Recommended · Science
Huge Numbers
Richard Elwes has made a career of persuading people who think they hate mathematics that they are wrong. His latest book attempts this with numbers so large they are, in a meaningful sense, incomprehensible.
Let us begin with a number you can write down. One million. Seven digits, perfectly manageable, the kind of quantity that appears in newspaper headlines and lottery jackpots without causing anyone to reach for a glass of water. One billion: ten digits. A trillion: thirteen. These are large numbers in the colloquial sense — most of us will never accumulate a million of anything very much — but they are not, in the mathematical sense, large at all. They are almost embarrassingly small. They barely register on the scale of things mathematicians find interesting.
A googol — one followed by a hundred zeroes — is somewhat larger. It exceeds, by a considerable margin, the estimated number of atoms in the observable universe, which runs to somewhere around ten to the power of eighty. A googolplex, one followed by a googol of zeroes, is larger still: so much larger that writing it out in standard notation is physically impossible, since doing so would require more space than the universe contains. These are numbers that genuinely begin to escape ordinary comprehension, and they are, in the territory that Richard Elwes explores in his new book, still very small.
Elwes is a mathematician at the University of Leeds and the author of several books — among them Maths in 100 Key Breakthroughs and Mathematics 1001 — that have established him as perhaps the most readable British expositor of mathematical ideas currently working in the popular science space. His gift is for the analogy that doesn't condescend: he finds ways to make abstract structures feel concrete without falsifying them, and he has an eye for the specific example that makes a general principle stick. He is also, crucially, funny — not in the way that popular science books sometimes attempt humour as a tactic to reassure anxious readers, but in the way that someone who genuinely finds his subject delightful tends to be funny, because delight is contagious.
In this new book, whose precise title is noted with a caveat in the footer, Elwes trains all of these gifts on the problem of very large numbers — specifically on the question of how large numbers can get before they escape not just intuition but the entire apparatus of conventional mathematical notation, and what it means that such numbers exist at all.
The universe contains roughly ten to the power of eighty atoms. Graham's number makes this look, by comparison, approximately like nothing.
The centrepiece of the book is Graham's number, which was for several decades listed in the Guinness Book of World Records as the largest number ever used in a serious mathematical proof. Graham's number is not large in the way that a googolplex is large. It is large in a way that defeats the standard notation for expressing large numbers — not just regular multiplication or exponentiation, but Knuth's up-arrow notation, which encodes towers of exponents, which encodes towers of towers, which encodes towers of towers of towers. Even this system collapses before Graham's number. The number requires a notation that describes how many times you apply a notation that describes how many times you apply a notation that describes — and so on, for sixty-four iterations before you reach anything that can be written down conventionally. The number of digits in Graham's number cannot be written in the observable universe. Neither can the number of digits in the number of digits. This is not a loose metaphor. It is the literal situation.
Elwes is very good at making this feel like a discovery rather than an abstraction. He traces the mathematical problem that gave rise to Graham's number — a combinatorics question about hypercubes, first posed by Ronald Graham and Bruce Rothschild in 1971 — and explains why a number this enormous appeared as a genuine mathematical bound rather than as an exercise in staggering the reader. The key insight, which Elwes delivers with characteristic clarity, is that Graham's number is not interesting because it is large. It is interesting because it is the answer to a real question, and that real questions can have answers this large tells us something surprising about the structure of mathematics itself.
Beyond Graham's number, Elwes enters territory that is less widely known but no less vertiginous. TREE(3) — the third term in a sequence defined by a problem in graph theory — makes Graham's number look, in a precise mathematical sense, infinitesimally small. The relationship between TREE(3) and Graham's number is not like the relationship between a million and a billion, or even a googol and a googolplex. It is not that TREE(3) has more digits. Graham's number's digits, and the digits of the digits of the digits, iteration upon iteration, do not approach TREE(3). The numbers occupy different scales of largeness entirely.
Elwes does not stop there. The book traces the history of large-number notation from Archimedes (who developed a system in The Sand Reckoner to express numbers large enough to count the grains of sand needed to fill the universe) through Cantor's transfinite cardinals, which move beyond counting into the infinite, and into the combinatorial wilderness of the present day. It is a history that doubles as an argument: that human beings have been extending their sense of numerical scale for as long as they have been thinking mathematically, and that each extension has revealed that the territory beyond the new horizon is vastly larger than the territory that prompted the extension.
There is a philosophical dimension to this that Elwes handles with admirable care. Large numbers raise genuine questions about the relationship between mathematical existence and physical reality. Numbers like TREE(3) cannot be instantiated in the physical universe in any form — they cannot be written, computed, or approximated in any material medium. Yet they are, in a robust mathematical sense, exactly determined. TREE(3) is not vague. It has a specific value. We simply have no means of approaching it. Elwes treats this not as a problem to be solved but as a feature of mathematics that tells us something important about what mathematics is — and why the question of what large numbers mean is, in the end, a question about the nature of mathematical truth.
For readers who have been told, or who have told themselves, that mathematics is not for them: this book is the argument that they are wrong. It requires no prior knowledge, no comfort with notation, no memory of school algebra. It requires only a willingness to follow a careful mind into territory that turns out, on arrival, to be considerably stranger and more beautiful than the landscape you left behind.