This answer is just adding flesh to @Frank Thorne's earlier answer. He noted that the idea of "addition and multiplication interacting" comes up in additive combinatorics. Perhaps the most obvious instance of this is in the study of the **sum-product phenomenon** (SPP).

Roughly speaking the SPP asserts that any subset of a field $F$ must ``grow'' under either addition or multiplication. The way one makes this rough statement precise depends on the field in question. Let me consider two instances:

**Suppose $F=\mathbb{R}$.** In this situation the central conjecture is due to Erdos and Szemeredi:

For every $\varepsilon\in (0,1)$, there exists $c>0$, such that for $A$ a finite subset of $\mathbb{R}$,
$\max(|A+A|, |A\cdot A|) \geq c |A|^{2-\varepsilon}$.

This conjecture is still open, however progress has been made. The strongest statement is (I believe) due to Solymosi, but it's also worth mentioning the work of Elekes. With a very simple argument, he connected SPP to questions in incidence geometry in the plane and to the idea of the **crossing number** in $\mathbb{R}^2$ to prove:

There exists $c>0$, such that for $A$ a finite subset of $\mathbb{R}$,
$\max(|A+A|, |A\cdot A|) \geq c |A|^{5/4}$.

One last remark - another way of thinking about the Erdos-Szemeredi conjecture is this: it says that a set $A$ of real numbers cannot simultaneously be both a geometric progression and an arithmetic progression (since, by results of Freiman and others, these are the classes of sets that do not grow under multiplication and addition, respectively).

**Suppose $F=\mathbb{Z}/p\mathbb{Z}$.** In this situation, the central result is due to Bourgain, Katz and Tao:

For every $\delta>0$ there exists $\varepsilon>0$ and $c>0$, such that for $A$ a finite subset of $\mathbb{Z}/p\mathbb{Z}$ with $|A| < p^{1-\delta}$, we have
$\max(|A+A|, |A\cdot A|) \geq c |A|^{1+\varepsilon}$.

The statement is slightly different to that in $\mathbb{R}$ because it is clear that sets that are almost as large as the field itself cannot possibly grow.

This result has been generalized in various ways to arbitrary finite fields. However in this more general setting one has to deal with the presence of finite subfields (again this does not crop up in $\mathbb{R}$), and so statements tend to be slightly technical. There is also a wealth of work giving values for $\varepsilon$ when $\delta$ is, say, $\frac12$, as well as a lot of work connecting this result to geometry over finite fields (in the spirit of the work of Elekes).

notcomputationally better) approach is to treat elements of $\mathbb{F}_{p^r}$ as matrices over $\mathbb{F}_p$: the additive and multiplicative structures coexist nicely then. This approach reduces much of the theory of finite fields concerned with the interplay of which you speak to representation theory. It is detailed in a cute article by Wardlaw: jstor.org/discover/10.2307/2690850 – Steve Huntsman Mar 16 '13 at 23:05