Mental Arithmetic and the Abacus

Doing arithmetic in your head is something you should work on, because it's often useful for working out whether you have enough money to buy two or more items, or to work out if a one-for-the-price-of-two offer is actually better value than a two-for-the-price-of-three offer on the same product. It also helps you perform much better with things like the Countdown Numbers Game. It may be that mental arithmetic skills are never tested in important exams in the country where you live, but don't limit yourself by the low expectations of your country: you should work at all these skills regardless.

Adding and subtracting

There is a simple trick you can use when faced with a sum like 34 + 48. If you add 2 to the 48 it becomes a much easier number to add to the 34: you can easily add 50 on in your head, getting 84, and then you can take the 2 away again to get the right answer. Alternatively you could ignore the 4 in 34 and just add 30 to the 48 (getting 78) and then add on the 4: that's the other way round from the way you would do it on paper. However, the Asians have a much better way of doing sums in their heads, and it involves an imaginary abacus: just take a look at this video.

The Abacus

In China and Japan it is common to use an abacus for calculating at high speed rather than using a calculator, and it's often faster too. People who practise a lot can actually use an imaginary abacus in their head and do the simpler kinds of arithmetic on it just as quickly as if they were working with a real abacus, so learning to use an abacus can be a stepping stone to being lightning fast at mental arithmetic. No one should be forced to learn to use an abacus in this way, but it's interesting and fun to have a go. I've written an abacus simulation program to let you have a shot at using one without having to buy it, so you can access it here: soroban. Right click on the link and open it in a new tab or window so that you can jump quickly back and forwards between the abacus and this page by keeping both open at the same time. A simple abacus would have nine beads in each column so that a column could hold any value from 0 to 9 where each bead is worth 0 or 1 depending on which end it's been pushed to. The soroban reduces the number of beads to 5 by using the trick of making one bead worth 5 normal beads, and that bead gets its own track away from the other four. So, one bead is worth 0 or 5 depending on where it is, and the other four beads in that column are each worth 0 or 1 depending on where they are: that means that a column can hold a maximum value of 9.
indentKeep the fingers of your right hand on the J K L ; keys and your thumb on the space bar: these keys are used to move the beads. The beads only count when they are pushed towards the inner horizontal bar that runs through the abacus: if they are pushed away from it they are worth 0, but if they are pushed towards it they are worth 1 (or 5 for the high bead). You can work it all out for yourself by playing with it and looking at the numbers that appear. The d and f keys are used to move the active column from side to side. With the scales set up as they initially are, the middle column is used for units, while the column to its left is for the tens and the column to the right is for tenths. Further to the left are hundreds, thousands, ten thousands and so on, while further to the right are hundredths, thousandths, ten thousandths, etc. This means that if all the colums are zero except for the one immediately to the left of the one in the middle, and if that column has one of the low beads up and the top bead down, the abacus is holding the number 60. Go and fiddle with it and see if you can follow it. Then I'll show you how you can use it to add and subtract numbers.
indentYou'll notice that the beads become darker when pushed towards the inner bar, and that's because they only count as having value when they've been pushed that way, so you should be learning to recognise the patterns of dark beads and the numbers they represent. The aim when using an imaginary abacus in your head is to see the numbers as these shapes rather than as normal digits, because then you can imagine beads being removed or added to turn them into different numbers. It will take many weeks or months before you can do this at any great speed, but the ability will come over time if you practise for just a few minutes every day. If you press capital T you can switch the abacus into "teach" mode and use the drills to improve your speed and accuracy, perhaps listening to music at the same time and trying to make the moves at regular intervals in time with the beat: this can lead to you doing a lot more practice than you intended, because you can easily lose track of time.
indentThe biggest difficulty you'll have is actually the first step of learning how to add and subtract simple little numbers to and from a single column, but you can click the "intro" button to get a reminder of how to add or subtract any particular number that you're having trouble with. I will give you a more friendly introduction to it here though, just to help you get your foot in the door.

The easiest number to add or take away is 5, because all you have to do is move the high bead and leave the low four beads as they are. If you're adding 5 and have to move the high bead up to do so, you'll need to add 1 to the column to the left as well, whereas if you're subtracting 5 and you move the high bead down, you'll need to take 1 away from the column to the left as well. Have a go at this and check the written numbers underneath to see why this is the case. You should be able to see that if you're adding five to 5, 6, 7, 8 or 9, the high bead will have to be moved up and the answer will be bigger than 10: that's why the column to the left has to have 1 added to it. If you're subtracting five from 4, 3, 2, 1 or 0, the high bead will have to move down and the answer will be less than 0, so a 1 has to be taken away from the column to the left to compensate (you're really taking 5 away from 14, 13, 12, 11 or 10, and the ten part has to be taken from that next column for this to be possible). Whenever you need to adjust the column to the left in this way, you should use the keys u/i/o/p/n to make this adjustment rather than moving the blue column cursor, and if adjusting it means that its value changes from 9 to 0 or from 0 to 9, you will need to adjust the next column to the left as well, and again this should be done without moving the blue column cursor: just use capital letters U/I/O/P/N. If more columns are affected, you will have to move the cursor, but that won't happen very often. The drills will actually teach you when to adjust the column to the left (pressing capital F changes the mode to control whether it tells you about mistakes made in the column to the left - to begin with you might not want to worry about that, so it's set to ignore errors of that kind until you switch it on by pressing F), so the most important thing for you to remember for now is simply that: when adding or taking away 5, you need to move the high bead. Go and try it now.

Adding 1 is usually fairly easy as well as you usually only need to move one of the low beads up, but if there is no bead there to move up, you need to move all the beads instead. If that results in the high bead moving up, then the next column to the left needs to have 1 added to it, as before. Go and practise the drill for adding 1 now. Once you've done that, try the drill for taking away 9: you'll see that it's more or less the same thing as adding 1, though the column to the left needs to be adjusted much more often to handle overflows. Expect to make loads of mistakes to begin with, but don't worry about it: within a week or two you'll get the hang of things, and then you'll just get faster and faster. Five minutes a day is all it takes: trust me on this, because it does work and it's well worth doing. Again, let the drills teach you when to adjust the column to the left: you will automatically get the hang of it. When adding 1 or taking away 9 you will either move one bead up, or all the beads will move if none of the low beads can be moved up.
indentAdding 2 and taking away 8 are very much the same thing as each other. If you can do it by pushing two of the low beads up, do so. If you can't, you have to push three of the low beads down instead and move the high bead. Try it out and see it with your own eyes. As always, use the drills to teach you when you need to make adjustments to the column to the left. When adding 2 or taking away 8 you will either move two low beads up, or three low beads down while moving the high bead..
indentAdding 3 and taking away 7 are very much the same thing as each other too. If you can do it by pushing three beads up, do so, but otherwise you must push two down and move the high bead. The drills will teach you when you need to adjust the column to the left. When adding 3 or taking away 7 you will either move three low beads up, or two low beads down while moving the high bead..
indentAdding 4 and taking away 6 are also very much the same thing as each other. If you can do it by moving all four low beads up, then do so, but most of the time you will need to move one low bead down while also moving the high bead. The drills will teach you when you need to adjust the column to the left. When adding 4 or taking away 6 you will either move all four low beads up, or one low bead down while moving the high bead.
indentAdding 6, 7, 8 or 9 and taking away 1, 2, 3 or 4 all involve following the above rules but with the beads moving in the opposite directions. Clicking the "teach" button followed by the "varied sums" button will give you lots of practice with all the different moves. It will all be very slow at first, but within a week or two you will find that you can work fairly quickly without making many mistakes. More practice will make you faster and more accurate all the time, so just give it five minutes every day and time will do the work for you. By the way, it is very important when adding and subtracting numbers on the soroban that you should work from left to right instead of the right to left way of doing sums on paper: your job is to become really quick at this so that you can do it all as fast as numbers are read out to you, and that means you have to work with those numbers in the order that they are read out, so you're going to deal with the big-value parts of numbers before the small-value parts: thousands are going to have to be done before hundreds, hundreds before tens, and tens before units. Your ultimate aim is to be able to do this in your head on an imaginary abacus. It's a good idea to practise visualising an abacus from early on, so go into the "varied sums" not only to improve your speed and accuracy at using an abacus, but every now and again you should try closing your eyes and imagining doing each sum on the abacus in your head before doing it on the actual abacus in front of you. You may be surprised at how easy it is to learn to do this. See the numbers as beads and imagine them moving just as they do on a real abacus: it isn't very hard to keep track of two or three colums, and many children in China and Japan learn to visualise six or more columns in this way, giving them enough space to work with that they can multiply two-digit numbers together on their imaginary abacus as quickly as using a calculator.

I may add more to this page later to show how multiplication and division are done on a soroban, but I certainly do intend to program a tutorial for both these things into the soroban itself at some point, though it'll have to wait for a while as there are many other parts of this Web site which need to be built more urgently. There's more than enough here to get you started in any case, and it will take you a few months to get really good at adding and subtracting numbers in your head, but feel free to e-mail me once you feel ready to move on to the next step. When enough people have reached that stage, I'll get the next bit programmed for you.