ABACUS: MYSTERY OF THE BEAD
The Bead Unbaffled - An Abacus Manual
The Leap of Faith
It's well know that over time the Japanese soroban has gone through a process of evolution. What began as a 2/5 bead abacus centuries ago later changed to an instrument with 1/5 beads before finally evolving into the 1/4 bead soroban we see today.
A modern day 1/4 soroban is a very thoughtful and efficient tool. But sometimes it can be difficult to understand how some of the process works. A good case in point is the problem 1000 - 1 = 999. It requires what I call the of leap of faith. Because it can't be seen, sometimes new students struggle with the technique and many more just accept it without actually understanding the process. In a sense what happens, happens behind the scenes. As a result "seeing it" is probably best illustrated using the help of an older style 1/5 bead soroban.
Of interest, some years ago I corresponded with the CEO of a well know soroban manufacturing company in Japan. When I asked her about the 1/5 bead soroban she told me that she grew up learning the 1/4 bead and had never gotten used to the older style. But then she went on to say that her mother, on the other hand, was never comfortable using anything but the older model 1/5 bead soroban. The following examples may shed some light as to why this might be.
Before moving on to the 1/5, let's first investigate the process on a 1/4 bead soroban.
Example: 1000 - 1 = 999 on a 1/4 bead soroban
Fig.1 1/4 bead soroban showing 1000. A quick look tells us that we don't have a value of 1 available to subtract from the unit rod.
Fig.2 Use the techniques for complementary numbers and subtraction illustrated at Abacus: Mystery of the Bead. Work from left to right and subtract 1 from the thousands rod; then in quick succession add 9 onto each of the rods on the right. This leaves us with the answer 999.
This is the way it's done on a 1/4 bead soroban. The trouble is for many this can't really be reconciled because what happens can't actually be seen.
Now let's explore what happens using a 1/5 bead soroban. The 1/5 is a great tool for seeing the process because we can place a value of 10 on any of its rods.
Example: 1000 - 1 = 999 on a 1/5 bead soroban
Fig.3 1/5 bead soroban showing 1000 (1 bead in the thousands column)
Fig.4 1000 again but in a different configuration. If we borrow one bead from the thousands column and place its value (1000) in the hundreds column the soroban still shows 1000 (500 above the beam + 500 below the beam).
Fig.5 Once again the soroban is showing 1000. If we borrow one bead from the hundreds column and place its value (100) in the tens column the soroban still shows a value of 1000 (900 + 100).
Fig.6 Still showing 1000. If we borrow one bead from the tens column and place its value (10) in the ones column the soroban shows a value of 1000 (900 + 90 + 10).
Fig.7 Finally subtract one bead from the ones column. What remains is the answer to the problem 999.
The above is essentially what happens when we take the "leap of faith" while solving a problem like 1000-1=999 on a 1/4 bead soroban. Because we can't actually see each step as we borrow from one column and add its value to the next, it can be difficult to conceptualize. It's one of the reasons why it took many years for the 1/5 bead soroban, which can show a value of 10, to evolve into the more efficient 1/4 bead soroban which only shows a value of 9. It's all a of leap of faith.
Other examples worth trying are:
1000 - 2 = 998
1000 - 5 = 995... and so on
But then it's a good idea to take it a step further and try;
1000 - 15 = 985
1000 - 27 = 973... and so on
1000 - 142 = 858
1000 - 287 = 713... and so on
But don't stop here. By all means take a calculator and your soroban and make up as many problems as you might need in order to learn the process. It's a good way to become conversant with the way it all works.
▪ Abacus: Mystery of the Bead
▪ Advanced Abacus Techniques
© January, 2010
Totton Heffelfinger Toronto Ontario Canada