Why is the decimal abacus more popular than the binary one?
An abacus is basically a device with beads in rows, each for 1 digit. A decimal abacus has 4 lower beads (signifying 1 unit) and an upper one (signifying 5 units). So 82 would be 3 beads plus the 5-bead on the tens row, and 2 beads on the ones. A decimal abacus just has 1 bead per row for each digit.
There are many shortcuts that make calculation on an abacus much faster for an experienced user. Is it that such shortcuts are limited on a binary one? It would be much quicker for a person to perform binary operations (even on more digits) than decimal ones, after practice. Converting decimal to binary numbers and back should also be easy on the abacus, but it could be the reason why it is not preferred.
If a binary abacus becomes popular, one could also create an octal one, with 1 4-bead and 4 1-beads, or even a hexadecimal one, with 3 4-beads and 4 1-beads.
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$\begingroup$Good. We are talking about the base and the length of a system for expression. The base is the number of individual symbols we have in one place, and the length is the number of places. When it comes to Number, we call it n-based system, where n is the measure of the base. Generally, if the base is large, the length would be small. But a larger base needs more complex operation. A small base will need long length, means more digits, for a number. More digits need more time or more loops to finish a job.
There must be a optimized size for base for a given problem. Then we need to understand how the human brain works, which is not a practical task though.
$\endgroup$ $\begingroup$Because of impedance mismatch, inertia, economies of scale, and technology limitations.
Better is not enough. To win over established solutions, things have to be so much better that it's worth teaching, learning, and manufacturing, which is an uphill thing for a binary abacus in a world of humans already used to and using decimal.
The abacus got popular because people like merchants just needed basic arithmetic to happen faster and more reliably.
Base 2 had a disadvantage from humanity's beginning:
There is no mindlessly obvious way for base 2 to scale well for manual human use: "9" or "nine" versus "1001" or "one zero zero one" or "one 'thousand' one" is much worse to write or say or read or listen to. Humans encode information in two-dimensional shapes when writing, and in a wide variety of sounds when speaking, so humans have more natural "digits" than just two to encode numbers with. Also, while normal finger counting is more like a weird "base 1" than base 10, most people have a total of 10 counting fingers, making transition into something based on 10 more intuitive. You can trivially solve this by just realizing that any larger power-of-two base like hexadecimal is just compact shorthand for binary, but that takes more abstract thought to understand and proactively creative yet logically sound thought for someone to initially make the connection like that.
So abacus inventors were probably already used to thinking in base 10 when they invented the abacus (base 2 doesn't feel more fundamental or superior until you're pretty deep into understanding math, computing, or information theory).
Then early and careful adopters of the abacus probably wanted to check if it worked and understand it while evaluating it, and that is easier to do if the stuff in the beads could be easily mapped to a base they were already familiar with.
Abacus users were in a world of base 10 users. Every time you need to use a binary abacus in a decimal world, you have to convert on input and output. In my experience this ends up being the slowest and most effortful part of using a binary abacus. If your human customer wants to buy something that costs \$52 and another thing that costs \$13, you'd be signing up for a multiplication problem just to get those numbers into your binary abacus and a division problem to get the numbers out. So if you're already used to basic arithmetic on a decimal abacus, you don't feel obvious benefits from a binary abacus until you're either regularly working with power-of-two bases, or until you've moved beyond addition and subtraction.
Basically, the core idea of a binary abacus has the following advantages:
It makes certain operations possible. Calculating square roots, for example, is something you can actually do on a binary abacus. Your typical merchant didn't need to compute square roots. Ditto logarithms and exponents.
It makes multiplication and division much easier. No memorization of multiplication tables needed, very little mental effort. No guessing, lookup tables, or multiplication needed during division. Just copy bit patterns and add or subtract, respectively.
It makes addition and subtraction simpler - there are less rules for the operations, because there is only one kind of bead, which represents a power of two - there are no special cases like "this bead represents five of those beads in the same register".
However, really quick and ergonomic addition and substraction movements are only possible with the innovation of replacing beads with buttons that can be toggled between two very obviously distinct states, with the same fairly gentle touch, not just individually but in sweeping motions that toggle multiple buttons.
This wasn't really possible until modern technology. It's close to impossible to pull off without modern mass-produced electronics, and would've taken specialized physical pieces until you could do it in software on modern computers, tables, and phones. By then we had calculators.
But once you do have that... well, on the binary abacus I coded, I just computed all powers of five up to 5^8 (five to the power of eight) by hand for this post (in this example, less significant bits are to the left, and each row is an increasing power of five):
This took me less than three minutes. That's including a couple of error checks I did along the way, by dividing by five and verifying I got back to the previous bit pattern. But I don't know how much time it would take a really well-trained user on either abacus type though. Proficient users of both decimal and binary abacuses could probably go much faster.
If all you had is traditional bead-on-rail abacus technology, you could still make a binary abacus and do this, but it would get annoying and tedious pretty quick I think, because additions are nicer when you can ripple a carry bit with one smooth motion (ditto subtractions and borrow, when checking work by dividing), and it takes more mental overhead to change the motion depending on whether the bit you're flipping is already up or down.
But since this is digital and I tuned it precisely for this, I just press and hold the left mouse button and move the mouse over multiple bits (or touch and swipe on touch screen) (or press and hold Shift, which I often do instead of the left mouse button, especially when I have to use a laptop trackpad) to toggle multiple bits. A friend of mine runs that website (I just implemented the ergonomic interactive web app in that earlier link), and there's a decent introductory PDF tutorial linked from the main page for some of the basic movements you can do.
Of course you also might not be able to store each intermediate result at the same time on traditional abacus technology, since beads-on-rails has less efficient space and the sizes of everything on a physical abacus have to be reasonably usable by humans.
So, nowadays, if I had to use an abacus, in many situations I'd probably prefer the binary one. I also sometimes use it for other purposes, like when I'm thinking through bit-twiddling algorithms. But this is only because I'm very fluent in binary already - the necessary operations are obvious and super simple to me, so for me most of the learning curve was already surmounted long long ago. Most people aren't, and I'd assume most people to find it uninteresting and non-trivial to learn, because they're used to thinking, writing, and communicating in decimal.
$\endgroup$ $\begingroup$Another thing to consider. An abacus does not care how many beads you move. Decimal abacus handy and need to do base 5 math for some reason, use it like when you do decimal math only move the carries over to the left at 5 and above instead of 10 and above.
This is why you can do hexadecimal math on a Chinese abacus since it has 5 ones counters and 2 fives counters.
Need binary? Just move fewer beads. No need to have a separate machine.
So Why is the decimal abacus more popular than the binary one? Couple of reasons.
- More people use decimal numbers on a day to day basis
- They can do binary operations on the same machine.
