(Note that this post contains spoilers for both the movie and the book; though if you've seen the movie, the spoilage of the book is not going to be enough to impair your appreciation of the book.)
Carl Sagan did a lot of work on the question of communication with aliens, both in his work on SETI for receiving and decoding, and on various projects for sending, most visibly the Pioneer plaque and Voyager golden record. Always the answer is mathematics and science, because always we must conclude that it is the only thing we know we'll share in common with pretty well any spacefaring (or even just interstellar-communicating) species. My first exposure to this concept was the seminal SF short story "Omnilingual", by H. Beam Piper. We see this referred to in the film three times:
- The frequency of the transmission is "hydrogen times pi". What kind of number is hydrogen? Well, Ellie is referring to the hydrogen line, which is a frequency itself -- thus neatly avoiding any problems of whether we're using the same units as the aliens. (To explain that: suppose I asked you to build a stick the length of a hydrogen atom times the number of protons in uranium, squared. If you work in metric and I use furlongs, our sticks will still end up the same length.)
- The prime numbers are proof that the message cannot be caused by a natural phenomenon. And therefore, proof that it originated from an intelligence, because it takes an intelligence to understand what prime numbers are.
- The primer found in the message, briefly glimpsed, becomes the key to deciphering it. It is a series of mathematical statements that can easily be broken down into components: addition, equality, truth, falsehood, and numbers. Combine that with a few pictures and you can start building up a complicated language. For instance, a representation of an atom, then the number six, then the symbol for equals, then an unknown symbol. What does the unknown symbol represent? (Check your answer.)
In the book, one of the things Ellie talks to the aliens about is God, and they reveal that, essentially, the Creator has chosen not to reveal himself until a species reaches a necessary level of understanding about the universe around them. How has this been done? Brilliantly, by using the same techniques of math as a universal language to hide a message to us in the structure of the universe itself.
The number pi is a fundamental constant of our universe. (Properly speaking, it is not really a constant because relativistically space is curved; however, it is nevertheless a fundamental number that any technological civilization must become aware of, even if only as the limit of the ratio of circumference to diameter as the curvature of space approaches zero.) What if you looked at the digits of pi and found, only a few million digits in, a series of 0s and 1s precisely 121 digits in length (121 being the product of primes, 11 times 11), such that if you drew them in an 11x11 grid, it depicted a circle? Furthermore, what if it turned out that this happened in any numerical base you evaluated pi in?
Technically speaking, that that particular sequence of numbers appears in pi isn't really that remarkable. We know that pi's digits go on forever and never repeat. It can thus be assumed that any given sequence of numbers is out there somewhere if you go far enough, because that's how infinity works (same as the oft-misunderstood monkeys and typewriters theory). Somewhere in pi is an ASCII representation of this blog post I'm typing now. Somewhere in pi is a complete description of every atom that makes your body. And this is true no matter what numerical base you work in. (Though even more technically speaking, this is not necessarily true; we can't mathematically prove pi contains every sequence of numbers. But it is widely believed nevertheless.)
What makes this discovery important, though, is that it occurs very early in pi, about the same number of digits in, in any base. It's all well and good to say theoretically any given sequence of numbers must appear somewhere in pi merely because it's infinite. But practically speaking, even if your computers could produce a million digits a second, the odds of ever finding a particular 121-digit sequence during the lifespan of the universe is very low, and the odds of finding the same sequence in each of many numerical bases approaches nil. One must very nearly conclude that it's there because it was put there. But pi cannot be changed by any being within the universe, and therefore, it must have been put there by a Creator whose existence is outside this universe. It's just like how the primes are proof that the signal is intelligent in origin, only now it's the universe itself that's proven to be intelligent in origin.
This is perhaps an even more dizzying discovery for Ellie to make than the fact that there are aliens.