Real numbers are of obvious importance in physics, and electrodynamics
is no exception. Measured or counted quantities are almost invariably
described in terms of real numbers or their embedded cousins, the
integers. Their virtue in physics comes from from the fact that they
form a (mathematical) *field*^{4.1} that is,
they support the mathematical operations of addition, subtraction,
multiplication and division, and it empirically turns out that physical
laws turn out to be describable in terms of algebraic forms based on (at
least) real numbers. Real numbers form a *group* under ordinary
multiplication and, because multiplication is associative and each
element possesses a unique inverse, they form a *division
algebra*^{4.2}

A division algebra is one where any element other than zero can be
divided into any other element to produce a unique element. This
property of real numbers is extremely important - indeed it is the
property that makes it possible to use algebra per se to solve for many
physical quantities from relations expressed in terms of products and
sums. The operational steps:

(4.1) |

are so pervasively implicit in our algebraic operations

In any event real numbers are ideally suited for algebra because they form a field, in some sense the archetypical field, whereby physical law can be written down in terms of sums and products with measurable quantities and physical parameters represented by real numbers. Other fields (or rings) are often defined in terms of either subsets of the real numbers or extensions of the real numbers, if only because when we write a symbol for a real number in an algebraic computation we know exactly what we can and cannot do with it.

Real numbers are the basis of real ``space'' and ``time'' in physics -
they are used to form an algebraic *geometry* wherein real numbers
are spatiotemporal *coordinates*. This use is somewhat presumptive
- spacetime cannot be probed at distances shorter than the *Planck
length* (
meters) - and may be quantized and
granular at that scale. Whatever this may or may not mean (close to
nothing, lacking a complete quantum theory of gravity) it makes no
meaningful difference as far as the applicability of e.g. calculus down
*to* that approximate length scale, and so our classical assumption
of smooth spacetime will be quite reasonable.

Are real numbers *sufficient* to describe physics, in particular
classical electrodynamics? The answer may in *some* sense be yes
(because classical measurable quantities are invariably real, as are
components of e.g. complex numbers) but as we will see, it will be *far easier* to work over a *different* field: complex numbers,
where we will often view real numbers as just the real part of a more
general complex number, the real line as just one line in a more general
complex plane. As we will see, there is a close relationship between
complex numbers and a two-dimensional Euclidean plane that permits us to
view certain aspects of the dynamics of the *real* number valued
measurable quantities of physics as the real projection of dynamics
taking place on the complex plane. Oscillatory phenomena in general are
often viewed in this way.