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Contact (1985)
Carl Sagan
Highly Rated!
Note: This work of mathematical fiction is recommended by Alex for hardcore fans of science fiction.

This is a fantastic novel; don't skip it just because you saw the movie. Mathematics plays an important role in the book, much more so than in the film. In both, Ellie Arroway detects a message from the star Vega using a massive array of radio telescopes. The detection and decryption of the message, of cours,e involves some mathematics. In particular, the message is first recognized as being the creation of intelligent beings rather than a natural phenomenon because it is a sequential list of prime integers. Hidden within this message is a movie, recognized as a three dimensional array of numbers because its length is the product of three primes. (Ellie notes that she knows of two uses of prime numbers in sending messages: one to make the message obvious as in this case, and the other to hide it as in a trap-door code.) In the movie, Jodie Foster (as Ellie) gives an interesting lecture on the prime numbers.

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.

Contributed by Piyush Singh

Can mathematicians truly rule out the possibility of pi actually being a recurring decimal?

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

Contributed by Ian Barral

An amazing work that has kept me intrigued for years. I've reommended many people to read this work and discussed the Pi bit many times including boring my children with it. At least they now know what base 11 is.

Contributed by Gene Ward Smith

Rather than giving me a sense of the numinous, the ending annoyed me because the logical reaction to Arroway's discovery is that the aliens are pulling a con job. First they proposed something which clearly makes no sense, and then they apparently jigger with her computer, to "confirm" that the impossible somehow happened, and God is sending us a Drake message in pi. A ruse on the part of the aliens is, I think, far more likely than that the circle really is there so early on (which is extremely unlikely) or that God put it there (which is impossible, since the digits follow from grade school arithmetic.) Unless you can change 1+1=2, you can't change pi, whose value is fixed by its definition in terms of arithmetic, and not, as Sagan would have it, "in the fabric of space and in the nature of matter", neither of which have anything to do with the value of pi.

The aliens gave us an IQ test, and we flunked. This may be to the good; possibly if we had passed they would regard us as dangerous. Or possibly they would have invited us to join galactic civilization. What's certainly clear is that the skeptical response to Arroway and the others by the authorities was warranted; their story was highly suspicious.

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.

Contributed by Nelson Walker

I just loved this book, and the whole idea that there could be ordered structure to pi. I reread the book once in a while just to read the conversation between arroway and her father, and the remainder of the book dealing with the analysis of pi. It reminds me of Rama Revealed - some kind of feeling that there is in fact some kind of definite order to the universe, if only we have the right tools to detect it.

Contributed by sacerdote

Just a pointer to a classic cartoon on the theme of embedded messages in pi.

Contributed by Simon Allen

Carl explored this idea with students that aliens might send 2D prints by using numbers that were coprime. I also recall he used a 3D construct to show how you could send a 3D graphic of a water molecule. I tried the 2D version on my astronomy group shown as a series of 1s and 0s. Sadly none of them were able to make any sense of it.

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Works Similar to Contact
  1. The Lure by Bill Napier
  2. Pi in the Sky by Rudy Rucker
  3. His Master's Voice by Stanislaw Lem
  4. Applied Mathematical Theology by Gregory Benford
  5. Calculating God by Robert J. Sawyer
  6. Pop Quiz by Alex Kasman
  7. Pieces of Pi by David Bartell
  8. Factoring Humanity by Robert J. Sawyer
  9. The Flight of the Dragonfly (aka Rocheworld) by Robert L. Forward
  10. The Center of the Universe by Alex Kasman
Ratings for Contact:
Mathematical Content:
3.73/5 (28 votes)
Literary Quality:
4.51/5 (32 votes)

GenreScience Fiction,
MotifAliens, Female Mathematicians, Math Education, Religion,
TopicComputers/Cryptography, Algebra/Arithmetic/Number Theory, Fictional Mathematics,
MediumNovels, Films,

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(Maintained by Alex Kasman, College of Charleston)