Large Numbers (1948)
A Classic Paper by J.E. Littlewood
As you may have read, the name Google originated from the misspelling by Larry Page and Sergey Brin of the word googol, a number whose value is 10^1001. This word, in turn, was coined by a child, the nephew of mathematician Edward Kasner (1878-1955) in 1940. As the story goes:
“Dr. Kasner's nine-year-old nephew [Milton] was asked to think up a name for a very big number, namely 1 with one hundred zeroes after it. He was very certain that this number was not infinite, and therefore equally certain that it had to have a name. At the same time that he suggested "googol" he gave a name for a still larger number: "googolplex". A googolplex is much larger than a googol, but is still finite, as the inventor of the name was quick to point out. It was first suggested that a googolplex should be 1, followed by writing zeros until you got tired. This is a description of what would happen if one tried to write a googolplex, but different people get tired at different times and it would never do to have Carnera a better mathematician than Dr. Einstein, simply because he had more endurance. The googolplex is, then, a specific finite number, equal to 1 with a googol zeros after it. “
Thus, a googolplex, 10^googol, is equal to 10^10^100. A googolplex is a famous example of an immense number. Other (larger) examples of such numbers include Skewes’s Number, Moser’s number and Graham’s number, all of which are difficult to express here given the limitations of mathematical notation.
Indeed, the problem of accurately expressing very large numbers goes back to 3rd century BC. In the work The Sand-Reckoner, Archimedes set out to define an upper bound for the number of grains of sand that fit into the universe. To do this, he had to estimate the size of the universe and invent a way to talk about extremely large numbers. The system he came up with is equivalent to that of modern exponentiation, although he lacked a suitable notation (such as a^b, its extension a^b^c and so on). The history of astronomy is in fact ripe with examples of scientists attempting to describe phenomena (distances, amounts, speeds, rates) at immense scales. In his classic paper Large Numbers, J.E. Littlewood (1885-1977) in 1948 mentions asking Sir Arthur Eddington (1882-1944) to what accuracy it was possible to measure the angular separation of widely separated stars, to which:
“The answer (given instantly) was 0”.1 which I for one found very surprising.”
0.1 arc seconds is approximately equivalent to 0.00002778 degrees, a level of separation commonly found in the observation of double stars and exoplanets. Unreasonably accurate, one might argue, given the magnitudes involved and the technological maturity of telescopes in the 1940s. However, large numbers are devious and, as we will see, supremely unintuitive.
The Practical Limitations of Exponentiation
“Professor Lehmer further tells me that numbers up to 2.7*10^9 can be completely factorised in 40 minutes; up to 10^15 in a day; up to 10^20 in a week; finally up to 10^100, with some luck, in a year.”
As Littlewood demonstrates again and again in his 1948 paper, the human mind is hopelessly inept at intuiting large numbers. To demonstrate this fact, he gives the following example. Consider types of numbers, described as type 1, 2, …, n:
A number of type 2.47 (between type 2 and type 3) may thus be written as N₂.₄₇ and be expressed as:
Littlewood’s claim is that numbers of type 2 and above (including 2.47) are “practically unaltered” by being squared, whereas numbers of type 3 and above are indeed, in a practical sense, “unaltered” by being raised, even to a power such as u (which he defines as the number of particles in the universe, 10⁷⁹). Type 2 numbers thus are large but can still be meaningfully influenced by squaring (albeit to much less of an extent than type 1 numbers). Type 3 numbers and higher, however, are so large that squaring or raising them to another large exponent doesn’t affect their magnitude significantly at the scale being considered. His evidence is the observation that:
Littlewood’s “principle of crudity”, suggested here, thus states that certain operations on sufficiently large numbers do not significantly change their overall scale in relation to magnitudes that are meaningful to human beings.
Large Numbers Deceive
Decades before Amos Tversky (1937-1996) and Daniel Kahneman (1934-2024)’s groundbreaking work on prospect theory, Littlewood in 1948 thus had noticed a fundamental reality about human cognition as it relates to large numbers of various kinds. As he wrote in 1948 (and which Tversky and Kahneman would call “non-linear probability weighting”), “Improbabilities are apt to be overestimated.” He leads off his argumentation with an example from cricket:
“A popular newspaper noted during the 1947 cricket season that two batsmen had each scored 1,111 runs for an average of 44.44. Since it compared this with the monkeys' typing of Hamlet (somewhat to the disadvantage of the latter) the event is worth debunking as an example of a common class. […] We have, of course, to estimate the probability of the event happening at some time during the season.
Take the 30 leading batsmen and select a pair A, B of them. At some moment A will have played 25 complete innings. The chance against his score then being 1,111 is say 700:1. The chance against B's having at that moment played 25 innings is say 10:1, and the further chance that his score is 1,111 is again 700:1. There are, however, about 30 x 15 pairs; the total adverse chance is 10 x 700²/(30 x 15), or about 10⁴: 1. A degree of surprise is legitimate.”
However, events with a likelihood of 10,000:1 indeed occur regularly. As magician Penn Jillette once pointed out, “million-to-one odds happen eight times a day in New York” (whose population is approximately 8 million). Other examples by Littlewood include:
“7 ships in Weymouth Harbour at the beginning of a 3 mile walk had become 6 when we sat down to rest: the 6 were riding parallel at their anchors, but the two-masted 7th had aligned itself exactly behind a mast of one of the 6. A shift of 5 yards clearly separated the masts. The chance against stopping in the right 10 yards is 600:1; that against the ship being end on about 60:1; in all about 4 x 10⁴:1; the event is thus comparable to the cricket average both in striking impact and real insignificance.”
and the even more surprising:
“Eddington once told me that information about a new (newly visible, not necessarily unknown) comet was received by an Observatory in misprinted form; they looked at the place indicated (no doubt sweeping a square degree or so), and saw a new comet. (Entertaining and striking as this is the adverse chance can hardly be put at more than a few times 10⁶)”
Often attributed to mathematician (and former professional magician) Persi Diaconis (1945-) and his advisor Frederick Mosteller (1916-2006), the law of truly large numbers states that with a large enough number of independent samples, any highly implausible result is likely to be observed. But, because humans never find it notable when a likely event occurs, only unlikely events are noticed and highlighted.
As Littlewood would later go on to claim (later referred to as “Littlewood’s law”) a person can expect to experience events with odds of one in a million at a rate of about one every month. Littlewood’s claim was later restated as “Littlewood’s Miracle Law” by Freeman Dyson (1923-2020) in a review of the book Debunked! ESP, Telekenesis, and Other Pseudoscience*:
The paradoxical feature of the laws of probability is that they make unlikely events happen unexpectedly often. A simple way to state the paradox is Littlewood’s law of miracles. John Littlewood [...] defined a miracle as an event that has special importance when it occurs, but occurs with a probability of one in a million. This definition agrees with our commonsense understanding of the word “miracle.”
Littlewood’s law of miracles states that in the course of any normal person’s life, miracles happen at a rate of roughly one per month. The proof of the law is simple. During the time that we are awake and actively engaged in living our lives, roughly for 8 hours each day, we see and hear things happening at a rate of about one per second. So the total number of events that happen to us is about 30,000 per day, or about a million per month. With few exceptions, these events are not miracles because they are insignificant. The chance of a miracle is about one per million events. Therefore we should expect about one miracle to happen, on the average, every month.
Littlewood’s classic 1948 paper was later republished in the book A Mathematician’s Miscellany* by the same author.
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Or, according to the official Google About page, “The name was a play on the mathematical expression for the number 1 followed by 100 zeros and aptly reflected Larry and Sergey's mission “to organize the world’s information and make it universally accessible and useful.”