1. Mathematics as a Second Language Mathematics as a Second Language Mathematics as a Second Language © 2010 Herb I. Gross Developed by Herb Gross and Richard A. Medeiros Arithmetic Revisited next

2. Origins of Place Value Origins of Place Value © 2010 Herb I. Gross next Roman Numerals and the Sand Reckoner Lesson 1 Part 2

3. Roman Numerals next © 2010 Herb I. Gross

4. The next evolutionary step in counting came from the Romans, and what we now call Roman numerals. next © 2010 Herb I. Gross

5. As a historical aside, the Romans and Egyptians invented quite similar systems for enumeration. However, because Roman, not Egyptian, numerals are taught in the elementary curriculum (at least in the Western World), we will only discuss the Roman version. next © 2010 Herb I. Gross

6. Realizing that it was hard to keep track of “too many” tally marks at a time and wanting to take advantage of the fact that we are born with ten fingers, when counting tally marks the Romans decided to cross them out in groups of ten. As a “short cut” they came up with the innovative idea that if you were going to cross out groups of ten tally marks, why write the tally marks in the first place? next

7. As a result, quite cleverly, they decided to use the “crossing out” symbol by itself to represent ten tally marks, and since the “crossing out” symbol looked so much like the letter ‘X’ in their alphabet; they eventually introduced the symbol (letter) X to represent ten tally marks. = X

8. In short, the Romans used the symbol X to replace | | | | | | | | | |. In a similar way, since a tally mark looked like the Latin letter ‘I’; the letter ‘I’ became the symbol (numeral) for representing a single tally mark.

9. © 2010 Herb I. Gross next With X and I as defined above, what number is named by XXIII? Practice Problem #1 Answer: 23 next

10. © 2010 Herb I. Gross Solution for Practice Problem #1 next Each X represents ten. Therefore, two X’s represents twenty; and since each I represents one, we see that XXIII represents twenty three. That is… X Hence, XXIII represents 20 + 3 or 23. XIII 10 + 101 + 1 + 1 +or 23

11. © 2010 Herb I. Gross next While XXIII is more cumbersome to write than the numeral 23, it is still more concise and easier to visualize than | | | | | | | | | | | | | | | | | | | | | | |. 1 Notes on Practice Problem #1 Notice the use of the adjective/noun theme here. For example, the nouns are I and X and the adjectives are the number of times each letter appears. 1 Recall our previous discussion in which we said that we judge progress by what the new concept replaces not by what it was later replaced by. note next

12. © 2010 Herb I. Gross next Notice also the new level of abstraction. Notes on Practice Problem #1 For example, when we look at XXIII it is clear that there are two X’s and three I’s, but unless we are told, there is no way that we can guess what number is represented by either X or I. next

13. © 2010 Herb I. Gross next For example, if the Romans had decided to use X to represent 5 tally marks 2, XXIII would still consist of two X’s and three I’s. Notes on Practice Problem #1 However, it would then have represent two 5’s and three 1’s or 13. 3 2 This most likely would have happened if the Romans had decided to use the number of fingers on one hand rather than both hands to keep track of the number of tally marks. notes next 3What number we elect to have X represent leads to the concept of different number bases and will be discussed in greater detail as an appendix to Lesson 2. next

14. Continuing in this way, the Romans continued to use letters of their alphabet whenever they exchanged ten of a particular denomination. They replaced ten X’s by the letter C. X X X X X X X X X X = C next The letters were not chosen at random. The letter ‘C’ is the first letter of the Latin word centum that means a hundred. © 2010 Herb I. Gross

15. A century is a hundred years; per cent means “for each (per) hundred;” there are one hundred cents in a dollar; and elite Roman soldiers were assigned in groups of a hundred, and their leader was called a centurion. next © 2010 Herb I. Gross

16. C C C C C C C C C C = M They replaced ten C’s by the letter M. next As the numbers they dealt with got larger, the Romans continued the process of choosing a new symbol (letter) whenever they amassed 10 of the previous denomination. © 2010 Herb I. Gross Just as the letter C was not chosen at random, neither was the letter M. More specifically, the letter ‘M’ is the first letter of the Latin word milla that means a thousand.

17. A millennium is a thousand years; a meter is a thousand millimeters; a gram is a thousand milligrams; and there was once a coin called a mill (that became obsolete due to inflation). A mill was a tenth of a cent (just as a cent is a tenth of a dime and a dime is a tenth of a dollar). Since there are a hundred cents in a dollar, there were a thousand mills in a dollar 4. next 4 In some states property taxes are assessed at a rate of “per hundred dollars.” Since the average home is assessed at many thousands of dollars (rather than hundreds of dollars), a fraction of a cent per hundred dollars can add up to a significant amount of tax money for a state or municipality. For this reason, even though there is no longer a coin called a mill, some states still use it in establishing a tax rate. Thus, even though we may not be comfortable with the notation, just as we are used to reading, for example, $27.34 as “27 dollars and 34 cents”, it would not have been any more difficult to have learned to read, say, $27.346 as “27 dollars and 346 mils”. This amount would be more than $ 27.34 but less than $27.35. note © 2010 Herb I. Gross

18. next What number is named by the Roman numeral MMMCXXIIII? Practice Problem #2 Answer: 3,124

19. next © 2010 Herb I. Gross Solution for Practice Problem #2 next In terms of our adjective/noun theme, we see that there are 3 M’s, 1 C, 2 X’s and 4 I’s. Knowing that M represents 1,000; C, 100; X, 10; and I, 1; we see that we have 3 thousands, 1 hundred, 2 tens and 4 ones, which in place value notation we write as… 3,124. MMM C XX II I I 3,000 100 20 4 +++ = 3,124

20. © 2010 Herb I. Gross next While it’s more cumbersome to write MMMCXXIIII than to write 3,124; the representation of this number as a Roman numeral is still a gigantic improvement over having to write 3,124 individual tally marks! Notes on Practice Problem #2

21. In order to be able to write numbers more compactly, the Romans were willing to sacrifice the luxury afforded by “trading in by tens” in order to invent “in between” denominations. For example, they used V to stand for five (probably because just as the number five is half of the number ten, the symbol V is half (in fact, the upper half) of the symbol X. next © 2010 Herb I. Gross

22. The Romans invented the symbols V to denote five, L to denote fifty, and D to denote 500. Thus, rather than write IIIIIII to denote seven, the Roman numerals representation would be written more concisely as VII, and the number we would write as 256 would be written as CCLVI. next © 2010 Herb I. Gross

23. As a further abbreviation, the Romans decided that if a symbol (numeral) was placed to the left of a symbol that denoted a greater denomination, it meant that the smaller denomination should be subtracted from the greater denomination. In this context, rather than write IIII they would write IV which would mean 4 not 6 5. next note 5 Using time as an analogy, we often refer to 5:59 as one minute before six, and to 6:01 as one minute after six.

24. In a similar way, rather than using either the numeral IIIIIIIII or VIIII, they would use IX. 6 6 Notice that this notion of incorporating “subtraction” into the process of representing a Roman numeral allows us to write numbers in an abridged form, but it runs counter to the experience of reality. For example, if you have a $10-bill and a $1-bill in your wallet, they represent $11 regardless of whether the $1-bill is placed on top of the $10-bill or below it. The point is that the Romans did not do their arithmetic using Roman numerals (they used the abacus). Roman numerals were used only for such things as counting (or for numbering pages). In that context a sequence such as IX, X, XI caused no problems. © 2010 Herb I. Gross next note As additional examples, the Romans used the symbols IV to denote 4, IX to denote 9, XL to denote 40, XC to denote 90, CD to denote 400, and CM to denote 900. next

25. © 2010 Herb I. Gross next Find the number represented by the modern Roman numeral CCCIX. Practice Problem #3 Answer: 309

26. next © 2010 Herb I. Gross Solution for Practice Problem #3 next C is the Roman numeral for 100, and there are 3 C’s. CCCIX 300 9 + = 309 However, while I is the Roman numeral for 1 and X is the Roman numeral for 10, the fact that I appears to the left of X means that we subtract 1 from 10.

27. © 2010 Herb I. Gross next Thus, in the original Roman numerals (that is, the numerals as they existed before the subtractive property was introduced) CCCIX and CCCXI would mean 311. As a more complicated illustration, in the original Roman numerals, MMCIIXIIXM and MMMCXXIIII both represent 3,124. However, it is much easier to read the number when the symbols are grouped in the form of MMMCXXIIII.

28. Sand Reckoner © 2010 Herb I. Gross

29. Because science and mathematics were in their relative infancy, there was no compelling need for the Romans to have invented denominations greater than a thousand. However, as time went on and technology increased, a need developed for expressing greater numbers. next © 2010 Herb I. Gross 23,000 143,000

30. However, since each new power of ten (that is, ten thousands, hundred thousands, millions, etc.) required the use of a different letter of the alphabet, it soon became apparent that using letters could eventually become as tedious as what occurred when people had to deal solely with tally marks. next © 2010 Herb I. Gross

31. next © 2010 Herb I. Gross Thus, the next evolutionary step in counting was the invention of the abacus, or as it was known in the western World, the sand reckoner. The sand reckoner consisted of vertical lines, arranged in a row, drawn in the sand. Each line represented a denomination that was a power of ten.

32. next © 2010 Herb I. Gross Since the sand reckoner was invented by Semitic people, and since Semitic people read and write from right to left, it was natural that they started naming the denominations from right to left. Thus, the line furthest to the right represented “ones”, the line immediately to its left represented “tens,” the next line to the left represented “hundreds,” and so on.

33. onestenshundredsthousands Sand Reckoner next © 2010 Herb I. Gross

34. Stones (or pebbles) were used as “markers” or “counters”. Thus, while a stone always stood for the adjective “one”, what denomination it modified depended on what line the stone was placed. next © 2010 Herb I. Gross For example, if the stone was placed on the third line (from the right), it stood for one hundred.

35. More specifically… next © 2010 Herb I. Gross 1 (a stone on the ones line) 10 (a stone on the tens line)100 (a stone on the hundreds line)1,000 (a stone on the thousands line) onestens hundreds thousands next

36. What number is represented on the sand reckoner below? Answer: 3,124 4213 Practice Problem #4 next © 2010 Herb I. Gross

37. next © 2010 Herb I. Gross Solution for Practice Problem #4 Reading from right to left, we have 4 stones on the first line, 2 stones on the second line, 1 on the third line, and 3 on the fourth line. Since the first line represents ones; the second line, tens; the third line, hundreds and the fourth line thousands; we have a total of 3,124. Notice that except for the appearance of our nouns, this problem parallels what we did using Roman numerals in the previous problem. next

38. IXCM © 2010 Herb I. Gross More visually… MMMMMM C XXXX IIIIIIII next

39. © 2010 Herb I. Gross next The sand reckoner is an improvement over Roman numerals in the sense that in Roman numerals we have to introduce a different letter to represent each new domination. With the sand reckoner, all we have to do is add one more line. That is, the lines we’ve drawn above take the place of our having to use I, X, C or M. 8 Notes on Practice Problem #4 note 8 The abacus is, in a sense, a “portable” sand reckoner. Namely, while the sand reckoner couldn't be carried from one site to another, the abacus could. That is, to obtain the abacus from the sand reckoner, replace the lines in the sand by wires and the stones by beads, string the beads on the wires, and enclose the entire system in a frame. As interpreted in the context of language, the wires of the abacus are the nouns and the beads are the adjectives. next

40. © 2010 Herb I. Gross Moreover, while when translated into Roman numerals, the number that is represented on the sand reckoner in this problem is MMMCXXIIII; the major difference is that in the Roman numeral format we can rearrange the symbols in any order wish. For example, we could have written MMXXCIIIIM or IIIIXXCMMM etc. However, we do not have this freedom with using the sand reckoner because all the lines look exactly the same except for their position. next

41. © 2010 Herb I. Gross next The Latin word for stone is “calculus”. note 9 In this context, every mathematics course that involves computation could have been called “calculus”. Thus, just as the English curriculum often lists courses by number as English 1, English 2, English 3, etc.; with the above as justification, one could have named all mathematics courses in the same way as Calculus 1, Calculus 2, Calculus 3, etc. next Hence, to calculate meant to do arithmetic by using stones; that is, by using the sand reckoner. 9

42. © 2010 Herb I. Gross In summary, the development of Roman numerals and the abacus (sand reckoner) made it much more convenient to represent “big” numbers. next For example, MMMCXXIII was a huge improvement over having to write 3,124 tally marks. And the sand reckoner was an improvement over Roman numerals in the sense that it was easier to draw more lines in the sand than to keep inventing new symbols to name the powers of ten.

43. © 2010 Herb I. Gross However, the evolutionary process was again brought into play when even the number of lines in the sand or wires on the abacus became too cumbersome for us to keep track of. 10 next note 10 For example, to talk about trillions, the abacus would have to have at least 13 wires (that is, we would need wires to hold the place of the 1's, 10, 100's, 1,000's, 10,000's etc). This could easily become very cumbersome to carry and/or to use.

44. © 2010 Herb I. Gross next What happened next is the subject of Part 3 of this lesson. hieroglyphics tally marks Roman Numerals Sand Reckoner next plateau?