This is basically what the Romans did. Except that since Arabic numbers didn't exist, the numbers were converted to an abacus-like representation, the calculations were performed, and the result was converted back.

The abacus also enjoys versions in China, Japan, India, Russia, Korea, and elsewhere. I concentrate on the Japanese variant, the soroban, since it is quite elegant in it's handling of decimal arithmetic. The soroban design reached it's modern form around 1920.

I have two instruments - one with five rods and large beads, and one with 27 rods. The smaller one is pictured. Note that it has a horizontal dividing bar, with a four beads below, and one above. The beads below are worth one each - counting zero through four, and the one bead above is worth five - counting zero or five. Beads gain value when they are placed toward the bar. The soroban is showing all fives.

It works just like Arabic numbers. The rod on the right is the ones column. The next to the left is tens. This soroban allows one to count from zero to 99,999. Addition, subtraction, multiplication and division are possible. With 27 rods, one can work with longer numbers and use multiple numbers at a time, storing temporary numbers, etc.

Once addition, subtraction, multiplication and division are mastered, essentially any computation can be performed. When I first studied this I noticed two very significant features. First is that addition and subtraction are about the same speed as on an electronic calculator (I had a scientific calculator in 1975). Basically, it only takes the time required to enter numbers. Secondly, after performing numerous calculations, it became apparent that I had not made any mistakes. I couldn't say this for pencil and paper methods. I couldn't even say this for calculator use. The error rate for calculators is increasing, as cheaper buttons are used.

As one practices, it may become apparent that an abacus can be visualized mentally. This is similar to playing chess without a board. Mental arithmetic seems to be about twice as fast as it is with the physical soroban, extending the high reliability by eliminating errors due to change of focus. After practicing mental arithmetic for awhile, I could extend my mental soroban to as many rods as I wished. For example, I multiplied two 9 digit numbers to arrive at an 18 digit result. This was remarkable, in that without the visualized soroban, I couldn't remember an 18 digit number. I found that I could use many soroban images to hold distinct numbers. There didn't seem to be any practical limit. Your mileage may vary.

In calculus, Taylor series expansions for trigonometry functions give formulas for computation. Since the calculator could verify a result, I decided to compute a nine digit sin(x) function, in my head. I started with a slightly random number. It was something like 23.7 degrees. I converted it to radians, then plugged it into the formula. I carried ten significant digits for all computations so that I would have nine good ones when finished.

I just sat there for about 35 minutes. I wrote down the answer, and checked it on the calculator. I was correct.

Oddly, I did find a limit to mental calculation, sort of. This was approximately the largest problem that I had patience to solve. I must have held at least six ten digit numbers in my head while wrestling with this monster.

Nearly thirty years later (without practice), I decided to teach the soroban to my seven year old son. It is, after all, an excellent way to learn and perform arithmetic. So, I bought the two instruments and the book, and started again. These pages come out of this newer journey.