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Contact (1985)  
Carl Sagan 
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 course involves some mathematics. In
particular, the message is first recognized as being the creation of
intelligen 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 trapdoor
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. 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 nonzero 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^n1) 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

(This is just one work of mathematical fiction from the list. To see the entire list or to see more works of mathematical fiction, return to the Homepage.) 
Ratings:  Have you seen/read this work of mathematical fiction? Then click here to enter your own votes on its mathematical content and literary quality or send me comments to post on this Webpage.  
Mathematical Content: 3.75/5 (18 votes) 
 
Literary Quality: 4.8/5 (20 votes) 

Categories:  

Genre  Science Fiction, 
Motif  Aliens, Female Mathematicians, Math Education, Religion, 
Topic  Computers/Crytpography, Algebra/Arithmetic/Number Theory, Fictional Mathematics, 
Medium  Novels, Films, 
Warning: Shameless Self Promotion Be sure to check out Reality Conditions, the new book of short mathematical fiction by the guy who runs this Website. 
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(Maintained by Alex Kasman, College of Charleston)