Archimedes had become fed up of people saying you couldn’t calculate the number of grains of sand on a beach.
In response to this nonsense (as he saw it) he invented new, enormous numbers. Then he calculated not just how many grains of sand there were on the beach, but how many there were in the universe.
The trouble Archimedes faced was the Greek number system. It was a primitive system in which letters became numbers: A = 1, B = 2, C = 3, etc.
Very large numbers were a problem, because there weren’t enough letters in the alphabet! The Greeks’ biggest number was a myriad, which in the modern, Hindu-Arabic number system, we write as 10,000.
Off with the old…
In The Sand Reckoner, Archimedes demolished the commonly held idea that the number of grains of sand on the shores around his home city of Syracuse could not be calculated.
In fact, he showed that he could produce numbers so large that they were bigger than the number of sand grains in the whole universe.
His calculation relied on his invention of what we now call exponents (often called powers, or index numbers). For example 104 is ten to the power of four. We usually call this ten thousand. The Greeks would have called it a myriad.
Archimedes’ New Number System
Archimedes introduced a new classification of numbers.
He said that ‘first order’ numbers went up to a myriad myriads, meaning 10,000 x 10,000.
We would write this as 100 million, or 100,000,000, or 108.
Numbers of the ‘second order’ went up to 100 million multiplied by 100 million – i.e. 108 x 108 or (108)2.
Numbers of the third order were those up to 108 x 108 x 108 – i.e. (108)3,
and so on.
Ultimately, Archimedes calculated that to count the number of grains of sand in the universe he needed numbers up to the eighth order, i.e. (108)8 = 1064, which is equal to:
This was the biggest number anyone would need in the universe Archimedes imagined. (By the way, electronic calculators will happily work with numbers as big as this.)
But Archimedes was not content with discovering this huge number. He went on to write numbers that dwarf it.
He moved from ‘orders’ of numbers, to what he called ‘periods.’
Mind Bogglingly Big Numbers
Archimedes said that numbers of the ‘first period’ will be those numbers up to the mind-bogglingly large:
This number is much too big for everyday electronic calculators to work with.
It could be written as 1 followed by 800 million zeros. If you were to publish this number as a book, the zeros would take up about 380,000 pages. That’s a long book!
Also, given that the number of atoms in the sun is 1 followed by just 57 zeros, you will have to work hard to find anything big enough to need 1 followed by 800 million zeros to describe it.
Archimedes’ Beast Number
However, Archimedes was still not ready to let things rest. He wanted to write even bigger numbers. He continued logically until he reached:
Archimedes called this number a myriad-myriad units of the myriad-myriadth order of the myriad-myriadth period. We’ll just call it Archimedes’ Beast Number. It’s one followed by 80 quadrillion zeros.
What could we use this number for practically?
Well, how about writing down the volume of the observable universe in cubic centimeters? Surely that would need a number close to the Beast Number? If we take it that the observable universe has a diameter of 93 billion light years, then…..
nope, the universe’s volume in cubic centimeters ‘only’ needs 35 followed by 85 zeros.
Hmmmmm. We’ll need to try harder.
Most of the universe is pretty much a vacuum. How about filling the universe with air. How many molecules of air would we need to fill the universe to Earth’s air pressure? The number of molecules needed must at least approach the Beast Number, mustn’t it?
Actually, no, we can fill the universe with just 1 followed by 106 zeros molecules of air.
How about the number of bacteria that has ever lived on Earth? Well, again… no, that’s only about 1 followed by 40 or so zeros.
Okay, one last try. Life is based on that well-known molecule, DNA. Given that each different human has different DNA, how many different human beings are possible genetically before we start creating humans with identical genotypes?
As far as we can calculate, the upper limit is believed to be somewhat less than 10106.
That’s a much bigger number than our earlier efforts: it’s 1 followed by a million zeros.
But again, it’s no match for the Beast Number.
Archimedes’ Number utterly dwarfs these huge, but more practical numbers. In fact, the Beast Number is nothing more than a measure of Archimedes’ towering mathematical ambition.
p.s. Bigger than the Beast
Before clicking the ‘publish’ button, it came to me that I ought to mention that the Beast Number does not even begin to approach infinity. In fact, it’s no closer to infinity than the number 1 is, because it’s still infinitely far away. No number we can name can get close to infinity.
And remember, the Beast Number is a rational number. The infinity of irrational numbers is even bigger than the infinity of rationals!