The best way to count
First, the hindu-arabic numeral system: it is a way to write numbers. So, for example, six would be wrtten as " \(6\) ". The arabic numerals are just: \(0, 1, 2, 3, 4, 5, 6, 7, 8,\) and \(9\). As opposed to the hindu-arabic numeral system which uses mostly decimal. For example, five hundred and sixty-seven in decimal is "five six seven", and then, if you use the arabic numerals to go from decimal to the hindu-arabic numeral system, you get " \(567\) ". WARNING! (Kinda.) \(567\) is NOT a number, it is just a popular way to write the corrasponding number.
The hindu-arabic numerals are a positional numeral system, which means that the value of each digit is affected by the position. Positional numeral system must've been a pretty revolutionary idea, at least, in comaprison to non-positional numeral systems, such as the roman numeral system. Positional numeral systems probably got into the top \(10\) most revolutionay mathematical ideas, 'cause you can write all infinately many numbers using just \(10\) digits (\(10\) arabic numerals), without getting out of hand quickly (just see \(100,000\) vs \(MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM\) to prove that). Also, the hindu-arabic numeral system is injective, every number has just one representation (e.g. nine as \(9\) vs \(IX\) or \(VIV\) or \(VIIII\) or \(IIIIIIIIII\)). All of these should make sense, and you probably take at least one if not all of these properties for granted.
But what if there was a bug with the hindu-arabic numerals? Something that makes them NOT the best way to write numbers? There is such a bug, filed as bug report number \(10\): why count up to \(10\) before using the next digit? (Side note! the first two-digit number in any base is known as the base of the number, so the hindu-arabic numeral system has a base of ten) The choice of \(10\) is comepletely arbitrary, (side note! Again, the base of any system would always be written as " \(10\) " in that base.) the only reason that it is a good base is because it is a positional numeral system, the same system would work for any integer base greater then \(1\), and if the only reason that \(10\) is a good base is because it is a positional numeral system, at least on my YouTube homepage, I see a lot of proposals for a dosenal system (base \(12\)), but if you can bother using a different base, then the question becomes: what is the best base?
Among (ඞ) all the proposals for a different base, one stands out: a better way to count by jan misali, proposing a seximal (base \(6\)) system. I was going to summerize it, but I'm doing this in incognito mode, too many ads. This is the best time to say it, this page is based off of the best way to count, you can either watch that video, or read this page, or both! (#NotSponsored, It's just a really good video.) But, just to be on the same page (no pun intended), jan misali is wrong, seximal is not the best base out there, so fasten your mathematical seatbelts (this page is going to be really, really long), because we're going down a deep rabbit hole, to discover
Binary¶
The actual best way to count.
Binary is an extreme case, because any smaller base would just break, using only two symbols (say, \(a\) and \(b\)) to represent all the natural numbers. Binary also decomposes numbers into powers of \(2\), doind a simmilar role for addition as primes do for multiplication, it's also a maximally efficient game of twenty questions, chopping the space of possibilities in half every digit. Binary has appeared in ancient Egyptian mathematics, in the \(16\)'th century works of leibniz, and in the Arecibo message, and, of coarse, in nearly every modern computer (which I know a LOT about). But binary is also the best base for human use. By nearly any metric for comparing bases, binary preforms best, and the best part is, it's all because \(2\) is the smallest integer greater than \(1\). In contrast, a base like seximal, misali's favorite, is still a very good base, but almost entirely because of a mathematecal coincidence: six contains factors of the first \(2\) primes, and is adjacent to the next \(2\) primes. seximal is a cheater in the world of bases, and a more thorough analysis exposes the shallowness of it's tricks and reveals a gaping emptiness underneith. Meanwhile, binary shines without the need for coincidences, it plays fair and square, and that fairness pays off in the end.
Because this is a reply to a better way to count, let's start out with misali's think of binary (click here to hear the actual audio clip): Base \(2\), binary!