> That was the positional number system, where a small set of symbols is sufficient to write even very big numbers.
Just the other day I was thinking about it and realised that positional number system enables us to represent infinite numbers with a finite set of symbols. Whoever (person/community) invented is a genius, as positional numbering unlocks so many further inventions down the line.
this is also true of clauses in grammar. the use of clauses makes it possible to write an infinite number of sentences using a finite number of words, each of which is drawn from a finite vocabulary, but whose assemblages are more infinite than the stars in the sky which so drew the attention of archimedes when he approximated the finite number of grains of sand that would be required to fill up the observable universe, a calculation which unfortunately he got low by several orders of magnitude due to an unfortunate lack of evidence for the hypothesis that the stars were other suns, not to mention the idea that some of the nebulae in the sky were entire galaxies of other stars, a fact which wasn't known until only a century ago
the previous sentence contains 125 words, and it should be evident that it could be extended indefinitely in any known human language (except possibly pirahã) without doing any violence to the rules of grammar, though perhaps great violence to the canons of courtesy to readers. if we use shannon's early estimate of 11.82 bits of entropy per english word, in english there are about 2¹⁴⁷⁸ ≈ 10⁴⁴⁵ perfectly unremarkable sentences of that precise length, a number which (it should be evident) grows exponentially with the sentence length
so, while i agree this concept is genius, it is part of the invention of language as we know it, and no isolated human tribe without language has ever been discovered. it probably dates to so-called behavioral modernity, at least 50000 years ago—probably longer than that
You can already represent infinitely many numbers with a finite set of symbols without positional notation, in unary. You can write "1 + 1 + 1 + 1" or "This number of stars: ****" or whatever. This idea (make N marks to represent the natural number N) has surely been understood about as long as counting has been understood.
The advantage of positional notation is greater efficiency at representing large numbers.
It allows us to represent numbers using log symbols. Symbols required = O(log10(N)). Or put another way, the max number we can represent with S symbols is exponential in S. N = O(exp(S)).
You can represent any arbitrarily large number with a single symbol without positional notation, though. Let's use *. The number 3 is ***. The number 9 is *********. That interpretation of GP's comment (which, on re-read, is probably correct) makes it even more trivially false. Another comment by the same poster clarifies that I was actually responding to what he meant.
Just the other day I was thinking about it and realised that positional number system enables us to represent infinite numbers with a finite set of symbols. Whoever (person/community) invented is a genius, as positional numbering unlocks so many further inventions down the line.