A store front display in NYC showing price tags with decimals.

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

6. 1 place value is 10000 1 place value is 1000 1 place value is 100 1 place value is 10 Place Values in a Decimal and the Expanded Form of a Decimal. Given a decimal 5.8147 place value is 1

7. 8 1 4 7 + + + + 5 10 100 1000 10000 8147 5 10000 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

8. 657 1204 1000 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 i.e. “One thousand two hundred four and six hundred fifty seven thousandths” 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”

9. 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”

10. 1 = × × × eg. 0 . 333 3 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 ….

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12. Comparing Decimals • Research shows that most students believe that • 0.287 is bigger than 0.35 • because • 0.287 has more digits than 0.35 • 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”.

13. Multiplying or Dividing Decimals by Powers of 10 Multiplying a decimal by 10n is the same as moving the decimal point to the rightn places, adding place holders if necessary. eg. 3.7615 × 103

14. Multiplying or Dividing Decimals by Powers of 10 Multiplying a decimal by 10n is the same as moving the decimal point to the rightn places, adding place holders if necessary. eg. 3.7615 × 103 = 37615

15. Multiplying or Dividing Decimals by Powers of 10 Multiplying a decimal by 10n is the same as moving the decimal point to the rightn places, adding place holders if necessary. eg. 3.7615 × 103 = 37615

16. Multiplying or Dividing Decimals by Powers of 10 Multiplying a decimal by 10n is the same as moving the decimal point to the rightn places, adding place holders if necessary. eg. 3.7615 × 103 = 37615

17. Multiplying or Dividing Decimals by Powers of 10 Multiplying a decimal by 10n is the same as moving the decimal point to the rightn places, adding place holders if necessary. eg. 3.7615 × 103 = 37615

18. Multiplying or Dividing Decimals by Powers of 10 Multiplying a decimal by 10n is the same as moving the decimal point to the rightn places, adding place holders if necessary. eg. 3.7615 × 103 = 37615

19. Multiplying or Dividing Decimals by Powers of 10 Multiplying a decimal by 10n is the same as moving the decimal point to the rightn places, adding place holders if necessary. eg. 3.7615 × 103 = 3761.5 eg. 743.28 × 105 = 74328000. Dividing a decimal by 10n is the same as moving the decimal point to the leftn places, adding place holders if necessary. eg. 743.28 ÷ 102 = 7.4328 eg. 3.7615 ÷ 103 = 0.0037615

20. 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)

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 ? 6.724 Solution We first put one of them above the other such that the decimal points are lined up.

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 ? 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. Next we compare the whole number portions – whichever has the larger whole number portion is the bigger decimal. 6.724 6.72189

24. 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. Next we compare the whole number portions – whichever has the larger whole number portion is the bigger decimal. 6.724 6.72189 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.

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. 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. 6.724 6.72189

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. 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. 6.724 6.72189

27. 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. 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.) 6.724 6.72189

28. 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. 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. 6.724 6.72189

29. 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. 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. 6.724 6.72189 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.

30. 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 ────── 34.16 + 2.3096 ─────── Correct, and we have to treat any empty space as a 0. Incorrect

31. Multiplication of Decimals 65 417 = ´ 2 100 1000 65 2417 = ´ 100 1000 ´ 65 2417 = ´ 100 1000 157105 = 100 , 000 • There are two ways to carry out this operation, • Converting the decimals to fractions • Example: • 0.65 × 2.417 = 1.57105

32. (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

33. 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)

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. Step 2. Move the decimal point in the dividend to the right by the same amount. (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.

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. Step 3. Put a decimal point above that one in the dividend. (click)

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.

38. 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

39. 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. (click)

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. Division of Decimals The process of long division is similar to that of dividing whole numbers with some modifications. Example: 2.556 ÷ 1.2 2 24 - 1 Step 4. Divide as if we are dividing whole numbers.

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 2 5 24 - 1 Step 4. Divide as if we are dividing whole numbers.

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 2 1 24 - 15 12 - 3 Step 4. Divide as if we are dividing whole numbers.

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 2 1 6 24 - 15 12 - 3 Step 4. Divide as if we are dividing whole numbers.

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 2 1 3 24 - 15 12 - 36 36 - Step 4. Divide as if we are dividing whole numbers. 0 Therefore 2.556 ÷ 1.2 = 2.13

45. 4 1 571 » ´ = = 571 0 . 571 7 1000 1000 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).

46. 0.571428 ··· 7 4 . 00000000 Converting Fractions to Decimals Conclusion The decimal expansion of a fraction a/b can be obtained by long division.

47. 3 17 is not terminating is terminating 11 625 21 7 7 is terminating = = 1920 640 7 1 2 5 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. • a product of 2’s only, • a product of 5’s only • a product of 2’s and 5’s only. Examples:

48. Example: The decimal expansion of 13 13 will have exactly 3 decimal places. = 3 1 40 2 5 Converting Fractions to Decimals 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? Theorem: If the fraction a/b is in its reduced form, and b = 2m5n then the decimal expansion of a/b is terminating with number of decimal places exactly equal to max{m, n}

49. 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.

50. Converting Fractions to Decimals More examples There is no obvious pattern on the cycle length, and a large denominator can have a small cycle length.