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, 2/3.

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, 1/3 = 0.3333333…, 3227/555 = 5.8144144144…, 2/1 = 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.

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.