David Darling - The Biggest Number in the World

The physical universe is vast beyond imagining. Even the nearest star lies at a distance almost impossible to grasp with our Earth-bound brains. The edge of the observable universe is inconceivably more remote: about 46 billion light-years, or 270 billion trillion miles, away. Yet were about to embark on a much greater voyage, not into the depths of space but into the farthest reaches of the mathematical cosmos.

Along the way well come across some extraordinary ideas, so alien to our normal way of thinking that the biggest challenge will be to find familiar words and concepts by which we can build bridges to understanding. Well venture far from home into regions of thought that, until now, few have seen or experienced. Our quest: nothing less than to find the edge of the numerical universe.

Surely, you might say, there is no such edge. Numbers go on forever. Even if we were to fill this book, or a library full of books, with 1 followed by zeros or all 9s on every line on every page, at the end you could name a bigger number simply by saying and 1. And thats true. The number line stretches away into the mists of infinite distance. But, as were about to find out, the search for an ultimately large number isnt confined to trekking slowly, step by step, down an endless road. There are some surprising, mind-bending alternatives to the often-repeated mantra theres no biggest number. Some of these will involve entering a shadowy land, still largely unexplored, between the finite and the infinite. Others will transport us into what are effectively parallel universes of maths, where different rules operate and what we thought was secure knowledge is easily overturned.

As with any expedition into the unknown we need to go well prepared. Well look at the history of large numbers and how the subject has been mapped out up to this point. Well delve into a few areas fascinating in themselves that are rarely broached in school or university curricula in order to equip ourselves for the great quest ahead.

Like mountaineers attempting to climb previously unconquered peaks, certain mathematicians throughout history have had the courage to try to scale new heights in the mighty ranges of towering numbers. Often theyve ventured alone, not relying on the intellectual, moral or financial support of others to help them in their ambition. These pioneers of a strange land have had to develop new tools and techniques to go beyond what was possible before. And the vistas with which theyve been rewarded are no less breathtaking and spectacular, in their way, than the views from the summits of Everest or the Matterhorn. These are the minds-eye spectacles that await us in the pages ahead.

We also have personal reasons for writing this book. Number theory and the mathematics of very large numbers, in particular is a passion of Agnijos. Its a subject thats fascinated him throughout his school career, which culminated in him taking first place in the 2018 International Mathematical Olympiad, and as a student at Cambridge. David has always enjoyed finding ways to explain difficult ideas to a wide audience. The book is the culmination of an unusual writing partnership, which began while David was tutoring Agnijo as a young teenager.

Theres a widespread suspicion that maths is cold and austere, somewhat aloof from the real world of people. But nothing could be further from the truth. Mathematics, along with music and art, is among the most human of enterprises, steeped in passion, tragedy, comedy and romance, wild and wonderful characters, and bold new ideas that constantly threaten the status quo. Nowhere is this drama of maths more evident than in the ultimate intellectual challenge: the search for the biggest number in the world.

Chapter 1

Of Sand and Stars

Are there more grains of sand on Earth or stars in the universe? With your eyes alone you can see at least a couple of thousand stars on a clear night well away from artificial lights, and nearer 4,000 if its moonless and your eyesight is keen. In a handful of sand are many more grains than that. But space is huge, dauntingly so, and powerful telescopes reveal that it contains a host of galaxies, each harbouring billions of stars. On the other hand, the deserts, beaches and ocean beds of our planet are home to sand particles in dizzying profusion. So, sand or stars, which wins in the numbers game?

A study carried out by researchers at the University of Hawaii in 2003 estimated the number of sand grains in the world to be 7.5 million trillion, or 75 followed by 17 zeros. As for stars, the figure they came up with, for the whole of the observable universe, was 70 thousand million trillion. Thats about ten thousand stars for every sand grain.

The Greek mathematician and scientist Archimedes was also interested in this kind of problem. In the third century bce he wrote a short treatise, addressed to Gelon, King of Syracuse, thats come to be known as The Sand Reckoner. Sometimes described as the first research-expository paper, because it combines both accuracy and clear language, aimed at the layperson, it asks: How many grains of sand would fit in the universe?

Figure 1.1 : Sand dunes in the Sahara, Libya.

The answer, of course, depends on how big is an average grain of sand and how big is the universe. Archimedes figured, very generously (to the point of being unrealistic), that one poppy seed could contain 10,000 grains of sand, which would make the grains almost microscopic in size. He also reckoned that 40 poppy seeds, side by side, would stretch across one Greek dactyl , or finger-width, equal to about three quarters of an inch (19 millimetres). A sphere one dactyl wide would then be able to hold in the region of 640 million sand grains.