by Simon Singh
During this period, Ramanujan began to develop new mathematical ideas in his spare time. He felt that they were innovative and important, but he had nobody to whom he could turn for advice and support. Desperate to explore mathematics in more depth and to have his work recognized, Ramanujan began to write to mathematicians in England in the hope that someone would mentor him or at least give him feedback on his newly discovered theorems.
One batch of letters eventually reached M. J. M. Hill at University College, London. He was mildly impressed, but admonished the young Indian for using outdated methods and making trivial mistakes. He wrote, in a schoolmasterly tone, that Ramanujan’s work needed to be “very clearly written, and should be free from errors; and he should not use symbols which he does not explain.” It was an unforgiving report card, but at least Hill responded. By contrast, both H. F. Baker and E. W. Hobson at the University of Cambridge returned Ramanujan’s papers without comments.
Then, in 1913, Ramanujan wrote to G. H. Hardy: “I have had no university education but I have undergone the ordinary school course. After leaving school I have been employing the spare time at my disposal to work at mathematics. I have not trodden through the conventional regular course which is followed in a university course, but I am striking out a new path for myself.”
When a second letter followed, Hardy found that Ramanujan had sent him a total of 120 theorems to consider. The young Indian savant would later say that many of these theorems were whispered to him in his sleep by Namagiri, an avatar of the Hindu goddess Lakshmi: “While asleep, I had an unusual experience. There was a red screen formed by flowing blood, as it were. I was observing it. Suddenly a hand began to write on the screen. I became all attention. That hand wrote a number of elliptic integrals. They stuck to my mind. As soon as I woke up, I committed them to writing.”
Upon receiving Ramanujan’s papers, Hardy’s reaction veered between “fraud” and so brilliant that it was “scarcely possible to believe.” In the end, he concluded that the theorems “must be true, because, if they were not true, no one would have the imagination to invent them.” Hardy dubbed Ramanujan “a mathematician of the highest quality, a man of altogether exceptional originality and power,” and he began to make arrangements for the young Indian, still only twenty-six, to visit Cambridge. Hardy took great pride in being the man who had rescued such raw talent, and would later call it “the one romantic incident in my life.”
The two mathematicians finally met in April 1914, and their resulting partnership gave rise to discoveries in several areas of mathematics. For example, they made major contributions toward understanding a mathematical operation known as partition. As the name implies, partitioning concerns dividing up a number of objects into separate groups The key question is, for a given number of objects, how many different ways can they be partitioned? The boxes below show that there is one way to partition one object, but there are five ways to partition four objects:
It is easy to find the number of partitions for a small quantity of objects, but it becomes trickier and trickier with more and more objects. This is because the number of possible partitions balloons in a rapid and erratic fashion. 10 objects can be partitioned in just 42 ways, but 100 objects can be partitioned in 190,569,292 ways. And 1,000 objects can be partitioned in an astonishing 24,061,467,864,032,622,473,692,149,727,991 ways.
One of Hardy and Ramanujan’s breakthroughs was to invent a formula that can be used to predict the number of partitions for very large numbers. The formula requires a great deal of effort to compute, so they also invented a rough-and-ready formula that gave a good estimate of the number of partitions for any given number of objects. Ramanujan also made an interesting observation that continues to provide food for thought today: If the number of objects ends with 4 or 9, then the number of partitions is always divisible by 5. To illustrate Ramanujan’s claim, 4, 9, 14, 19, 24, and 29 objects generate 5, 30, 135, 490, 1,575, and 4,565 partitions, respectively.
Ramanujan’s achievements were numerous, complex, and brilliant, and his genius was recognized in 1918 when he was elected one of the youngest fellows in the history of the Royal Society. Sadly, while his move to Cambridge enabled his mind to embark on incredible adventures, the harsh English winters and the change in diet took their toll on Ramanujan’s health. Toward the end of 1918, he left Cambridge and was admitted to a private nursing home, Colinette House in Putney, London. It was against this background that the conversation linking Ramanujan to Futurama took place.
According to Hardy: “I remember once going to see him when he was lying ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. ‘No,’ he replied, ‘it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.’”
The two men were clearly not comfortable engaging in small talk or gossip. As usual, their exchange revolved around numbers, and it can be unpacked and expressed as follows:
1,729
=
13 + 123
=
93 + 103
In other words, if we had 1,729 small cubelets, we could arrange them as two cubes with dimensions 1 × 1 × 1 and 12 × 12 × 12, or we could arrange them as two cubes with dimensions 9 × 9 × 9 and 10 × 10 × 10. It is rare that numbers can be split into two cubes, and even rarer that they can be split into two cubes in two different ways . . . and 1,729 is the smallest number that exhibits this property. In honor of Ramanujan’s comment about Hardy’s taxicab, 1,729 is known in mathematical circles as a taxicab number.
Prompted by Ramanujan’s off-the-cuff remark, mathematicians have asked a related question: What is the smallest number that is the sum of two cubes in three different ways? The answer is 87,539,319, because
87,539,319
=
1673 + 4363
=
2283 + 4233
=
2553 + 4143
This number, which is also labeled a taxicab number, crops up in a special extended Futurama episode titled “Bender’s Big Score” (2007). When Fry hails a cab, the number on the roof is 87,539,319. It is, of course, very appropriate that the taxicab number (in the normal sense) is a taxicab number (in the mathematical sense).
Thus, by repeatedly referencing 1,729 and including 87,539,319, the Futurama writers are paying tribute to Ramanujan, whose story is largely unknown outside the world of mathematics. It is an inspiring story of a natural genius plucked from obscurity by a Cambridge don, yet it ends tragically. While suffering from various ailments, including vitamin deficiencies and tuberculosis, Ramanujan returned to India in 1919 in the hope that a warmer climate and a more familiar vegetarian diet might restore his health. After barely a year in India, he died on April 26, 1920, at the age of thirty-two.
Nevertheless, Ramanujan’s ideas remain at the heart of modern mathematics, and always will. This is partly because the language of mathematics is universal and partly because mathematical proofs are absolute. Unlike ideas in the arts and humanities, mathematical theorems do not go in and out of fashion. As Hardy himself pointed out: “Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. ‘Immortality’ may be a silly word, but probably a mathematician has the best chance of whatever it may mean.”
These Futurama references to taxicab numbers can all be traced back to one writer, Ken Keeler, who ranks as one of the most mathematically gifted writers on either The Simpsons or Futurama. According to Keeler, his fascination with mathematics was largely inspired by his father, Martin Keeler, a medical doctor whose favorite hobby was playing games with numbers. Whenever the family went to a restaurant and received the bill at the end of the meal, he would check it for prime numbers, and his children were expected to join in. On one particular occasion, Ken recalls asking his father if there was a quick way to add up square numbers. For example, what is the su
m of the first five square numbers, or the first ten square numbers, or the first n square numbers? Dr. Keeler thought about it for a short while and then correctly responded with the correct formula: n3/3 + n2/2 + n/6. Keeler’s formula can be checked with an example, such as n = 5:
Sum of the first five square numbers: 1 + 4 + 9 + 16 + 25 = 55
Dr. Keeler’s formula:
This is not a seriously challenging problem for a mathematician, but Dr. Keeler was not a mathematician. Moreover, he solved the problem using a radical and highly intuitive approach. A brief and moderately technical explanation in Ken Keeler’s own words appears in appendix 3.
His father’s playful approach to mathematics was partly responsible for Ken Keeler’s decision to study applied mathematics at college and then pursue a doctorate in the subject. However, after completing his PhD, he was torn between a career in research and trying his hand at comedy writing, his other great passion. Although he landed a job as a researcher at AT&T Bell Labs in New Jersey, he had already sent his résumé to the producers of Late Night with David Letterman. That proved to be the turning point. He was invited to join the writing team, left his research job, and never looked back. Keeler then had stints writing for Wings and The Critic, before becoming part of the Futurama team, working alongside half a dozen other mathematically inclined writers. Nowhere else in Hollywood would Keeler’s love of the number 1,729 have been so fully appreciated.
One of Keeler’s other mathematical contributions to Futurama is the Loew’s 0-Plex, which first appeared in “Raging Bender” (2000). Loews built a reputation in the twentieth century for operating some of the world’s biggest multiplex movie theaters, but the 0 prefix implies a major scaling up of their operations in the thirty-first century. 0 (pronounced aleph-null) is a mathematical symbol that represents infinity, so the name of the movie theater implies that it has an infinite number of screens. According to Keeler, when the Loew’s 0-Plex made its debut on Futurama, the draft script included a comment that this movie theater with infinitely many screens “still wouldn’t be big enough to show Rocky and all its sequels at once.”
Although the symbol 0 will be unfamiliar to most readers, there is another symbol for infinity, ∞, that we come across in high school. Hence, you might ask what is the difference between ∞ and 0. In short, ∞ is a broad-brush symbol for the concept of infinity, whereas 0 applies to a particular type of infinity!
The concept of a “particular type of infinity” might sound impossible, particularly as the earlier story of Hilbert’s Hotel demonstrated two clear conclusions:
(1)
infinity + 1
=
infinity
(2)
infinity + infinity
=
infinity
It would be easy to jump to the conclusion that there is nothing bigger than infinity, and that all infinities have the same bigness, so to speak. However, there are actually different sizes of infinity, and this can be demonstrated using a fairly simple argument.
We begin by focusing on the set of decimal numbers that sits between 0 and 1. This includes simple decimals such as 0.5 and also numbers that have many more decimal places, such as 0.736829474638…. There are clearly an infinite number of these decimals, because for any given decimal (e.g., 0.9), there is a bigger one (0.99), and then a bigger one (0.999), and so on. Next, we can consider how the infinity of decimals between 0 and 1 compares with the infinity of counting numbers, 1, 2, 3,…. Is one type of infinity bigger than the other, or are they the same size?
To find out which, if either, infinity is larger, let us imagine what would happen if we tried to match all the counting numbers against all the decimal numbers between 0 and 1. The first step would be to somehow make a list of all the counting numbers and a separate list of all the decimal numbers between 0 and 1. For this particular argument, the list of counting numbers should be in numerical order, while the list of decimals numbers can be in any order. The lists are then written down side by side, with a one-to-one matching.
Counting numbers
Decimal numbers
1
0.70052…
2
0.15432…
3
0.51348…
4
0.82845…
5
0.15221…
⋮
⋮
Hypothetically, if we can match the counting numbers and the decimal numbers in this way, then there must be the same number of each, and the two infinities would therefore be equal. However, establishing a one-to-one correspondence turns out to be impossible.
This becomes clear in the final stage of our infinity investigation, which involves creating a number by taking the first digit of the first decimal number (7), the second digit of the second decimal number (5), and so on. This generates the sequence 7–5–3–4–1…. Then, by adding 1 to every digit (0 → 1, 1 → 2, …, 9 → 0), we generate a new sequence, 8–6–4–5–2…. Finally, this sequence can be used to construct a decimal number, 0.86452….
This number, 0.86452…, is interesting because it cannot possibly exist in the supposedly complete list of decimal numbers between 0 and 1. That seems like a bold claim, but it can be verified. The new number cannot be the first number on the list, because we know that the first digit won’t match. Similarly, it cannot be the second number, because we know that the second digit won’t match, and so on. More generally, it cannot be the nth number, because the nth digit won’t match.
Slight variations of this argument can be repeated to show that there are lots of other numbers that are missing from the list of decimals. In other words, when we try to match up the two infinities, the list of decimals between 0 and 1 is doomed to be incomplete, presumably because the infinity of decimal numbers is greater than the infinity of counting numbers.
This argument is a simplified version of Cantor’s diagonal argument, a watertight proof published in 1892 by Georg Cantor. Having confirmed that some infinites are bigger than others, Cantor was confident that the infinity that describes counting numbers was the smallest type of infinity, so he labeled it 0, with (aleph) being the first letter of the Hebrew alphabet. He suspected that the set of decimals between 0 and 1 illustrated the next and bigger type of infinity, so he labeled it 1 (aleph-one). Larger types of infinity, for they also exist, are logically named 2, 3, 4, ….
Thus, although Futurama’s Loew’s 0-Plex movie theater has an infinite number of screens, we now know that it is only the smallest type of infinity. Had it been an 1-Plex movie theater, it would have had even more screens.
Futurama does make one more reference to Cantor’s categorization of infinities. Mathematicians describe 0 as countably infinite, because it describes the scale of infinity associated with the counting numbers, whereas larger infinities are dubbed uncountably infinite. As noted by David X. Cohen, the latter term receives a casual mention in an episode titled “Möbius Dick” (2011): “We go briefly into this weird four-dimensional universe and there are many, many copies of Bender floating around all doing a conga line and then he comes back to reality and says, ‘That was the greatest uncountably infinite bunch of guys I ever met.’”
CHAPTER 16
A One-Sided Story
In “Möbius Dick”, the Planet Express ship is traveling through the galaxy and inadvertently enters the Bermuda Tetrahedron, a spaceship graveyard containing dozens of famous lost vessels. The Planet Express crew decide to investigate the region, whereupon they are attacked by a fearsome four-dimensional space whale, which Leela nicknames Möbius Dick.
The space whale’s name is both a play on Herman Melville’s novel Moby-Dick and a reference to a bizarre mathematical object known as a Möbius strip or Möbius band. The Möbius strip was discovered independently by the nineteenth-century German mathematicians August Möbius and Johann Listing. Using their simple recipe, you can build one for yourself. You will require:
(a) a strip of paper,
(b) sticky tape.
First, take the strip and twist one end through half a turn, as shown below. Then tape the two ends together to create the Möbius strip. That is all. A Möbius strip is essentially just a loop with a twist.
So far, the Möbius strip does not seem very special, but a simple experiment reveals its remarkable property. Take a felt-tip pen and draw a line around the strip without taking the pen off the paper, without crossing any edges, and continuing until you get back to where you started. You will notice two things: It takes two circuits to get back to where you started, and you will have drawn along every section of the strip. This is very surprising, because we assume that a piece of paper has two sides and you can only draw on both of them if your pen is allowed to leave the paper or go around an edge. So what happened in the case of the Möbius strip?
Sheets of paper have two sides (a top side and a bottom side), and loops of paper also typically have two sides (an inner side and an outer side), but the Möbius strip has the unusual property of only having one side. The two sides on the initial strip of paper were transformed into one side when the half twist was introduced prior to joining the ends together. This unusual property of a Möbius strip has provided the basis for my third all-time favorite mathematical joke:
