You’re right. There are two types of 0: the internal syntax denoting digit shift, like in 507, and the number representing nothing, as in 0. So in our number system, o isn’t just syntax, it’s also a number.
Which is precisely what enables the positional system at the conceptual level: not only you can have "gaps" in the middle, but it enables you to distinguish between 1 and 10 and 100.
Babylonians, for example, didn't conceptualize 0 as a numeric quantity just like 1 or 5. So, when they invented a sign for gaps in the numbers, they didn't use it at the ending positions, so their "I" could mean 1, or 60, or 3600, or..., depending on the context. Neither was it used to signify the quantity of zero: empty space was used instead.
Yes, sorry, my point didn’t come across very well.
What’s quite interesting is to ask pupils to assign numbers to their fingers and thumbs:
1 2 3 4 5 6 7 8 9 ... X?
It’s a good way of introducing positional number systems, and the... paradox? quandary? quirk?.. of how base N never includes a symbol for N, and how N in base N is always ‘10’.
See also: asking pupils to divide 12345* by 67, taught in the context of arithmetic and logical shifting.
Fun Fact: The babylonians used a base 6 (really, base 60) system. We still see echoes of this with 360 degrees in a circle (invented by Babylonians) as well as 60 minutes in an hour, 12/24 hours in a day, etc. Even the 7 day week is viewed as a 6 day work week plus an extra day for ceremonies (A "Sabbath" was invented by the Babylonians as well, although in a different form than the Jewish Sabbath - the Babylonian Sabbath barred work on the seventh day because it was considered unlucky).
The theory is they did this because of their hands, but used one hand to count digits and the other hand to count groups of digits: for each of the 4 fingers they are divided into 3 segments. When you are counting, you can point to the four fingers with your thumb, thus giving 12 and a pointer to which finger you are on. Then in your other hand you can keep track of how many units of 12, so 5*12 = 60. A base 60 system. But there was no digit for zero, although of course they had a way of dealing with zero, it was just not considered a number.
I wish we were taught to think on various base number systems from a young age. I imagine there could be elegant solutions to many problems currently being hidden just out of sight.
I could be mistaken, but I believe in the US that elementary school mathematics is now addressing the concept of arithmetic base at an earlier age. Unfortunately I've mostly heard about this in the context of parents complaining about the curriculum probably because they don't understand the topic.
I taught base-10 to my kids (pre school, ie 3/4) by teaching them the concept of number bases, and considering aliens with different numbers of fingers ... and robots of course (my robots only speak binary!). Seemed to work somewhat.
Mind you, despite having relatively good teaching in mainstream UK schooling they could all definitely have done/do with more headroom in their school maths.
I like reading about pre-decimal number handling in Europe. This text kind of made me wish for some historical drama movie going all in on doing that tired "genius concentration flow breakthrough" trope (it's always the same sequence of excessive cross fades) with expert handling of the counting board the text describes. And particularly into illustrating how noting down the results in Roman numerals would effectively be like the duality of working in RAM and disk persistence we know from computers. And that's the point were I suddenly realize that they didn't even talk in the same numeral system as we do, probably a wild mix of terms like dozens or score and so on for countables, Latin for years and who knows what more.
And, to make things even more weird, spoken Latin, as opposed to their written numerals, seems to be refreshingly decimal except for those crazy duode- and unde- anomalies? From this perspective, it seems almost impossibly unlikely that they kept non-decimal systems around for so long.
12 comments
[ 3.2 ms ] story [ 43.6 ms ] threadBecause we have ten digits on our hands. Maybe a shrug or open palms to mean “nothing”.
But base ten has nine digits: 1 to 9 and a 0 for nothing.
Babylonians, for example, didn't conceptualize 0 as a numeric quantity just like 1 or 5. So, when they invented a sign for gaps in the numbers, they didn't use it at the ending positions, so their "I" could mean 1, or 60, or 3600, or..., depending on the context. Neither was it used to signify the quantity of zero: empty space was used instead.
What’s quite interesting is to ask pupils to assign numbers to their fingers and thumbs:
1 2 3 4 5 6 7 8 9 ... X?
It’s a good way of introducing positional number systems, and the... paradox? quandary? quirk?.. of how base N never includes a symbol for N, and how N in base N is always ‘10’.
See also: asking pupils to divide 12345* by 67, taught in the context of arithmetic and logical shifting.
(*base 67)
The theory is they did this because of their hands, but used one hand to count digits and the other hand to count groups of digits: for each of the 4 fingers they are divided into 3 segments. When you are counting, you can point to the four fingers with your thumb, thus giving 12 and a pointer to which finger you are on. Then in your other hand you can keep track of how many units of 12, so 5*12 = 60. A base 60 system. But there was no digit for zero, although of course they had a way of dealing with zero, it was just not considered a number.
Mind you, despite having relatively good teaching in mainstream UK schooling they could all definitely have done/do with more headroom in their school maths.
And, to make things even more weird, spoken Latin, as opposed to their written numerals, seems to be refreshingly decimal except for those crazy duode- and unde- anomalies? From this perspective, it seems almost impossibly unlikely that they kept non-decimal systems around for so long.
Interesting how binary numbers creep into the multiplication algorithms used to operate on roman numerals.