However, transcendental real numbers also play an important role in the book, without any analogue in the film. Ellie's intelligence is exemplefied by her reaction to learning about the decimal expansion of the number Pi. (We also learn a bit about her math teacher -- bad as usual.) Then, towards the end of the book, there is an absolutely beautiful, amazing piece of fictional mathematics: she finds a message hidden in the decimal expansion of the number pi. If you are going to read the book, then you'll find out there. If you do not plan to read the book, check this out. (The link will take you to a short description I wrote of the conclusion of the novel Contact as well as a criticism from Mike Hennebry (NDSU) concerning the implications.)

Strangely, despite Sagan's outspoken skepticism and agnosticism, the other underlying theme of this book is religious. Though science and religion seem very different at the beginning of the book, by the end they are almost the same. Whatever your views on religion and science, reading this thought provoking book with an open mind will provide you with ample opportunity to question your beliefs.

Yes, they can. First, you have to realize that if the decimal expansion of a number repeats then the number must be rational (i.e. the ratio of two integers). In fact, it is easy to see if the decimal expansion is all zeroes from some point on, because if the number q is all 0 from the nth place on, then when you multiply q by 10^n you get an integer, and so q is the integer (q*10^n) divided by the integer 10^n. It is only slightly more complicated if the number q has a repeating, but non-zero tail. In this case, even though 10^n * q is not an integer, you can pick the number n so that (10^n * q) - q has a tail that is all zeros. (Think about it, if the portion that repeats is n digits long, then 10^n*q and q both have exactly the same "tail" from some point on and so their difference ends in all zeros.) This means that Q=(10^n*q)-q is a rational number. But then we can solve for q to get q=Q/(10^n-1) which is also a rational number.

That's the relatively easy part, noticing that any number which has a decimal expansion with a tail that repeats is rational. The harder part is showing that pi is not a rational number. This is rather difficult to prove, and was not known until 1768 when Lambert, using advanced techniques for his day, showed that the number e raised to any rational power is irrational, and concluded from this that pi is also irrational. (See this biography for more details about Lambert and his proof.) A modern, and very short, proof of the irrationality of Pi can be found here.

In any case, since we know that any number with a repeating decimal tail is rational, the fact that pi is irrational means that it does not have a repeating decimal tail. This does not mean that there is no simple pattern to it. For instance, the number .1010010001000010000010000001.... for which the number of zeroes increases by one each time is irrational, but there is obviously a simple pattern to it. Still, even though we have many different formulas for computing the digits of pi (including one by Borwein et al that can compute an arbitrary digit in the hexadecimal or binary expansion without computing the earlier digits), the expansion of pi appears essentially random and therefore generates a great deal of interest at the boundaries of philosophy and number theory.

Hope that helps! -Alex

Gene, I think you're not being open minded enough. Yes, it is hard to imagine how anyone could put a message into the decimal expansion of pi...but that is exactly why it seems amazing to me. I'm not claiming I believe it is possible or that I understand what it would mean. Rather, I'm saying that if someone showed me it was true, I would be amazed because I cannot imagine how it would be possible. It would force me to rethink my worldview. As you say, it is not something about changing the physics of the universe, which I could more easily imagine, but rather changing mathematics itself! Okay, if someone tried to convince me right now that this was the truth, I would approach it with a great deal of skepticism. But, the purpose of the story is to make you think "what if...?" Try to open up enough to the possibility that you can be impressed by it rather than rejecting it outright and you may find yourself in touch with the "numinous" as well.