:β] equals [ab:α+β]. It is worth noting that if b = 1 then the product equals [a:α+β]. In other words, the effect of mul­ti­plying [b:β] by a com­plex number of mag­ni­tude 1 and phase α is to rotate [b:β] coun­ter­clock­wise by the angle α.