In a new book, mathematician Richard Elwes shares how mathematical boundaries have been broken time and time again, thanks to the creative explorers of the biggest numbers known.
Huge Numbers
Richard Elwes
Basic Books, $32
Try to think of a really big number. What comes to mind? A trillion? A quintillion? Or perhaps a googol (10100, or a 1 with 100 zeroes after it) or even a googolplex (a 1 followed by a googol zeroes).
Prepare for those numbers to suddenly seem imperceptibly tiny. In the book Huge Numbers, mathematician Richard Elwes lays out incomprehensibly large figures, so enormous that mathematicians had to devise new notation just to write them down. It’s a delightful survey of the field of googology, the study of large numbers.
Big numbers may seem like a niche interest. But, Elwes writes, “small numbers are the exceptions; big numbers are the rule.” That’s because numbers go on and on, indefinitely getting larger. Pick any huge number you can think of. There are more numbers of a magnitude bigger than that value than there are smaller numbers.
Elwes leads the reader through a morass of ever-increasing numbers. That includes the enormous figures that can arise from exponential growth, such as $2 × 1034 (a 2 with 34 zeroes after it), the amount that a Russian court fined Google in 2024 due to a rapidly ballooning financial penalty. But also sequences of numbers that explode even faster, so quickly that they break standard mathematics and demand new arithmetic rulebooks to explain them.
But Elwes’ narrative starts unexpectedly, with numbers like 4 and 5. Humans have an innate number sense that allows us to distinguish small quantities, below about 5, without counting them. Above that threshold, things become fuzzy. But the technique of counting enables us to surpass our innate limitations. Elwes continues in this vein, explaining numerical technologies, developed throughout the history of mathematics, that invoke language, notation and eventually computers to allow humans to wrangle bigger and bigger quantities.
From there, Elwes dives into the huge numbers present in science, particularly in physics: the immense size of the cosmos and the uncountable years that lie ahead as the universe slowly burns out. The litany of large numbers at this point may make the reader’s eyes glaze over. But Elwes has barely begun.
To describe the large numbers relevant to the universe, scientific notation is generally sufficient. In this system, large numbers are represented by a 10 with an exponent, which indicates a 10 multiplied by itself a given number of times, so 3 × 106 represents 3,000,000, or 3 million.
But at some point, this system reaches its limit. A single exponent can only hold so many zeroes before it becomes difficult to interpret, like 1010000000000. Next, mathematicians turn to towers of powers, exponents which themselves have exponents. In that notation, 1010000000000 becomes a tower of three tens. But eventually those towers become too tall to write down. A type of notation called Knuth arrows takes over. 5↑↑4 represents a tower 4 fives tall. Even that notation runs out eventually, requiring Knuth mountains.
And onward and upward to numbers that are represented only by mathematical functions, labeled with mathematicians’ names or quirky monikers: Goodstein numbers, Rayo’s number, busy beaver numbers, Fish’s number 7. Here, numbers are so mind-blowingly large as to become untethered from reality, relating, for example, to the capabilities of hypothetical computers called Turing machines outfitted with magical powers.
At times, the reader may lose grasp of the thread, as Elwes goes on tangents for which it’s unclear how they relate to large numbers. Eventually he circles back, but as the numbers get more complex, the thread gets more difficult to catch.
But the patient reader willing to stick with Elwes will be rewarded with a new appreciation for numbers and a vastly expanded frame of reference for what it means to be truly, unfathomably, large. It’s a joy to marvel at how boundaries have been broken time and again, thanks to the creative, intrepid explorers of the biggest numbers known.
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