The dozenal system

Today we use a decimal (base 10) number system, but not all cultures have done the same throughout time. The Mayans, for instance, used both quinary (base 5) and vigesimal (base 20) systems, while the Babylonians used a sexagesimal (base 60) system. In this video, Dr James Grime discusses the dozenal (base 12) system, and how our lives might be easier if we decided to use it.

Video source: Numberphile / YouTube.

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JAMES GRIME: There are people out there who want to change the way that you count. They want to change the world so that we stop counting using 10 symbols and start counting using 12 symbols. I love them. I think these guys are great.

Let’s talk about what we know so far. So, the way we count is the decimal system. It uses 10 symbols. So, let’s just write those out, first of all. 0, 1, 2 … 8, 9. Once we get to this point, we start to have to make our numbers using combinations of the symbols we’ve already got. 12, 13, 14 … 22, 23. Now, if we had been born with six fingers on each hand, then we would have 12 fingers, and maybe we would have started using something called the dozenal system where it’s based on 12 symbols. And there are some people out there who think this is the system we should be using. They think we should have six fingers on each hand. No, they’re disappointed we don’t have six fingers on each hand, because they think it’s a much easier way to count.

So first of all, if we’re going to use 12 symbols, I need two extra. Let’s have a look at what they suggest. For 10, we’re going to use a symbol like this, which is called ‘dek’. And for 11, we’re going to have a symbol like this, which is called ‘el’. So now it reads 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, dek, el. This [10] would be 12. In fact, they say ‘do’. So this is do, do 1, do 2, do 3, do 4, do 5, do 6, do 7, do 8, do 9, do dek, and do el. So, do el would be 12 and 11 more. So that’s 23. And then you keep counting. 2 do is 24, so: 2 do, 2 do 1, 2 do 3, 2 do 4, and so on. 

Just like decimal system, really. It’s the exact same idea. There’s no difference between the dozenal system and the decimal system. That would be 3 do, this would be 4 do. You keep going and you would get 9 do, then you would get dek do, el do, and then you’d get do do. Do do. 12 12s. 12 12s, that’s 144, 12 12s, or gross. So if we write that out, 12 12s will be written like this [100]. It’s 144, or gross, or gro. So that would be a gro. Something like this [1000] would be a mo. So if we take this year, right, it’s 2012, which is a good year to pick, since we were talking about 12. So if we took the year 2012, in the dozenal system, it’s written like this [1 1 el 8]. That is mo gro el 8. That’s the current year if you count in base 12.

Now there is a reason why these guys want to count this way. Because they say it’s going to make arithmetic much easier. I want to make this clear. For serious maths, this system would not make any difference. But for day-to-day life, for your regular shopping, these guys reckon that this 12 system would make like much easier because of the number of factors that 12 has. 

So 10 has—let’s see. How many factors does it have? It has 1; 10 has 2, 5; and 10 itself. But 12 has much more factors. It has 1, it has 2, it has 3, it has 4, it has 6, and it has 12 itself. The reason this is useful is it will make calculations using thirds and quarters much easier than they are in decimal.

Let’s do something like a 4 times table. If we did that in decimal, it would look like this. OK, it’s 4, and then 2 times 4 is 8, 3 times 4 is 12, 4 times 4 is 16, 5 times 4 is 20. 

If we did it with the dozenal system, then we would have 4, and then we would have 8, then we would have do, which is their word for 12. Then we would have do 4, do 8, 2 do, 2 do 4, 2 do 8, 3 do, and so on.

This pattern is a much easier pattern to teach children. 4, 8, 0, 4, 8, 0, 4, 8, 0. The patterns are much easier to see. But the real strength of it is when you start dividing. If we wanted something like 1/3, you know in decimal that this is 0.3333 … it goes on forever, which is fairly ugly. But in the dozenal system, it suddenly becomes really nice. If I want to write 1/3, that’d be 4 out of do. 4/12 is 1/3. 4 out of do, which is equal to 0.4. 1/3 written in this system is 0.4. It’s not something that goes on forever. That’s the sort of thing that decimal does. It’s horrible. In dozenal, it doesn’t happen. 

Let’s do something else. Let’s do a quarter. A quarter of 12, a quarter of do, is 3 out of do, which would be 0.3. 1/2 would be 6 out of do, which is 0.6. If you want to do 1/6, that would be 2 out of do, which is 0.2. Suddenly, all these become much simpler. You can see now why they like it. Suddenly, all those horrible things that happen in decimal don’t happen in dozenal. 

If we had been born with six fingers on each hand and this is the system we learnt, and then I came to you and said no, I’ve got an idea, let’s not use base 12, let’s use base 10. You would laugh in my face. Why would you use base 10? It’s not as good.

Now, naturally, we actually count in twelves. Historically, weights and measures were in twelves. So you have feet and inches, 12 inches to a foot. And currency was counted in twelves as well, because it was natural. It made dividing by halves, dividing by thirds, dividing by quarters and sixths much more easy and much more practical. And that’s the system we had. Then in the French Revolution, the French got obsessed with decimalising everything. They wanted to make weights and measures easier to use, so they wanted to decimalise it, divide it into 100 because we use the decimal system. Now, they had a choice. They could have decimalised everything, or they could have changed the way we counted from 10 to 12. If they had done that, they could have kept all the weights and measures the same and all the calculations would have been as easy to do in dozenal as they are today in decimal. Now, they chose to make things metric and keep the system. Maybe they made the wrong choice. In fact, in the French Revolution, they went so mad with decimalisation that they had a decimal week, they had a decimal calendar, and a decimal clock as well. They did that for a few years, but it didn’t take off.

BRADY HARAN: What would pi be? Would pi be the same?

JAMES GRIME: So, if we’re talking about serious maths, something like pi, it would look a lot different, and it would involve these dek symbols and el symbols. And if we look at it, it looks weird and unusual. But if you were born with this system, it would be fine. It would be normal. There’s no significant difference in the mathematics. The real difference this would make would be to just your regular life. It might make arithmetic easier for children as well. 

BRADY HARAN: But the one thing, I guess, people are going to say is, when you’re very small and you’re learning about numbers, your fingers and your hands are a really useful learning aid. You’d be taking that away from children right at the start.

JAMES GRIME: Well, to answer that, there are some cultures that still use base 12, and what they do is they count on the segments of your fingers. 1, 2, 3, 4, 5, 6, 7, 8, 9, dek, el, do. Just as easy to use.

Ethnomathematics

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