Oneworld
DR EVIL (aka ) walked up to a blackboard in front of a packed auditorium at the Massachusetts Institute of Technology and wrote a single 1. It was a slow but tactical start to the 2007 , an event described on its posters as: “Two competitors. One chalkboard. Largest integer wins.”
The Mexican Multiplier (aka ) wasn’t happy with such a slow start, so he filled in the blackboard with 1s. Dr Evil countered by turning almost every 1 into the factorial sign “!”. This clever mathematics trick soon transformed the figures on the board into impenetrably large numbers, containing more digits than there are particles in the universe. The pair had moved into the world of googology – the study of extremely large numbers.
“If you love journeying into imagined mathematical worlds, then you’ll find this book pure escapism”
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Googology is a mind-boggling subject, and the topic of a wonderful new book, The Biggest Number in the World: A journey to the edge of mathematics by and . The co-authors met when Banerjee was a teenage mathematics prodigy and Darling, an eminent science writer, became his tutor. Banerjee went on to get a perfect score at the International Mathematical Olympiad, one of only two out of 594 contestants from more than 100 countries to do so.
The Biggest Number in the World starts out by tackling such “normal” big numbers as the number of stars in the observable universe (70 billion trillion) or the number of ways to shuffle a pack of playing cards (8.0658 × 1067). But then it quickly introduces the tools needed to write far larger values.
Numbers are like interstellar travel, write the duo. If spacecraft are too slow – and they are – it could take us tens of thousands of years to reach other solar systems. The same is true when writing massive numbers, so we need the mathematical equivalent of better propulsion methods, such as .
This notation is named after computer scientist Donald Knuth, and emerges from a simple pattern. Recall that multiplication is simply repeated addition (5 × 3 = 5 + 5 + 5) and exponentiation is just repeated multiplication (53 = 5 × 5 × 5). Up-arrow notation uses a single up arrow to mean exponentiation, so that 5↑3 = 53. Two arrows then mean repeated exponentiation 5⇈3 = 5↑(5↑5) = 55^5.
This pattern continues, with each new arrow meaning the steps before should be repeated, resulting swiftly in immense numerical power. The number 2↑↑↑↑4, for example, is so large it is “beyond the capacity of the universe to display in digital decimal form”, write the authors.
If you are someone who needs real-world applications to get excited about mathematics, this probably isn’t the book for you. But if you love journeying into imagined mathematical worlds and simply exploring, then it is pure, unadulterated escapism.
In different hands, the book could have been too jargon-packed to hold the attention of the average reader, but Darling and Banerjee have done a brilliant job mixing big ideas and analogies with the tools needed to understand up arrows and other approaches to notation, each more powerful than the last.
Darling and Banerjee don’t shy away from their book’s title, and eventually deliver a number so large it is very likely to earn the accolade of biggest number in the world. This number is so huge that when the Mexican Multiplier wrote it on the blackboard in the Big Number Duel, it was considered to be a knockout blow.
on his quest for unfathomably large numbers at New Scientist Live on 8 October
