 The dustboard was a standard computational tool for many centuries in many cultures. Unfortunately, mathematics was often used by charlatans, astrologers, numerologists, and other untrustworthy characters, so the dusty hands of the calculator had a negative connotation. But how else could one compute? So let's talk about the Fabric Revolution. The previous writing materials were either very hard to write on, stone, or expensive, leather. And what this meant was that you didn't use writing material to perform computations, but around the first century a new material appeared in China, paper. Now, during the Tang Dynasty, Imperial China and the Islamic states fought several border skirmishes, and in one of them a group of paper makers was captured, and in the 9th century a thriving paper industry existed in Samarkand, now in Uzbekistan. In 1427 Samarkand resident Shamshi Al-Khashi wrote The Calculator's Key, a detailed compendium of computational methods using the new medium. The work included sections of the basic operations of arithmetic, including the use of decimal fractions, and it also includes examples of extracting roots. We'll start with a square root. Al-Khashi gives the example of finding the square root of 331,781. Since we're taking the square second root, we'll break the number into two digit blocks, which Al-Khashi calls cycles. We'll start with the leftmost block. It helps to ignore all columns to the right of the working block. So we'll cover up those digits. The largest number whose square is less than 33 is 5, and so we write 5 immediately above and some distance below the 33. And the reason is we're going to fill in this intermediate space with a bunch of computations. So we'll subtract 5 squared 25 from 33, leaving 8. Now at the time paper was still fairly expensive, so Al-Khashi didn't write the subtrahend, but we will for clarity. Finally, he doubles the found digit to 10 and writes the digits offset by one place above the lower 5. And now we're working in the second cycle, so we'll reveal those numbers. So let's take a look at the next cycle, and at this point we should read our number as 817. So to make it easier to read, we'll copy those two digits down. Now since each cycle contains two digits, it's also helpful to think about this bottom number as 100. Again, we'll continue to ignore these cycles to the right of the working cycle. The next digit of the root will be the largest number that allows the procedure that follows. The product of the next digit and 100 plus the next digit must be less than the remaining part of the root, 817. Now by trial and error, we find that this number is 7. So we'll write down a 7. We'll add it to the lower number to get 107. Then we find 7 times 107 is 749, which will subtract from 817, leaving 68. Again, we'll write down the subtrahend, but Al-Khashi didn't. And as before, we'll double the found digit to 14 and shift. And it's helpful to think about adding 7 to 107 to get 114. And we're in the third cycle, so let's go ahead and reveal those numbers. And incorporating the digits in the last column, this is 6881. And now we got the next digit of the root. And again, we want the product of the next digit, whatever it is, and 1140 plus the next digit to be less than the remaining part of the root, 6881. And again, by trial and error, we find the next digit is 6. So we multiply 6 by 1146, that gets us. And we subtract, leaving 5. Now in an earlier section of the calculator's key, Al-Khashi discusses decimal fractions. So we almost certainly knew the procedure could be continued to find decimal digits of the root. Instead, he presented an approximation for the fractional part. If we continued, our next step would be to add 6 to 1146 to get 1152. So we add 1 to get 1153, and then divide into the remainder to get the fraction 5, 1153. And so the square root of 331,781 is approximately 576 and 5, 1153. Now today, paper is relatively cheap. However, for the first few centuries, paper was still relatively expensive. Cheaper than the alternatives, but still not something we would casually discard. And this promoted techniques that save space on the page. And so Al-Khashi's actual work omitted many intermediate steps and the rewriting of digits. So we wrote it out this way. But that was because we wrote down the subter hand. We recopied the menu end and wrote down a couple of intermediate steps. If we get rid of those, we can write the essential features in fewer lines, which saves paper. And so in Al-Khashi's work, only these lines appear. The significance of this effect can't be overstated. For centuries, algorithms that used less paper were favored over algorithms that might have been easier to understand, use, or teach. Since today paper is cheap, there's no reason to save paper at the cost of understanding. And this means two things. We can try to create better algorithms for basic arithmetic. Or there may be no point in reinventing the wheel. We can find them in historical sources. In other words, the history of mathematics is useful as well as important. And finally, we'll note one other thing. Al-Khashi's method of finding square roots is essentially identical to that of Liohe with some changes. The main one being that Al-Khashi's method is designed to work on paper and not on a counting board. And the fractional approximation that Al-Khashi gives has no equivalent in Liohe's work.