The Simpsons and Their Mathematical Secrets

by Simon Singh

  Similarly, Google’s ubergeek bosses are huge fans of e. When they sold stock in 2004, they announced that they planned to raise $2,718,281,828, which is $1 billion multiplied by e. That same year the company erected the following billboard advertisement:

  The only way to find the name of this website was to search through all the digits of e to discover a sequence of ten digits that represented a prime. Anyone with sufficient mathematical ingenuity would have discovered that the first ten-digit prime, which starts at the ninety-ninth digit of e, is 7427466391. Visiting the website www.7427466391.com revealed a virtual signpost that pointed toward another website that was a portal for those who wanted to apply for positions at Google Labs.19

  Another way to express admiration for e is to memorize its digits. In 2004, Andreas Lietzow from Germany memorized and then recited 316 digits while juggling five balls. However, Lietzow was spectacularly trumped on November 25, 2007, when Bhaskar Karmakar from India, unencumbered by balls, set a new world record by reciting 5,002 digits of e in 1 hour 29 minutes and 52 seconds. That same day he also accurately recited 5,002 digits of e backward. These are incredible feats of memory, but we can all memorize ten digits of e by learning this mnemonic: “I’m forming a mnemonic to remember a function in analysis.” The numbers of letters in each word represent the digits of e.

  And, finally, the writers of The Simpsons are passionate about e. Not only does it appear as part of a book title in “MoneyBART,” it also receives a special mention in “The Fight Before Christmas” (2010). The final segment of the episode is in the style of Sesame Street, so it ends with the traditional sponsorship announcement. However, instead of something along the lines of “Today’s episode of Sesame Street has been brought to you by the letter c and number 9,” viewers were treated to “Tonight’s Simpsons episode was brought to you by the symbol umlaut, and the number e; not the letter e, but the number whose exponential function is the derivative of itself.”

  CHAPTER 12

  Another Slice of π

  In “Marge in Chains” (1993), Marge is arrested for shoplifting after she walks out of the Kwik-E-Mart having forgotten to pay for a bottle of bourbon. She is put on trial and is represented by the attorney Lionel Hutz, a man with a dubious reputation. Before Marge’s trial begins, Hutz admits that it is likely to be an uphill battle because of his poor relationship with the judge: “Well, he’s had it in for me ever since I kinda ran over his dog . . . Well, replace the word kinda with the word repeatedly, and the word dog with son.”

  Hutz’s strategy for defending Marge is to discredit Apu Nahasapeemapetilon, proprietor of the Kwik-E-Mart, who witnessed the alleged theft. However, when he calls Apu to the witness box and suggests that his memory might be flawed, Apu’s response is to point out that he has a perfect memory: “In fact I can recite pi to forty thousand places. The last digit is 1.”

  Homer is not impressed, and merely thinks to himself: “Mmm . . . pi(e).”

  Apu’s extraordinary claim that he has memorized π to forty thousand decimal places only makes sense if mathematicians had determined π to at least that degree of accuracy. So, when the episode was broadcast in 1993, what was the state of play with respect to calculating π?

  We saw in chapter 2 how mathematicians, from the ancient Greeks onward, used the polygon approach to establish increasingly precise values for π, which eventually gave them a result accurate to thirty-four decimal places. By 1630, the Austrian astronomer Christoph Grienberger was using polygons to measure π to thirty-eight decimal places. From a scientific perspective, there is literally no point in identifying any more digits, because this is sufficient for completing the most titanic astronomical calculation conceivable with the most refined accuracy imaginable. This statement is not hyperbole. If astronomers had established the exact diameter of the known universe, then knowing π to thirty-eight decimal places would be sufficient to calculate the universe’s circumference accurate to within the width of a hydrogen atom.

  Nevertheless, the struggle to measure π to more and more decimal places continued. The challenge took on an Everest quality. The number π was an infinite peak in the mathematical landscape, and mathematicians tried to scale it. There was, however, a change in strategy. Instead of using the slow polygon approach, mathematicians discovered several formulas for determining the value of π more quickly. For example, in the eighteenth century Leonhard Euler discovered this elegant formula:

  It is remarkable that π can be deduced from such a straightforward pattern of numbers. This equation is known as an infinite series, because it consists of an infinite number of terms, and the more terms included in a calculation, the more accurate the result. Below are the results of calculating π using one, two, three, four, and five terms of Euler’s series:

  The approximations approach from below the true value of π, with each result becoming slightly more accurate as each extra term is introduced. After five terms, the estimate is 3.140, which is already accurate to two decimal places. Then, after one hundred terms, π can be determined accurately to six decimal places: 3.141592.

  Euler’s infinite formula is a reasonably efficient method for calculating π, but subsequent generations of mathematicians invented other infinite series that approached the true value of π even more rapidly. John Machin, who was professor of astronomy at London’s Gresham College in the early eighteenth century, developed one of the fastest, albeit less elegant, infinite series.20 He shattered all previous records by measuring π to one hundred decimal places.

  Others exploited Machin’s infinite series with even greater verve, including an English amateur mathematician named William Shanks, who devoted most of his life to calculating π. In 1874, he claimed to have calculated 707 digits of π.

  In honor of his heroic achievement, the science museum in Paris known as the Palais de la Découverte decorated its Pi Room with an inscription of all 707 digits. Unfortunately, in the 1940s it was discovered that Shanks had made an error while calculating the 527th decimal place, which impacted on every subsequent digit. The Palais de la Découverte called in the decorators and Shanks’s reputation took a knock. Nevertheless, 526 decimal places was still a world record at the time.

  After the Second World War, mechanized and electronic calculators took over from the pencil and paper used by Shanks and previous generations of mathematicians. The power of technology is illustrated by the fact that Shanks spent a lifetime calculating 707 digits of π, 181 of which were wrong, while in 1958 the Paris Data Processing Center performed the same calculation without error on an IBM 704 in forty seconds. Although π’s digits were now set to tumble at an accelerating rate, the level of excitement among mathematicians was tempered by the realization that even computers could not tackle an infinite task.

  This fact was a plot point in the 1967 Star Trek episode “Wolf in the Fold.” In order to exorcise an evil energy force that has occupied the USS Enterprise’s computer, Spock issues the following command: “Computer—this is a Class A compulsory directive. Compute to the last digit the value of π.” The computer is so distraught by this request that it cries out “No” over and over again. Despite its distress, the computer must obey the directive, and the resulting computational impossibility somehow purges the circuits of the evil force.

  Spock’s genius in “Wolf in the Fold” more than makes up for some appalling innumeracy displayed by Captain James T. Kirk in another episode earlier that same year. In “Court Martial,” one of Kirk’s crewmen has gone missing on board the Enterprise, and nobody is sure if he is alive or dead. Kirk, who would be held responsible for the crewman’s fate, decides to use the computer to search for the missing man’s heartbeat. He explains his plan: “Gentlemen, this computer has an auditory sensor. It can, in effect, hear sounds. By installing a booster, we can increase that capability on the order of one to the fourth power.” Of course, 14 is still 1.

  Shortly after the French computer scientists calculated 707 digits in less than a minu
te, the same team used a Ferranti Pegasus to calculate 10,021 digits of π. Then, in 1961, the IBM Data Processing Center in New York computed π to 100,265 digits. Inevitably, bigger computers led to more digits, and the Japanese mathematician Yasumasa Kanada calculated π to two million decimal places in 1981. The eccentric Chudnovsky brothers (Gregory and David) built their own DIY supercomputer in their Manhattan apartment and broke the billion-digit barrier in 1989, but they were overtaken by Kanada, who cracked fifty billion digits in 1997 and then one trillion digits in 2002. At present, Shigeru Kondo and Alexander Yee are top of the π chart.

  This duo reached five trillion digits in 2010, and then doubled the record to ten trillion digits in 2011.

  Thus, returning to the courtroom, Apu could easily have had access to the first forty thousand decimal places of π, because mathematicians had calculated beyond this level of accuracy by the early 1960s. However, is it also possible that he could have memorized forty thousand decimal places?

  As mentioned previously, in the context of e, the best approach for remembering a handful of digits is to rely on a phrase such that each word contains the relevant number of letters. For example, “May I have a large container of coffee” gives 3.1415926. “How I wish I could recollect pi easily today!” gives one more digit. The great British scientist Sir James Jeans, in between pondering deep questions about astrophysics and cosmology, invented a phrase that offers seventeen digits of π: “How I need a drink, alcoholic of course, after all those lectures involving quantum mechanics.”

  Several memory experts have extended this technique. They can recount π by telling themselves long, elaborate stories, with the number of letters in each word reminding them of the next digit of π. This technique enabled Canadian Fred Graham to break the 1,000-digit barrier in 1973. By 1978, American David Sanker cracked 10,000 digits, and in 1980, an Indian-born British mnemonist named Creighton Carvello recited π to 20,013 digits.

  A few years later, British taxi driver Tom Morton also tried to memorize 20,000 digits, but he stumbled at 12,000 digits, because there was a printing error on one of the cue cards that he was relying on during his preparation. In 1981, the Indian memory expert Rajan Mahadevan broke the 30,000-digit barrier (31,811 digits to be precise), and Japanese mnemonist Hideaki Tomoyori set a new world record of exactly 40,000 digits in 1987. Today, the record holder is Chao Lu from China, who memorized 67,890 digits in 2005.

  However, it was Tomoyori’s 40,000-digit record that was in place when the script for “Marge in Chains” was being finalized in 1993. Hence, Apu’s claim to have memorized π to 40,000 digits was a direct reference and tribute to Tomoyori, who was the world’s most famous and successful π memory expert at the time.

  This episode was written by Bill Oakley and Josh Weinstein. According to Weinstein, the overall plot of “Marge in Chains” had already been outlined by the time it was assigned to Oakley and himself: “We were the junior writers, so we were assigned scripts that other people didn’t want to do. Scripts revolving around Marge are very hard to write. By contrast, Homer is instantly funny, and so is Krusty. But Marge is really hard work, so her storylines were often pawned off on the new guys, like us.”

  Weinstein and Oakley took the basic storyline for “Marge in Chains,” developed the plot details, wrote the core jokes, and handed in their draft script. Importantly, when we met, Weinstein was anxious to point out that this version of the script contained absolutely no mention of π.

  He explained that the scene with Apu in the witness box began with the attorney Lionel Hutz asking the same question that still appears in the transmitted episode: “So, Mr. Nahasapeemapetilan, if that is your real name, have you ever forgotten anything?”

  However, instead of claiming that he could recite π to forty thousand places, Apu revealed that he had been famous across India for his incredible memory. In fact, in the original script, Apu stated on oath that he had been known as Mr Memory and had appeared in over four hundred documentary films about his mental ability.

  It is perhaps not surprising that the original script for “Marge in Chains” had no mention of π or forty thousand digits, as neither Oakley nor Weinstein have mathematical backgrounds. So, when did the mathematical references appear in the script?

  As usual, the first draft script was dissected and discussed by the rest of the writing team in order to refine the story and inject additional humor wherever possible. At this point, Weinstein and Oakley’s colleague Al Jean saw an opportunity to add some mathematics to the episode. Thanks to his lifelong interest in mathematics, Jean was aware that the world record for memorizing π was forty thousand decimal places, so he suggested altering the script so that Apu makes a claim that matches the memorization record. And, to give the claim some credibility, Jean suggested that Apu should cite the forty-thousandth decimal place.

  Everyone agreed that this was a good idea, but nobody happened to know the forty-thousandth decimal place of π. Worse still, it was 1993, so the World Wide Web was only sparsely populated, Google did not exist, and searching Wikipedia was not yet an option. The writers decided they needed some expert advice, so they contacted a brilliant mathematician named David Bailey, who at the time was working at the NASA Ames Research Center.21 Bailey responded by printing out all forty thousand decimal places of π and mailing them to the studio. Here are the digits from the 39,990th through to the 40,000th decimal place, and you can see that Apu is correct when he says that the last digit in his memorized sequence is 1:

  The fact that Bailey made his contribution as a mathematician based at NASA was referenced three years later in “22 Short Films About Springfield” (1996). When Barney Gumble, Springfield’s favorite drunk, stumbles into Moe’s Tavern, he finds that Moe has some bad news for him: “Remember when I said I’d have to send away to NASA to calculate your bar tab? . . . The results came back today. You owe me seventy billion dollars.”

  Apu’s line about π in “Marge in Chains” also influenced another episode, namely “Much Apu About Nothing” (1996). In this episode, Apu reveals some of his backstory, and his past has to be compatible with someone who would be interested in memorizing π to 40,000 decimal places. Hence, when he recalls his journey from India to America, Apu tells Marge: “I came here shortly after my graduation from Caltech. Calcutta Technical Institute. As the top student in my graduating class of seven million.”

  Although the Calcutta Technical Institute is fictional, there is a technical institute near Calcutta named the Bengal Institute of Technology, which perhaps could claim to be the inspiration for Apu’s alma mater. It has the acronym BIT, which is highly appropriate for a college that specializes in computer science and information technology. We also learn that Apu went to America to study at the Springfield Heights Institute of Technology, which has a rather less fortunate acronym. Under the supervision of Professor Frink, Apu spent nine years completing his PhD in computer science by supposedly developing the world’s first tic-tac-toe program, which could only be beaten by the best human players.

  David S. Cohen, who wrote “Much Apu About Nothing,” decided that Apu should be a computer scientist rather than a mathematician, because Cohen himself had been a graduate student in computer science at the University of California, Berkeley, and had shared classes with several Indian students. In particular, Apu’s backstory is based on the life of one of Cohen’s closest friends at Berkeley, Ashu Rege, who went on to work for NVIDIA, a pioneering computer graphics company.

  Pi has made one more notable appearance on The Simpsons. In the concluding scenes of “Lisa’s Sax” (1997), we learn that Homer bought Lisa a saxophone in order to nurture her nascent genius. However, before investing in a musical instrument, Homer and Marge considered sending Lisa to Miss Tillingham’s School for Snotty Girls and Mama’s Boys. In a flashback, we see Homer and Marge visiting the school, where they encounter two child prodigies in the playground, who have invented their own lyrics to a hand-clapping song:

  Cros
s my heart and hope to die,

  Here’s the digits that make π,

  3.14159265358979323846....

  Al Jean was the writer responsible for deftly crowbarring this mathematical reference into the episode. At first hearing, it seems like an uncontroversial recitation of the world’s most famous irrational number, but on further consideration I began to wonder why π was being expressed in a base-10 decimal form.

  Base 10 is our standard number system, with the first decimal place representing tenths (1/101), then each subsequent decimal place representing hundredths (1/102), thousandths (1/103), and so on. Our number system developed in this manner because the human hands between them have ten digits.

  However, if you take a close look at the hands of the characters in The Simpsons, you will notice that they have only three fingers and a thumb on each hand, so eight digits in total. Therefore, counting in Springfield should rely on the number 8, which should lead to an entirely different system of counting (known as base 8), which in turn should result in a different way of expressing π (3.1103755242...).

  The mathematics of base 8 are not important, particularly as The Simpsons, like us, rely on base 10. Nevertheless, there are two outstanding questions that must be addressed. First, why do the residents of Springfield only have eight digits on their hands? And, second, why does the universe of The Simpsons rely on base 10, when the characters have only eight digits?

  The mutation that results in only eight digits in The Simpsons dates back to the early days of animation on the big screen. Felix the Cat, who debuted in 1919, had only four digits on each hand, and Mickey Mouse shared this trait when he made his first appearance in 1928. When asked why his anthropomorphized rodent had missing digits, Walt Disney replied: “Artistically five digits are too many for a mouse. His hand would look like a bunch of bananas.” Disney also added that simplified hands meant less work for the animators: “Financially, not having an extra finger in each of 45,000 drawings that make up a six-and-one-half-minute short has saved the Studio millions.”