2. A store front display in NYC showing price tags with decimals. The Use of Decimals in the Real World

7. Place Values in a Decimal and the Expanded Form of a Decimal. Given a decimal 5.8147 place value is 1 10000 1 is valueplace 1000 1 is valueplace 100 1 is valueplace 10 1 is valueplace

8. Place Values in a Decimal and the Expanded Form of a Decimal. Given a decimal 5.8147 We can therefore rewrite it in an expanded form which can then be converted to a mixed number 10000 7 1000 4 100 1 10 8 5  10000 8147 5

9. How a decimal is read? A decimal is read as if they were written in fraction form except that the decimal point is read “and”. We don’t use the word “and” in any other places. Example: 1204.657 is read as “One thousandtwo hundredfourandsix hundred fifty seventhousandths” i.e. This method works only for short decimals, and when there are many digits after the decimal point, such as 2.71828, the scientists and engineers will call it “Two point seven one eight two eight” 657 1204 1000

10. How is a decimal read? A problem for discussion Is there any chance of confusion when a student reads the decimal 1204.38 as “One thousand and two hundred and four and thirty eight hundredths” ? Response In daily uses, it seldomly causes a problem, but it doesn’t mean that confusion will never happen either. For instance, consider the number 36,800.041 If you read this as “Thirty six thousand and eight hundred and forty one thousandths”, then the person who heard this may interpret it as “3600.841”

11. Remarks 1. The decimal notation is not unique throughout the world even up to this day. For example, the British uses 3·1416 (with the dot higher) while the French and German use 3,1416 2. When we change a fraction into a decimal, the representation is not always terminating. and we will see later that even terminating decimals have non-terminating representations such as 0.25 = 0.2499999 ….  333.0 3 1 eg.

12. dot comma Momayyez unknown

13. Comparing Decimals Research shows that most students believe that 0.287 is bigger than 0.35 because a)0.287 has more digits than 0.35 b)0.287 is read as “two hundred eighty seven thousandths” which sounds larger than “thirty five hundredths”, particularly when they are not familiar with the fact than “one thousandth” is really smaller than “one hundredth”.

14. Multiplying or Dividing Decimals by Powers of 10 Multiplying a decimal by 10 n is the same as moving the decimal point to the right n places, adding place holders if necessary. eg. 3.7615 × 10 3

15. Multiplying or Dividing Decimals by Powers of 10 Multiplying a decimal by 10 n is the same as moving the decimal point to the right n places, adding place holders if necessary. eg. 3.7615 × 10 3 = 3 7 6 1 5

16. Multiplying or Dividing Decimals by Powers of 10 Multiplying a decimal by 10 n is the same as moving the decimal point to the right n places, adding place holders if necessary. eg. 3.7615 × 10 3 = 3 7 6 1 5

17. Multiplying or Dividing Decimals by Powers of 10 Multiplying a decimal by 10 n is the same as moving the decimal point to the right n places, adding place holders if necessary. eg. 3.7615 × 10 3 = 3 7 6 1 5

18. Multiplying or Dividing Decimals by Powers of 10 Multiplying a decimal by 10 n is the same as moving the decimal point to the right n places, adding place holders if necessary. eg. 3.7615 × 10 3 = 3 7 6 1 5

19. Multiplying or Dividing Decimals by Powers of 10 Multiplying a decimal by 10 n is the same as moving the decimal point to the right n places, adding place holders if necessary. eg. 3.7615 × 10 3 = 3 7 6 1 5

20. Multiplying or Dividing Decimals by Powers of 10 Multiplying a decimal by 10 n is the same as moving the decimal point to the right n places, adding place holders if necessary. eg. 3.7615 × 10 3 Dividing a decimal by 10 n is the same as moving the decimal point to the left n places, adding place holders if necessary. eg. 743.28 × 10 5 = 74328000. eg. 3.7615 ÷ 10 3 = 0.0037615 eg. 743.28 ÷ 10 2 = 7.4328 = 3761.5

21. Ordering Decimals Given two decimals, how do we determine quickly which one is larger? The method is rather easy to learn from just a few examples. Example Which one is larger, 6.724 or 6.72189 ? Solution We first put one of them above the other such that the decimal points are lined up. (click to see the animation)

22. Ordering Decimals Given two decimals, how do we determine quickly which one is larger? The method is rather easy to learn from just a few examples. Example Which one is larger, 6.724 or 6.72189 ? Solution We first put one of them above the other such that the decimal points are lined up. 6.724

23. Ordering Decimals Given two decimals, how do we determine quickly which one is larger? The method is rather easy to learn from just a few examples. Example Which one is larger, 6.724 or 6.72189 ? Solution We first put one of them above the other such that the decimal points are lined up. 6.72189 6.724

24. 6.72189 Ordering Decimals Given two decimals, how do we determine quickly which one is larger? The method is rather easy to learn from just a few examples. Example Which one is larger, 6.724 or 6.72189 ? Solution We first put one of them above the other such that the decimal points are lined up. 6.724 Next we compare the whole number portions – whichever has the larger whole number portion is the bigger decimal.

25. Ordering Decimals Given two decimals, how do we determine quickly which one is larger? The method is rather easy to learn from just a few examples. Example Which one is larger, 6.724 or 6.72189 ? Solution We first put one of them above the other such that the decimal points are lined up. 6.72189 6.724 Next we compare the whole number portions – whichever has the larger whole number portion is the bigger decimal. In this case, they are both equal, so we have to compare the digits in the first column on the right of the decimal point.

26. Ordering Decimals Given two decimals, how do we determine quickly which one is larger? The method is rather easy to learn from just a few examples. Example Which one is larger, 6.724 or 6.72189 ? Solution We first put one of them above the other such that the decimal points are lined up. 6.72189 6.724 Next we compare the whole number portions – whichever has the larger whole number portion is the bigger decimal. In this case, they are both equal, so we have to compare the digits in the first column on the right of the decimal point.

27. Next we compare the whole number portions – whichever has the larger whole number portion is the bigger decimal. In this case, they are both equal, so we have to compare the digits in the first column on the right of the decimal point. 6.72189 Ordering Decimals Given two decimals, how do we determine quickly which one is larger? The method is rather easy to learn from just a few examples. Example Which one is larger, 6.724 or 6.72189 ? Solution We first put one of them above the other such that the decimal points are lined up. 6.724

28. In this column, the two digits are also equal, so we have to keep moving to the right until we can find a column that has two different digits. (Please click to see animation.) Ordering Decimals Given two decimals, how do we determine quickly which one is larger? The method is rather easy to learn from just a few examples. Example Which one is larger, 6.724 or 6.72189 ? Solution We first put one of them above the other such that the decimal points are lined up. 6.72189

29. In this column, the two digits are also equal, so we have to keep moving to the right until we can find a column that has two different digits. Ordering Decimals Given two decimals, how do we determine quickly which one is larger? The method is rather easy to learn from just a few examples. Example Which one is larger, 6.724 or 6.72189 ? Solution We first put one of them above the other such that the decimal points are lined up. 6.724 6.72189

30. In this column, the two digits are also equal, so we have to keep moving to the right until we can find a column that has two different digits. Ordering Decimals Given two decimals, how do we determine quickly which one is larger? The method is rather easy to learn from just a few examples. Example Which one is larger, 6.724 or 6.72189 ? Solution We first put one of them above the other such that the decimal points are lined up. 6.72189 6.724 Now we see that the upper digit in the highlighted column is larger, there for the corresponding number (i.e. the upper one) is larger than the lower one.

31. Addition and Subtraction of Decimals These two operations are respectively similar to the addition and subtraction of whole numbers, except that the numbers are aligned by the decimal points rather than the last digits (counting from the left). Example 34.16 + 2.3096 ────── Incorrect 34.16 + 2.3096 ─────── Correct, and we have to treat any empty space as a 0.

32. There are two ways to carry out this operation, (I)Converting the decimals to fractions Example: 0.65 × 2.417 Multiplication of Decimals = 1.57105 1000 417 2 100 65  1000 2417 100 65  1000100 241765    000,100 157105 

33. (II) Ignore the decimal points first and multiply the two numbers as if they were whole numbers. In the end we insert the decimal point back to the answer in the proper position. Example: 0.65 × 2.417 can be first treated as 65 × 2417 = 157105, the decimal point is then re-inserted to the product such that “the number of decimal places in the answer is equal to the sum of the number of decimal places in the multiplicands.” In this particular case, 0.65 has two decimal places and 2.417 has three decimal places. Hence their product should have 5 decimal places, and this means that 0.65 × 2.417 = 1. 57105

34. Division of Decimals The process of long division is similar to that of dividing whole numbers with some modifications. Example: 2.556 ÷ 1.2 Step 1. Move the decimal point in the divisor to the right until it becomes a whole number. (click)

35. Division of Decimals The process of long division is similar to that of dividing whole numbers with some modifications. Example: 2.556 ÷ 1.2 Step 1. Move the decimal point in the divisor to the right until it becomes a whole number. Step 2. Move the decimal point in the dividend to the right by the same amount. (click)

36. Division of Decimals The process of long division is similar to that of dividing whole numbers with some modifications. Example: 2.556 ÷ 1.2 Step 1. Move the decimal point in the divisor to the right until it becomes a whole number. Step 2. Move the decimal point in the dividend to the right by the same amount.

37. Division of Decimals The process of long division is similar to that of dividing whole numbers with some modifications. Example: 2.556 ÷ 1.2 Step 1. Move the decimal point in the divisor to the right until it becomes a whole number. Step 2. Move the decimal point in the dividend to the right by the same amount. Step 3. Put a decimal point above that one in the dividend. (click)

38. Division of Decimals The process of long division is similar to that of dividing whole numbers with some modifications. Example: 2.556 ÷ 1.2 Step 1. Move the decimal point in the divisor to the right until it becomes a whole number. Step 2. Move the decimal point in the dividend to the right by the same amount. Step 3. Put a decimal point above that one in the dividend.

39. Step 1. Move the decimal point in the divisor to the right until it becomes a whole number. Step 2. Move the decimal point in the dividend to the right by the same amount. Step 3. Put a decimal point above that one in the dividend. Division of Decimals The process of long division is similar to that of dividing whole numbers with some modifications. Example: 2.556 ÷ 1.2

40. Step 2. Move the decimal point in the dividend to the right by the same amount. Step 3. Put a decimal point above that one in the dividend. The process of long division is similar to that of dividing whole numbers with some modifications. Example: 2.556 ÷ 1.2 Step 4. Divide as if we are dividing whole numbers. (click) Division of Decimals

41. Step 2. Move the decimal point in the dividend to the right by the same amount. Step 3. Put a decimal point above that one in the dividend. Division of Decimals The process of long division is similar to that of dividing whole numbers with some modifications. Example: 2.556 ÷ 1.2 Step 4. Divide as if we are dividing whole numbers. 2 2 42 4 - 1

42. Step 2. Move the decimal point in the dividend to the right by the same amount. Step 3. Put a decimal point above that one in the dividend. Division of Decimals The process of long division is similar to that of dividing whole numbers with some modifications. Example: 2.556 ÷ 1.2 Step 4. Divide as if we are dividing whole numbers. 2 2 42 4 - 1 5

43. Step 2. Move the decimal point in the dividend to the right by the same amount. Step 3. Put a decimal point above that one in the dividend. Division of Decimals The process of long division is similar to that of dividing whole numbers with some modifications. Example: 2.556 ÷ 1.2 Step 4. Divide as if we are dividing whole numbers. 2 2 42 4 - 1 51 5 1 1 21 2 - 3

44. Step 2. Move the decimal point in the dividend to the right by the same amount. Step 3. Put a decimal point above that one in the dividend. Division of Decimals The process of long division is similar to that of dividing whole numbers with some modifications. Example: 2.556 ÷ 1.2 Step 4. Divide as if we are dividing whole numbers. 2 2 42 4 - 1 51 5 1 1 21 2 - 3 6

45. Step 2. Move the decimal point in the dividend to the right by the same amount. Step 3. Put a decimal point above that one in the dividend. Division of Decimals The process of long division is similar to that of dividing whole numbers with some modifications. Example: 2.556 ÷ 1.2 Step 4. Divide as if we are dividing whole numbers. 2 2 42 4 - 1 51 5 1 1 21 2 - 3 63 6 3 3 63 6 - 0 Therefore 2.556 ÷ 1.2 = 2.13

46. Converting Fractions to Decimals Exploration: Find the first 3 digits in the decimal expansion of 4 / 7. We first consider the following From long division we have 4000 ÷ 7 = 571 r3, hence It is not hard to see that we can actually insert the decimal point first in the quotient and then continue the long division as long as we want (see next page). 5710 1000 571 1000 1 571 7 4. 

47. Converting Fractions to Decimals Conclusion The decimal expansion of a fraction a/b can be obtained by long division. 0.571428 00000000.47 ···

48. Converting Fractions to Decimals Fact: The decimal expansion of any fraction a/b is either terminating or repeating. Theorem: If the fraction a/b is in its reduced form, then its decimal expansion is terminating if and only if b is one of the following forms. (1) a product of 2’s only, (2) a product of 5’s only (3) a product of 2’s and 5’s only. Examples: is not terminating 11 3 is terminating 625 17 is terminating 17 52 7 640 7 1920 21 

49. Converting Fractions to Decimals Theorem: If the fraction a/b is in its reduced form, and b = 2 m 5 n then the decimal expansion of a/b is terminating with number of decimal places exactly equal to max{m, n} Now we know what kind of fractions will have terminating decimal expansions, but can we predict how many decimal places there will be in the expansion? Example: The decimal expansion of 13 52 13 40 13  will have exactly 3 decimal places.

50. Converting Fractions to Decimals One more question: If we know that a certain fraction has repeating decimal expansion, can we predict its cycle length? Unfortunately there is no formula to calculate the precise cycle length. All we know is an upper bound and a small (not too helpful) property. Theorem If p is a prime number other than 2 and 5, then the cycle length of 1/p is at most (p – 1), and the cycle length must divide (p – 1). Example: The cycle length of 1/31 is at most 30, and it must divide 30. In fact, the cycle length of 1/31 is 15.

51. Converting Fractions to Decimals More examples prime number pcycle length of 1/p 76 112 136 1716 1918 2322 2928 3115 373 415 There is no obvious pattern on the cycle length, and a large denominator can have a small cycle length.

52. Converting Fractions to Decimals More facts (optional) 1.If p is a prime other than 2 or 5, then the cycle length of 1/(p 2 ) is at most p(p – 1) and the cycle length must divide p(p – 1). 2. If p and q are different primes other than 2 and 5, then the cycle length of 1/pq will be at most (p – 1)(q – 1) and divides (p – 1)(q – 1). Example: cycle length of 1/7 is 6, cycle length of 1/49 is 42 (= 7×6). Example: Cycle length of 1/(7×11) is less than 6×10 = 60, and must divide 60. It turns out that the cycle length of 1/77 is only 6.

53. Converting Decimals to Fractions From the previous theorem, we see that only repeating or terminating decimals can be converted to a fraction. Procedures: (1)terminating decimal, eg. 0.35742 = 35742 / 100000 The number of 0’s in the denominator is equal to the number of decimal places. (2)repeating decimals of type I, eg 0.2222 ··· = 2 / 9 0.47474747 ··· = 47 / 99 0.528528528 ··· = 528 / 999

54. Converting Decimals to Fractions Procedures: 2)repeating decimals of type I, eg. 0.2222 ··· = 2 / 9 0.47474747 ··· = 47 / 99 0.528528528 ··· = 528 / 999 3)repeating decimals of type II, eg. 0.0626262 ··· = 62 / 990 0.00626262 ··· = 62 / 9900 0.000344934493449 ··· = 3449 / 9999000

55. Converting Decimals to Fractions Procedures: 3)repeating decimals of type II, eg. 0.0626262 ··· = 62 / 990 0.00626262 ··· = 62 / 9900 0.000344934493449 ··· = 3449 / 9999000 4) repeating decimals of type III, eg. 0.576666 ··· = 0.57 + 0.006666 ···

56. Converting Decimals to Fractions Procedures: 3)repeating decimals of type II, eg. 0.0626262 ··· = 62 / 990 0.00626262 ··· = 62 / 9900 0.000344934493449 ··· = 3449 / 9999000 4) repeating decimals of type III, eg. 0.576666 ··· = 0.57 + 0.006666 ···