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

2. Development of Place Value Development of Place Value © 2010 Herb I. Gross next Invention of Zero Lesson 1 Part 3

3. Although the “nouns” on the sand reckoner all looked alike, they could still be distinguished from one another by their relative positions. This property of the sand reckoner is essential in understanding the next stage of abstraction in the development of our number system, known as place value. next © 2010 Herb I. Gross The development of our place value system for representing numbers ranks as one of the great inventions of the human mind. next Place Value

4. Thus, again based on our having ten fingers, symbols (numerals) called digits 1 next © 2010 Herb I. Gross next 1 The word “digit” is also used as a name for a finger. Perhaps the word “digit” was also used to represent the numerals 0,1,2,3,4,5,6,7,8, and 9 because we tend to count to ten using our fingers. ; numbers we know as one, two three, four, five, six, seven, eight, and nine, and are denoted by the symbols 1, 2, 3, 4, 5, 6, 7, 8 and 9, respectively. That is, as in the sand reckoner, the digits were placed next to one another in a row with the digits playing the role of the stones. note next

5. For example, in the new system, the number represented on the sand reckoner as… © 2010 Herb I. Gross would now appear as… 4213

6. The key point is this… next © 2010 Herb I. Gross 1, 2,3 4,5,6 7,8,9 In this new invention, while the digits 1, 2, 3, 4, 5, 6, 7, 8 and 9 serve as the adjectives; the nouns (that is, the denominations) for the first time were invisible. That is, the denominations could only be determined by where a digit was placed; hence, the name place value. 2 2 As a play on words, we may say that a digit has both a face value and a place value. For example, the “face value” (adjective) of 2 is always 2. However, what the digit 2 modifies depends on the “place” that it is in. For example, in 23 its place value is tens while in 234 its place value is hundreds, note next

7. Inventions are not made “in a vacuum”. © 2010 Herb I. Gross Special Note When we talk about the ideas that go into the concept of place value, notice how many were invented earlier. next 1, 2, 3…

8. For example… next © 2010 Herb I. Gross Place value is abstract, but so are tally marks. Place value is based on “trading in” by tens, but so are Roman (and Egyptian) numerals. Place value is based on each denomination being worth ten of the denomination immediately to its right, but so is the sand reckoner. Special Note next

9. What made place value different is that for the first time one could not recognize a denomination until one saw what digit was in its place. © 2010 Herb I. Gross Special Note

10. Consider the following four numbers as represented on a sand reckoner… next © 2010 Herb I. Gross Importance of Zero as a place holder We can see how each of these numbers is different by looking at which lines the stones are on. next

11. For example, in the top left above, the line with the two stones is modifying “tens”; © 2010 Herb I. Gross In each case the adjective is 2, but we need to see the line in order to know what 2 represents. next while the line with the two stones in the top right is modifying “thousands”.

12. However, if we use the place value representation of these numbers, the lines become invisible, and what we would see in each row is the sequence of digits 2 followed by 3 but without the denominations the 2 and the 3 are modifying. © 2010 Herb I. Gross In other words, when we write 2 3 it is impossible to tell what denominations are being modified by 2 and 3. next

13. For this reason, we introduce a place holder to make it completely clear which noun is being modified by which adjective, even though the nouns remain invisible. The digit, or symbol, that one uses for this purpose is 0, and the name we give this symbol is zero. 3 © 2010 Herb I. Gross next 3 It is easy to confuse 0 with “nothing”. That is, just as 3 is a digit that tells us that we have three of a particular denomination, 0 is a digit that tells us that we have none of a particular denomination. For example, the place holder, 0, allows us to distinguish the difference between 40 and 400 in the sense that we can see that in 40 the 4 modifies tens while in 400 it modifies hundreds. note

14. In any case, with the introduction of 0 as the place holder, the above four numbers have their familiar place value appearance 23, 203, 2003, and 2300, respectively. next © 2010 Herb I. Gross next 2320320032300

15. © 2010 Herb I. Gross next Write each of the following as a Roman numeral… (a) 23 (b) 203 (c) 2,003 (d) 2,300 Practice Problem #1 Answer: XXIII, CCIII, MMIII, MMCCC. next

16. © 2010 Herb I. Gross Solution for Practice Problem #1 next In all four cases, we have two of one denomination and three of another. That is… In the first case, it’s 2 tens and 3 ones, in which case the Roman numeral would be XXIII. In the second case, it’s 2 hundreds and 3 ones, in which case the Roman numeral would be CCIII. In the fourth case, it’s 2 thousands and 3 hundreds, in which case the Roman numeral would be MMCCC. In the third case, it’s 2 thousands and 3 ones, in which case the Roman numeral would be MMMIII.

17. However, we need 0’s to distinguish 2,300 from 2,003, because the nouns are named solely by the position of the digits. © 2010 Herb I. Gross Notes on #1 The nouns I, X, C and M are visible, so there is never a danger of us confusing, say, MMCCC with MMIII because CCC doesn’t look like III. next

18. In summary, the invention of zero as a place holder is one of the ranking achievements of the human mind. © 2010 Herb I. Gross Notes on #1 next The importance of zero for the development of mathematics, science and technology cannot be overstated, especially in terms of how it, in combination with our adjective/noun theme, simplifies the study of arithmetic.

19. The advent of place value made it easier to represent (as well as comprehend) greater numbers, and as science and technology improved, numbers became sufficiently large so that even by trading in by tens, the process eventually became tedious. © 2010 Herb I. Gross For example, think of how tedious it would be to try to read a number such as… 8945600893049875. next Representing Large Numbers

20. Thus, the next evolutionary step in developing our number system was the subtle discovery that we could just as easily “trade in” by thousands as we could by tens. next © 2010 Herb I. Gross By way of analogy, when we see the word “cat” we recognize it without having to “sound out” each letter. Similarly, when we see the 3-digit place value numeral 647 we recognize it immediately as representing “six hundred forty-seven,” without sounding out the denominations. That is, we don’t say “Let's see, the 6 is in the hundreds place, the 4 is in the tens place and the 7 is in the ones place”. next

21. Thus, rather than use denominations such as “ones”, “tens” and “hundreds” and the adjectives 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 to modify them, more comprehensive denominations such as thousands, millions, billions, trillions, quadrillions… were introduced, and one thousand adjectives were used to modify them. © 2010 Herb I. Gross next These one thousand adjectives are the numerals 0 (written as 000), 1 (written as 001), 2 (written as 002), 3 (written as 003)… up to and including 999.

22. More specifically, starting with the digit furthest to the right, we count from right to left in groups of three digits (including 0’s). In this way, the hard-to-read numeral 8945600893049875 becomes the easier to interpret numeral 8,945,600,893,049,875. © 2010 Herb I. Gross next Namely… hundreds tens ones h t o 89 4 56 0 08 9 30 4 98 7 5 quadrillionstrillionsbillionsmillionsthousands“units”

23. next © 2010 Herb I. Gross next hundreds tens ones h t o 8 quadrillions 9 4 5 trillions 6 0 0 billions 8 9 3 millions 0 4 9 thousands 8 7 5 “units” We read the number as… 8 quadrillion,945 trillion,600 billion, 893 million,49 thousand,875 “units” Note that as long as the denominations are visible there is no need to write the 0’s. That is, we could have written hundreds tens ones h t o 8 quadrillions 9 4 5 trillions 6 billions 8 9 3 millions 4 9 thousands 8 7 5 “units” next

24. The 0’s are needed when the denominations are not visible, and we use commas rather than vertical bars to separate the denominations. © 2010 Herb I. Gross next That is, starting with the digit furthest to the right, we count from right to left in groups of three digits (including 0’s). In this methodology we would write the number as… 8 945 600 893 049 875,,,,, next

25. Notice that each large denomination is further modified by the number of ones, tens and hundreds; e.g., hundred millions, or ten quadrillions. © 2010 Herb I. Gross Notes For example, in the numeral, 8,945,600,893,049,875 the ‘6’ is in the hundreds place and the hundreds place is modifying “billions”. Hence, we see that ‘6’ is modifying a hundred billion. next

26. © 2010 Herb I. Gross Notes The word million is associated with the third large denomination from the right, and as such it may be thought of as a thousand thousands (in much the same way that we can look at, say, 200 as being 200 ones, 20 tens or 2 hundreds). next No matter how large the noun (denomination) is, we never have to deal with an adjective that is greater than 999.

27. © 2010 Herb I. Gross Notes The units are the name of the noun we are describing. Thus, if we are talking about grains of sand, our number would now be 8,945,600,893,049,875 grains of sand. next The prefixes “bi”, “tri”, “quad”, “quint”, “sept”, “oct” etc., give rise to the denominations billion, trillion, quadrillion, quintillion, sextillion, septillion, and octillion etc…

28. © 2010 Herb I. Gross next Does the adjective 364 in 364,987 mean the same thing as the adjective 364, in 364,000,987? Practice Problem #2 Answer: Yes. next

29. © 2010 Herb I. Gross Solution for Practice Problem #2 In both cases, the adjective is 364. However, in 364,987 the adjective 364 is modifying “thousands”, but in 364,000,987 it is modifying “millions”. In essence in either case, the “face value” is 364, but the place values are different (i.e., thousands in one case and millions in the other). Again, it's like saying that a blue pencil doesn’t look like a blue shirt, but that the adjective “blue” means the same thing in both cases. next

30. © 2010 Herb I. Gross next Write two hundred trillion, thirty-seven thousand as a place value numeral? Practice Problem #3 Answer: 200,000,000,037,000 next

31. © 2010 Herb I. Gross Solution for Practice Problem #3 In this situation our denominations (nouns) are units, thousands, millions, billions, trillions, etc.; and our adjectives are the numbers “none” (000) to nine hundred ninety- nine (999). So we have… 200 trillion 37 thousands. next

32. To prepare for place value notation, we may say that the number is 200 trillions, no billions, no millions 37 thousands and no units. In terms of our chart, we have… © 2010 Herb I. Gross next hundreds tens ones h t o 2 0 0 3 7 trillionsbillionsmillionsthousands“units” Solution for Practice Problem #3

33. And if we now wish to make the nouns invisible, we must replace the “empty spaces” by 0’s. That is, we may first write… next © 2010 Herb I. Gross next hundreds tens ones h t o 2 0 0 3 7 trillionsbillionsmillionsthousands“units” Solution for Practice Problem #3 0 0 0 0 and then write… 200,000,000,037,000.

34. When we ask a person a question such as “How much is three 2’s?”, we usually hear “6” as the answer. The person probably heard the question as if it were “How much is the sum of three 2’s?”, in which case “6” is the correct answer. That is, 2 + 2 + 2 = 6. next © 2010 Herb I. Gross Special Note However, there are times when we want to compute the product of three 2’s; that is 2 × 2 × 2. next

35. In this context, we would just like to introduce a bit of new notation, called exponential notation. Just as we use 3 × 2 as an abbreviation for 2 + 2 + 2, we use the notation 2 3 as an abbreviation for 2 × 2 × 2. © 2010 Herb I. Gross Exponential Notation In the expression 2 3, we call 2 the base and 3 the exponent. This idea will be explored in this course, in our lesson on exponents. next

36. © 2010 Herb I. Gross next (a)Express 3 4 in place value notation. (b) Express 4 3 in place value notation. Practice Problem #4 Answer: (a) 81 (b) 64 next

37. © 2010 Herb I. Gross Solution for Practice Problem #4 (a) 3 4 is an abbreviation for… 3 × 3 × 3 × 3 or 81. (b) 4 3 is an abbreviation for… 4 × 4 × 4 or 64. next The point of Practice Problem #4 is to illustrate that while 3 × 4 and 4 × 3 denote the same number, 3 4 and 4 3 denote different numbers. In summary, it’s important to remember which number is the base 3 4 3 4 and which is the exponent.

38. There is a saying that the more things change, the more they remain the same. In the subject under discussion, we have now come full circle. We began the evolution from tally marks to place value by showing how the use of tally marks rapidly becomes too cumbersome to represent numbers. As an example, while it was already difficult at sight to distinguish an array of 29 tally marks from an array of 30 tally marks, in the language of place value we can immediately distinguish between much larger numbers, such as 23,456 and 23,457. © 2010 Herb I. Gross So what is the point of our introducing exponential notation at this time?

39. However, even with nouns (denominations) that allow us to keep track of numbers a thousand at a time, with today’s technological development we still run into the same problem as with tally marks. The only difference is that this time, the problem is with the use of 0's rather than with the use of tally marks. As an example that arises in molecular chemistry, the approximate value of Avogadro’s Number is written in place value notation as a 6 followed by twenty-three 0’s; that is… 600,000,000,000,000,000,000,000. next © 2010 Herb I. Gross

40. next © 2010 Herb I. Gross Notes Each chemical compound has what is called a molecular weight. A quantity of the compound having mass (measured in grams) equal to its molecular weight is called a mole of the substance. The number of molecules in a mole is the same for all substances, and it is that “constant of nature” that is known as Avogadro’s Number.

41. next © 2010 Herb I. Gross Notes For example, a mole of water is 18 grams (or roughly, half an ounce). Hence, roughly speaking, in the language of place value, there are 600,000,000,000,000,000,000,000 atoms in about half an ounce of water. Even with the commas, is it immediately obvious that there are 23 zeroes in the representation? Or, even if it is, is it immediately obvious that the number would be read as 600 sextillion?

42. Thus, as civilization progressed (or, more accurately, as science and technology became more advanced), we found ourselves having to go beyond place value. next © 2010 Herb I. Gross More specifically, the next step was the birth of exponential notation. next

43. © 2010 Herb I. Gross In general, it is tedious to convert a number such as 7 23 into an equivalent place value numeral. In essence, we first have to know the place value equivalent of 7 22 and then multiply this by 7. However, there are three numbers that lend themselves nicely to exponential notation, two of which are rather “trivial” (that is, they are of little interest).

44. next (a) Write 0 1,000 as a place value numeral. (b) Write 1 1,000 as a place value numeral. Practice Problem #5 Answer: (a) 0 (b) 1 next

45. © 2010 Herb I. Gross Solution for Practice Problem #5 (a) For any number, n, n × 0 = 0. In other words, for any non-zero whole number n, 0 n = 0. next 0 × 0 × 0 = (0 × 0) × 0 = 0 × 0 = 0 0 × 0 × 0 × 0 = (0 × 0 × 0) × 0 = 0 × 0 = 0 etc. Hence… 0 × 0 = 0

46. © 2010 Herb I. Gross Solution for Practice Problem #5 (b) For any number, n, n × 1 = n. In other words, for any non-zero whole number n, 1 n = 1. next 1 × 1 × 1 = (1 × 1) × 1 = 1 × 1 = 1 1 × 1 × 1 × 1 = (1 × 1 × 1) × 1 = 1 × 1 = 1 etc. Hence… 1 × 1 = 1

47. For reasons that will become more obvious later in the course, we define any non-zero number to the 0 th power to be 1. © 2010 Herb I. Gross Notes on #5 next One way to see this is to think of 2 n as meaning that we start with 1 and then multiply it by 2 “n” times. Thus we think of 2 1 as being equal to 1 × 2; 2 2 as being equal to 1 × 2 × 2; etc. In this context 2 0 means we start with 1 and then multiply it by 2 “zero” times.

48. Another way is to look at the following chart… © 2010 Herb I. Gross Notes on #5 next 2424 =2 × 2 × 2 × 22323 =2 × 2 × 2 2 =2 × 22121 =2?=? = 16 = 8 = 4 = 2 = ?

49. Notice that as we read the above chart from top to bottom, next © 2010 Herb I. Gross next 2424 =2 × 2 × 2 × 2 2323 =2 × 2 × 2 2 =2 × 2 2121 =2 ?=? = 16 = 8 = 4 = 2 = ? …in the first column the exponent decreases by 1 each time. …in the last column we divide each entry by 2 to get to the next entry. next

50. In order to ensure that this pattern continues, the chart must be completed as follows… © 2010 Herb I. Gross Notes on #5 next 2424 =2 × 2 × 2 × 2 2323 =2 × 2 × 2 2 =2 × 2 2121 =2 ?=? = 16 = 8 = 4 = 2 = ? 2=2 ÷ 2 = 1 0

51. © 2010 Herb I. Gross next Write 10 4 as a place value numeral. Practice Problem #6 Answer: 10,000 next

52. © 2010 Herb I. Gross Solution for Practice Problem #6 10 2 = 10 × 10 = 100 next 10 3 = (10 × 10) × 10 = 10 2 × 10 = 100 × 10 = 1,000 10 4 = 10 3 × 10 = 1000 × 10 = 10,000

53. In place value notation, every time we multiply a whole number by 10, we simply annex a zero at its end. © 2010 Herb I. Gross Notes on #6 next For example, 42 × 10 = 420. In terms of our adjective/noun theme, by annexing a zero to 42 to form 420, the 4 which was modifying tens is now modifying hundreds etc.

54. © 2010 Herb I. Gross Notes on #6 next To see this a bit more visually, think in terms of Roman numerals. Notice that ten I’s become one X, ten X’s become one C and ten C’s become one M. Thus, to multiply by ten, we simply replace each numeral by the next greater one. That is… X X X X I I C C C C X X

55. next © 2010 Herb I. Gross Notes on #6 next Did you notice the pattern? 10 1 = 10 10 2 = 100 10 3 = 1,000 10 4 = 10,000 Namely, the number of 0’s that followed the 1 was exactly the same as the exponent. 10 1 = 10 10 2 = 100 10 3 = 1,000 10 4 = 10,000 next

56. © 2010 Herb I. Gross next Use exponential notation to express 100,000,000,000,000,000,000,000. Practice Problem #7 Answer: 10 23 next

57. © 2010 Herb I. Gross Solution for Practice Problem #7 We begin by counting the number of 0's that follow the 1 in the place value numeral 100,000,000,000,000,000,000,000 and determine that it is 23. next Hence, we have a 1 followed by 23 zeroes; and based on our last note to the previous problem, we see that this can be written as 10 23.

58. © 2010 Herb I. Gross Review next In the expression 10 23, 23 is the number of 0’s that follow the “1”, not the number of 0’s that follow “10”. More generally, if n denotes any whole number, 10 n represents the number which in place value notation is a 1 followed by n zeroes. In the expression 10 23, 10 is called the base and 23 is called the exponent, and we read 10 23 as “10 raised to the 23 rd power,” or “10 to the 23 rd power” for short. Caution

59. next © 2010 Herb I. Gross Notes on #6 next So while it would be difficult to distinguish, say, between a 6 followed by 23 zeroes and a 6 followed by 26 zeroes; it is not difficult to distinguish between 6 × 10 23 and 6 × 10 26. Notice that since 10 n is a 1 followed by n zeroes, a number such as 6 × 10 n is a 6 followed by n zeroes. And since Avogadro’s Number is a 6 followed by 23 zeroes, we may write it in the form 6 × 10 23.

60. © 2010 Herb I. Gross In closing this saga of the evolution of whole number representation from hieroglyphics to exponential notation, it is naive to view our own generation as the “ultimate in gracious living”. It is certainly natural for us to consider our own generation as the most advanced “state of the art” generation in comparison to past generations, which we might consider technologically more primitive.

61. next © 2010 Herb I. Gross next At the same time, we should not forget that when future generations look back at us, they may find that we were primitive relative to their “state of the art” society. So, as the cliché goes, the beat goes on, and it will be left to future generations to continue the chronicle of human achievement by building upon the heritage that has been left to them by the previous generations.

62. © 2010 Herb I. Gross next What’s next? Hieroglyphics Tally Marks Place Value Roman Numerals Sand Reckoner Exponential Notation