6CHAPTER 1. NUMERATION SYSTEMS1.2Systems of numerationThe Romans devised a system that was a substantial improvement over hash marks, because it useda variety of symbols (or ciphers) to represent increasingly large quantities. The notation for 1 is thecapital letter I. The notation for 5 is the capital letter V. Other ciphers possess increasing values:X = 10L = 50C = 100D = 500M = 1000If a cipher is accompanied by another cipher of equal or lesser value to the immediate right of it,with no ciphers greater than that other cipher to the right of that other cipher, that other cipher’svalue is added to the total quantity. Thus, VIIIsymbolizes the number 8, and CLVIIsymbolizesthe number 157. On the other hand, if a cipher is accompanied by another cipher of lesser value tothe immediate left, that other cipher’s value is subtracted from the ﬁrst. Therefore, IVsymbolizesthe number 4 (V minus I), and CMsymbolizes the number 900 (M minus C). You might have noticedthat ending credit sequences for most motion pictures contain a notice for the date of production,in Roman numerals. For the year 1987, it would read: MCMLXXXVII. Let’s break this numeral downinto its constituent parts, from left to right:M = 1000+CM = 900+L = 50+XXX = 30+V = 5+II = 2Aren’t you glad we don’t use this system of numeration? Large numbers are very diﬃcult todenote this way, and the left vs. right / subtraction vs. addition of values can be very confusing,too. Another major problem with this system is that there is no provision for representing thenumber zero or negative numbers, both very important concepts in mathematics. Roman culture,however, was more pragmatic with respect to mathematics than most, choosing only to develop theirnumeration system as far as it was necessary for use in daily life.We owe one of the most important ideas in numeration to the ancient Babylonians, who werethe ﬁrst (as far as we know) to develop the concept of cipher position, or place value, in representinglarger numbers. Instead of inventing new ciphers to represent larger numbers, as the Romans did,they re-used the same ciphers, placing them in diﬀerent positions from right to left. Our own decimalnumeration system uses this concept, with only ten ciphers (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9) used in”weighted” positions to represent very large and very small numbers.