# The Howling Fantods

David Foster Wallace News and Resources Since March 97

• • • ## Dubious Math in Infinite Jest

Dubious Math in Infinite Jest by Mike Strong

Before I outline the errors, I must make a logistical note.  Because I am writing this using a simple text editor, I am somewhat symbolically challenged.  Exponentiation will be denoted using a caret (e.g., 2^3 = 8).  Also, I will not even attempt to depict an integral sign or the usual mathematical symbol for combinations, a pair of elongated parenthesis.  I hope this makes the explanations no less understandable.

The first, and perhaps the most interesting error that I noted, appears on page 259 of IJ.  The narrator states that the odds of a 108 game tennis match ending in a 54-match-all tie are 1 in 2^27.  This is incorrect by about seven orders of magnitude; in fact, such an outcome is much more likely than the narrator suggests.  The problem of determining the odds is an exercise in probabilty and combinatorics, a fancy mathematical term for systematic counting.  By the laws of probability, the odds of a 108 game match ending in a tie are

(the number of outcomes resulting in a 54-54 tie) / (the total number of possible outcomes).

The denominator in this expression is easy to calculate.  It is simply 2^108 (an example will follow).  The numerator can be found using the concept of combinations.  That is, the correct anwer is the total number of ways team A can be assigned exactly 54 victories out of the 108 matches.  (Of course, if team A wins exactly 54 matches, so must team B.)  In combinatorics this is referred to as "the number of combinations of 108 things taken 54 at a time."  Although the derivation is well beyond the scope of this discussion, it can be shown that the number of combinations of n things taken m at a time is

n! / ((n-m)! m!).

Therefore, the numerator in the probabilty expression above is 108! / (54! * 54!), and the correct odds are

(108! / (54! *54!))/(2^108) or approximately 0.0766.

This is considerably greater than 1/2^27, or about 0.00000000745.  To illustrate this result, consider the following enumerable example:  What are the odds of a 4 game match ending in a 2-2 tie?  Again, the method above predicts that odds are

(4! / (2! * 2!))/(2^4)  =  (24 / (2 * 2))/(16)  = 6/16.

The answer suggested by the narrator of IJ would be 1/(2^1) = 1/2.  To see which is correct, we can list all 16 possible outcomes as follows:

AAAA   ABAA                BAAA                BBAA

AAAB   ABAB                BAAB                BBAB

AABA   ABBA                BABA                BBBA

AABB    ABBB                BABB                BBBB

Clearly, there are six outcomes resulting in a 2-2 tie (AABB, ABAB, ABBA, BAAB, BABA, and BBAA), so the odds are, indeed, 6/16 as the method above predicts.

The second significant instance of dubious math in IJ occurs in a footnote on pages 1023 and 1024.  In this section Mike Pemulis describes to Hal how the Mean Value Theorem for integrals can be used to distribute megatons of thermonuclear weapons among Eschaton combatants.  The Mean Value Theorem for integrals states that, for a function  f(x) that is continunous on the interval from x = a to x = b,

the integral from a to b of  f(x)dx = f(x')(b - a)  for some value x'  between  a and b.

In effect, this theorem simply states that the area underneath the curve described by the function  f(x) from x = a to x = b is exactly the same as the area of a rectangle whose width extends from a to b and whose height has a value of  f(x') for some value x'  between a and b.  Now, while Pemulis' description of the theorem is essential correct, the problem in this footnote is the way the theorem is supposedly applied.  Specifically, the Mean Value Theorem for integrals is a theoretical tool for proving the existence of this particular x'.  It does not, however, offer any method of finding the value of  x'.  Therefore, it is difficult to imagine how the Mean Value Theorem for integrals could be employed in Pemulis' Eschaton calculations.  Incidently, the third, minor error, which may very well be typographical, also occurs on page 1024 in this footnote.  Note that the abstract statement of the Mean Value Theorem for integrals which appears in the text (i.e.,  f(x)dx = f(x')(b - a) ) is missing the sign for the integral from a to b.

The final mathematical error that I noted occurs in a footnote on page 1063.  Again, Mike Pemulis is lecturing Hal, but this time he is helping Hal  prepare for the college board exams.  Pemulis states  that for the function x^n, the derivative is nx + x^(n-1).  In fact, the correct expression is nx^(n-1).  This, too, may be a typographical error.

As I have said, I have no theories to explain the existence of these errors.  I would, however, be interested in the thoughts of others.

Mike Strong
Last Updated on Tuesday, 16 March 2010 10:43