 Until very recently, most people lived near where they were born in small communities, so you rarely needed surnames. If two people had the same name, you might distinguish them by a profession, Smith or Taylor, a location, Hill, Atwood, or a parent, Williamson Fitzgerald. And this brings us to one of the greatest mathematicians of the medieval period who was named Leonard. Since it came from Pisa, he was referred to as Leonardo Pizzano, Leonard from Pisa. However, since his father was named Benaccio, he would later be called Vibinacci. In 1202 Leonardo published Liber Abache. The title means Book of the Abacus, but the abacus is nowhere discussed. But since the abacus was a general computational tool, a better title might be Book of the Calculator, where Calculator is a person who does computations. Leonardo gave extensive instructions for working with Hindu-Arabic numerals. While other algorithms existed before Leonardo, he has proved particularly influential. Leonardo described several methods for division. One method led to a novel form of representing fractions. Suppose N can be written as a product of A, B, C, and so on, not necessarily prime. Leonardo wrote the rule for this division as, and the quotient would be found by dividing by A, B, and C, and so on, recording the remainders above each divisor. Leonardo gave the rules for division by composite numbers under 100, but he also described how to find the rules, essentially using the prime factorization, but not always. For example, let's use the rule to divide by 12, then let's interpret our answer in modern form. So we divide 175 by 3 to get 58 with remainder 1, and we'll record that 1 above the 3. We then divide 58 by 4 to get 14 with remainder 2, and we record the remainder above the 4 and our quotient next to it. To read this number, note that we're actually calculating 175 times a third times a fourth. So this first division, 175 by 3 to get 58 with remainder 1, that's really this first product, which gives us 58 and 1 third. This second division, 58 by 4 to get 14 with remainder 2, that's really performing the whole number part of this product, while the fractional part will pick up an extra factor of 4. And so in modern notation, this quotient could be expressed as 1 3 times fourths plus 2 fourths plus 14. Now we can use whatever rules we want, so let's find two different rules for division by 48 and then divide. So 48 has several factorizations, and each distinct factorization gives a different rule. First we'll use 48 equals 6 times 8, giving us the rule. We'll divide by 6, we'll record the remainder, then we'll divide 265 by 8 to get 33 with remainder 1, and we can rewrite our answer. Now dividing by 6 and then by 8 is a little daunting because we have to divide by fairly large numbers, so as an alternative we can use a factorization into smaller factors like 3 times 4 times 4, which gives us the rule. Now dividing by 3 gives, dividing by 4 gives, and dividing by 4 again gives, and so we have our quotient, which we can rewrite in modern terms as.