Calculating Machines --- Abacus --- Napier's bones --- Slide Rule --- Logarithms --- Calculator
Many of the early number systems were hard for doing even simple arithmetic. So people developed a machine to do it for them. This started as stones in lines dawn in the sand. In fact, the word 'calculate' comes from the Latin word for 'stone', calculus. Eventually people developed the abacus. There are different types of abacus, so we will start with the simplest. Every bar has nine beads, and zero is when all the beads are at the top. The abacus has to be flat on a table, or all the beads would slid to the bottom, of course. But on this page, I will talk about 'pushing up' or 'pushing down' beads, meaning pushing them to the top or bottom of the screen.
Enter a number from 1 to 9999999 to see it on a simple abacus, or enter a number to count with.
|This is how an abacus looks before you start to enter a number. All the gaps are at the bottom, representing zero.|
|Now some beads have been pushed to the bottom. They don't really change colour, of course, but this is to make them easier to see. There are two beads in the tens column, and three beads in the units column, so this shows 23.|
|Now we need to add 41 to the 23 that is already on the abacus. Four beads are pushed down on the tens column and one bead on the units column. These are coloured blue to make them easy to see. To find the result, count the number of beads in each column - six tens and four units, making 64.|
|It gets harder if we need to 'carry'. Imagine adding 67 and 52. It's easy to set up 67 on the abacus.|
|We can add the two units of 52. However, when we try to push down five beads in the tens column, we can't. We've only pushed down four beads, and we've run out of beads.|
|So we push up all ten beads in the tens column. This is ten times ten, or a hundred, so we must also push down one bead in the hundreds column.|
|We now have plenty of beads available in the tens column, so we can push down the remaining one bead in the tens column, giving the answer, 119.|
|You can also use an abacus to subtract numbers. If the sum is 97 - 45, you enter 97 in the normal way.|
|Now you push up (rather than down) four of the tens beads, and five of the unit beads. This gives the answer, 52. The coloured beads above the gap show which beads have been pushed up.|
|You may need to 'carry' with subtraction as well. The sum is 52 - 18. Enter 52 in the normal way.|
|When you try to subtract 18, you have a problem with the units column. You can take off two beads, but then run out.|
|So you push down all ten beads in the units column, which means you must push up one bead in the tens column, to balance it.|
|Now you can push up the remaining six units beads.|
|Push up the one bead in the tens column (remember the original number than we were subtracting was 18!) The answer is 34.|
Here is an abacus for you to play with. Click on a bead to push beads into the gap. Click on a bead above the gap to add a number, and below the gap to subtract a number. You can use the abacus leaving the beads green, as if it was a real abacus, or you can chose colours. For example, to add two numbers, chose a colour, such as blue, then enter the number on the abacus by clicking on the beads. Then change the colour, and add in the second number's beads. If you make a mistake or 'carry' then the colours get messed up. If you get muddled, you can click on Restart to start again. You may find it easy to click one bead at a time, rather than move several at once. You can choose your own numbers to enter into the abacus, or you can ask for a sum, addition of subtraction, easy or hard. For the hard sums, it's best to work from right to left, doing the units first, then the tens, and so on.
An abacus uses a number of beads to represent a number, so it is a unary system. But a number of beads in one column are a different number to the same number in a different column, so it is a positional system. It doesn't really have a zero. If there is no value in one column, then there are no beads there.
The simple abacus has ten beads per column. It isn't really used any more for calculation, although children sometimes use them to learn about numbers. Abacuses are still used in the Far East, but they look more like the abacus below.
The zero position is for all beads to be away from the central bar, as the beads on the left are. The top two beads represent five each, and the bottom beads represent one. The units column has a single 'one' bead and no 'five' beads, so this is one. The tens column has one 'five' bead and two 'one beads, representing 70. The hundreds has a 'five' bead alone, so that is 500. Then there is 3000 and 60,000. So the total number is 63,571.
This sort of abacus is easier to use, as the human eye finds it a lot easier to detect five beads or less, rather than larger numbers up to ten. You can quickly see the difference between 7 (a 'five' and two 'one's) and 8 (a 'five' and three 'one's), but 7 and 8 look similar on the simple abacus. You can see that the Romans would like this sort of abacus, as they had a symbol for five as well as a symbol for one. You may wonder why there are two fives as well as five ones, allowing a value up to fifteen in a single column. It's for much the same reason as the simple abacus having ten beads in a column. It allows you to store a number before having to carry it.
Below is an abacus of this type for you to play with. Click on the beads to move them.
© Jo Edkins 2007 - Return to Numbers index