Today’s earlier post got into only the beginnings of abacus operation — mainly how to add numbers. To demonstrate how they have more utility than just adding and subtracting, this Sideband tackles a multiplication problem.
This also illustrates a property of abacus operation that doesn’t arise with addition. With pen and paper, we multiply right-to-left to make carrying easier. Because of the way an abacus works, multiplication has to work left-to-right.
The process is simple enough, but has lots of steps!
Let’s say we have a big-ish number, for example the speed of light, which is 299,792,458 meters per second, and we want to know how far light goes in one hour.
There are 60 seconds per minute and 60 minutes per hour, so there are 60×60 seconds per hour. Our multiplication problem, then, is to multiply the speed of light by 3,600.
299,792,458 × 3,600 = ???
(Spoiler Alert: The answer is 1,079,252,848,800.)
We begin by setting the abacus to the first number:
The multiplier, 3600, is 36×100, so a simple pre-step is to just move the number to the left two columns, leaving two zeros in the right-most two columns:
Now we have to multiply that by 36, which is 6×6.
It’s also 4×9, or any other combination of 2×2×3×3, but 6×6 gives us the best balance of what we multiply and how many times we have to do it. Two and three are easy, but require more iterations. It’s down to multiplying by six (twice) or by four and nine. Dealer’s choice.
Starting on the left, the first column is a 2, which we multiply by 6 to get 12. There’s a zero to the left, so it’s easy to add the 1 there.
Moving to the right, a 9, which we multiply by 6 to get 54. We change this column to a 4 and carry the 5 to the left:
The third column has another 9, which times 6, gives us another 54, so change this column to a 4 and carry the 5:
Now we have a 7, times 6, is (the magical) 42, so change this to a 2 and carry the 4:
The fifth digit is yet another 9, so change this column to a 4 and carry the 5:
Next up is a 2, times 6, is 12, so (since it’s already a 2), just carry the 1:
Only three more to go. The seventh digit is a 4, times 6, is 24, so all we have to do is carry the 2:
The eighth digit is 5, times 6, is 30, so change to a 0 and carry the 3:
The last digit is 8, times 6, is 48, so all we do is carry the 4:
And we’re (halfway) done. As a self-check, 299,792,458 × 6(00) is, indeed, 179,875,474,8(00).
Now we have to do it all over again, starting from the left.
Very quickly, column one: 1×6=6:
Column two: 7×6=42, change to 2, carry the 4:
Column three: 9×6=54, change to 4, carry the 5:
Column four: 8×6=48, keep the 8, carry the 4:
Column five: 7×6=42, change to 2, carry the 4 (which causes another carry):
Column six: 5×6=30, change to 0, carry the 3:
Column… well, you get the idea. Let’s just cut to the end:
And there it is, the number of meters light travels in one hour:
Maybe not. 😀
But there is something Zen-like about manipulating the stones. Part of the fun of an abacus is the physicality of it. It’s somehow very satisfying.
Stay speedy, my friends!