My Base 11 Abacus Bugbear



So the last two days have been a learning adventure for me. I read this post by @srcav and it has sent my curiosity into overdrive. I’ve long considered getting an abacus to help my children get to grips with basic numeracy. My eldest is six and youngest is four, so it feels like their little brains might be ready for it now. I also found this in my maths training room which brought everything back into the fore again:


That was as much as I’d thought about it. However, discovering that the ‘standard’ (Western) abacus is in fact in Base 11, not Base 10 (i never looked very closely) has led to a very simple but apparently difficult question. WHY?! Almost everything in our culture is in Base Ten (counting… for a start). There are a few exceptions, like time (base 60) and programming stuff (hexadecimal ASCII is base 16, and binary is base 2), but considering an abacus is intrinsically linked to a number system, it seems more than a little odd that a counting tool is in the ‘wrong’ (least appropriate)¬† base.

Time for a brief break and a history lesson.

The first ‘abacus’ was base 60 and designed by the Sumerians around 2500 BC. Their version was Base 60. They were really rather clever, and are also credited with finding the area of triangles, volume of cubes, multiplication tables and creating the concept of place value. Very nice.

Then in 5BC similar devices cropped up around Greece (from where the word Abacus is derived) and central America.

The Abacus started to take the form of the instantly recognisable beads and sticks when the Romans had a punt at it. It’s generally believed that the Chinese got in on the act somewhere along the line and modified the Roman design.

And it’s the Ancient Chinese version that is still in use today… in China at least. Two other significant modifications (including the terribly conceived Base 11) came later, but we’ll get to that.

So the Chinese Abacus is known as the Suanpan Abacus. And it looks like this:


The lower deck are known as the ‘earth (or water) beads’ and the upper deck the ‘heaven beads’… Stop sniggering at the back.

Lower beads have a value of ‘1’, and upper beads a value of ‘5’. And you read it from right to left in terms of quantity, so column two (from the right to left) is your ‘tens’ column etc.

The Suanpan can be used both as a Base 10 (yay!) system or a Hexadecimal (Base 16… were you listening earlier?) system. So if you want to play around with it for maths AND computing, this is the droid you’re looking for.

Here is how you would use one column in a decimal system (zero would be ‘heaven up, earth down’):


And as hexadecimal:


And here’s how you can do sums:


Simple… ish. Maybe a video will help:

The Japanese thought it was all a bit meh, so they redesigned it and created the next significant incarnation: The Soroban Abacus.:


Notice it has only FOUR earth/water beads, and only ONE heaven bead in each column. Hence it works only in a Decimal Base 10 system. Like so:


This system seems to make more sense (to me at least).

Here’s a handy tutorial on how to do addition using a Soroban Abacus:

The Soroban Abacus is still widely used in schools today (not in the West) and has even been used as a basis for a watch:


I want nothing to do with a Soroban Base 10 watch that tells Base 60 time with Base 12 hours. I think my head might explode.

So, back to the abacus you’re perhaps most familiar with…


Ten beads on each row. Which makes it Base 11.

The closest cousin to the ‘School Abacus’ – it really is officially called that, is the Russian Abacus:


Ten beads on each row? Check. It differs in that it has bent wires to keep beads separate and is purposefully split into groups of four beads and a divider of two different coloured beads (black in the picture). This is apparently the first abacus normally used vertically (like the School abacus).

The Russian Abacus was taken to France in 1820 and is therefore the most likely reason we have the bizarre bastardised Base 11 abacus in production today.

Now there are some other Abacuses, such as this (frankly amazing) binary abacus and the ridiculous Bagua Abacus (see below) but I think I’ve digressed PLENTY at this point.Round_Bagua_Chinese_Abacus_Circular_Suanpan_ChinaDaily

Back to the issue.

You can’t use a Base 11 abacus for counting very efficiently. You can’t use it for addition, multiplication, subtraction etc without going against more instinctive methods, whereas a Base 10 abacus would compliment the ‘typical’ methods of performing numerical operations that you would learn at school. So why are they Base 11?! No-one seems to have questioned their efficiency, or people have only used them for counting perhaps, without really unlocking their true potential like in Japan, or India, or China.

I tried to track down a Base 10 abacus and really struggled to find anything apart from the Soroban, which I still think might be a little counter intuitive for a 4 or 6 year old.

Eventually I found this on ebay:


– which funnily enough doesn’t fit into ANY of the categories I’ve just spent however long telling you about. Furthermore I can’t find ANYTHING about whether this is an official “breed” of abacus, or just something someone made because it made more sense. And in a time where phrases like basic numeracy and a ‘deep’ understanding of addition, subtraction, multiplication and division are being thrown around like confetti, without any real strategy as to how to achieve those things, might it be time to bring back the Abacus? Properly? In Base 10?? It’s a nice physical representation of an abstract concept, much like numicon, but with more dynamics.

Either way, if you watch the video clip below, I hope you’ll agree that using it sits nicely in the ‘intuitive’ part of your brain:

So i bought a few, although I suspect they’re the size of a packet of tic tacs. We’ll see. I’ll let you know how my children and I get on.