1. Decimals as an Extension of Percents © Math As A Second Language All Rights Reserved next #8 Taking the Fear out of Math 0.25 25%

2. In our discussion of percents we pointed out that people, for whatever reason, were most comfortable with scales that went from 0 to a power of 10. We happened to focus on percents (that is, per hundred). It would have been just as logical to choose a scale that went from 0 to 10 (perhaps calling it a “perdec”) or a scale that went from 0 to 1,000 (perhaps calling it “permill”), etc. © Math As A Second Language All Rights Reserved next

3. © Math As A Second Language All Rights Reserved next As an illustration, let’s suppose that there was a quiz with 7 equally weighted questions and you got the correct answer to 6 of the problems but omitted answering the seventh question. Then the fractional part of the test that you did correctly was 6/7. If the perfect score was 10, then your exact score would be given by… 6/76/7 of 10 = 6/76/7 × 10 = 60 / 7 = 60 ÷ 7 next

4. © Math As A Second Language All Rights Reserved next To begin to capture the flavor of how decimals are related to our present discussion, let’s carry out the division using the traditional algorithm… 760 8 – 56 4 4/74/7

5. next © Math As A Second Language All Rights Reserved next In terms of our corn bread model, this tells us that if a perfect score is represented by 10 (the corn bread is presliced into 10 equally-sized pieces), 6 / 7 is worth more than 8 pieces but less than 9 pieces (in fact, it is exactly 8 4 / 7 pieces). Less than 6 / 7 Greater than 6 / 7

6. © Math As A Second Language All Rights Reserved next Next suppose that instead of a perfect score being 10, it is 100. In this case, we would obtain… 60 / 7 of 100 = 60 / 7 × 100 = 600 / 7 = 600 ÷ 7 next Performing the division, we would find that… 7600 8 – 56 40 5/75/7 5 – 35 5 next

7. © Math As A Second Language All Rights Reserved next This tells us that if a perfect score is represented by 100 (the corn bread is presliced into 100 equally-sized pieces), 6 / 7 is worth more than 85 pieces but less than 86 pieces (in fact it is exactly 85 5 / 7 pieces). Less than 6 / 7 Greater than 6 / 7

8. The observation we wish to make now is that part of the above division problem was done previously when we assumed that the perfect score was 10. © Math As A Second Language All Rights Reserved next 7600 8 – 56 4040 R 55 – 35 5 The highlighted portion shows what we could have started with when we did the computation. next

9. In a similar way if the perfect score was represented by 1,000, our division problem would look like… © Math As A Second Language All Rights Reserved next This shows us that 6 on a scale of 7 is greater than 857 on a scale of 1,000 but less than 858. It is exactly 857 1 / 7 per 1,000. 76000 8 – 56 4040 R 15757 – 35 5050 – 49 1

10. We can now appreciate what the role of a decimal point might mean here. In terms of decimals, the first digit to the right of the decimal point modifies “per 10”, the second digit to the right of the decimal point modifies “per 100” etc. In other words, in decimal form, 6 / 7 can be represented by… © Math As A Second Language All Rights Reserved next 76.0006.000 0.8 – 56 4040 5757 – 35 5050 – 49 1

11. The fact that the digit furthest to the right (7) occurs in the third column to the right of the decimal point tells us that 857 is modifying “thousandths”. © Math As A Second Language All Rights Reserved next 76.0006.000 0.8 – 56 4040 5757 – 35 5050 – 49 1

12. And if we want to see how many parts per 10,000 is represented by 6 / 7, we could annex another 0 in the previous computation and simply continue to the next step as shown by the highlighted numerals. © Math As A Second Language All Rights Reserved next 76.0006.000 0.8 – 56 4040 5757 – 35 5050 – 49 1 0 0 1 – 7 3 next

13. Switching to “short division” to save space, we can continue the division process to obtain… © Math As A Second Language All Rights Reserved next 76. 0 4 0 5 0 1 0 3 0 2 0 6 0 0 0. 8 5 7 1 4 2 … …and from the above we can see a problem that arises when we want to express the common fraction as an equivalent decimal fraction. Namely, when we started the division the dividend consisted of a 6 followed by nothing but 0’s. next

14. However, as indicated by the numeral (6) above, after we obtained the quotient 0.857142, the dividend was again a 6 followed by nothing except 0’s. © Math As A Second Language All Rights Reserved next 76. 0 4 0 5 0 1 0 3 0 2 0 6 0 0 0. 8 5 7 1 4 2 … In other words, when 6 / 7 is represented as a decimal, the cycle of digits “857142” will repeat endlessly. next

15. The fact that the decimal representation never comes to an end means that we have to “invent” a notation that takes the place of us having to write the decimal “endlessly”. © Math As A Second Language All Rights Reserved next Notes What we do is place a bar over the repeating cycle. For example, if we write 0.6173, it means 0.617361736173...endlessly. next

16. In summary, only the digits that are under the bar repeat endlessly. We will discuss endless representations in greater detail in a later presentation. © Math As A Second Language All Rights Reserved next Notes The fact that a decimal representation is endless doesn’t worry us in the “real world”.

17. Even the very best measuring devices have a limited number of decimal place accuracy. © Math As A Second Language All Rights Reserved next Notes For example, 0.6173 represents a rate of 6,173 per 10,000 while 0.6174 represents a rate of 6,174 per 10,000. So having to choose between 0.6173 and 0.6174 as an estimate for 0.6173 involves an error of less than 1 part per 10,000.

18. While 0.6173 represents an endlessly repeating decimal, both 0.6173 and 0.6174 do end. © Math As A Second Language All Rights Reserved next Notes Because they end, we can look at the digit furthest to the right of the decimal point to determine that 0.6173 represents 6,173 ten thousandths, and 0.6174 represents 6,174 ten thousandths.

19. © Math As A Second Language All Rights Reserved next Enrichment Note #1 In general our intuitive ideas are based on things that we have experienced, and in mathematics no matter how many steps there are in a process, it is still a finite number. An Obsolete Hybrid Decimal Notation

20. © Math As A Second Language All Rights Reserved next Enrichment Note #1 So, for example, while a number such as 50,000,0000,000,000,000,000 is a very large number, it is still less than 50,000,0000,000,000,000,001. In essence there is a noticeable difference between being “very huge” and being infinite. Thus, the idea of a decimal having endless many digits is far more subtle than a decimal that has 50,000,0000,000,000,000,000 digits. next

21. © Math As A Second Language All Rights Reserved next Enrichment Note #1 So it should not come as a big surprise that in the “old” days people felt uncomfortable with expressing a non- ending decimal by putting a bar over the endlessly repeating cycle. Specifically, they coped with this notion by using a hybrid notation that lasted well into the 18th century.

22. © Math As A Second Language All Rights Reserved next Enrichment Note #1 Rather than represent 6 / 7 as 0.8 R4, they would write it as 0.8 4 / 7. The fraction was not a separate entry, but rather it was attached to the digit immediately to its left. In other words, 0.8 4 / 7 was used as an abbreviation for 8 4 / 7 tenths. 1 next note 1 1While the notation might have been confusing, it allowed people to avoid having to come to grips with endlessness. More specifically… 0.8 4 / 7 = 8 4 / 7 per 10 = 60 / 7 per 10 = 60 / 7 ÷10 = 60 / 7 × 1 / 10 = 60 / 70 = 6 / 7

23. © Math As A Second Language All Rights Reserved next Enrichment Note #2 Earlier in our discussion we talked about eventually having to round off endless decimals when we are making real world measurements. This introduces an error but it does convert the endless decimal into an “almost equivalent common fraction”. next

24. © Math As A Second Language All Rights Reserved next Enrichment Note #2 For example, if we look at the endless decimal 0.857142 and round it off to 0.857, we obtain almost equivalent common fraction 857 / 1,000. The error in this approximation is given by… next 0.857142 – 0.857 =0.857142 – 0.857 = 0.000142857 …and in most real-life applications this error is less than the limitations that are present in our measuring devices. next

25. © Math As A Second Language All Rights Reserved next The corn bread model makes it easier for students to visualize this rather abstract concept. For example, in the case where our measuring instrument can only give us guaranteed accuracy to the nearest tenth, it is equivalent to the case in which our corn bread is presliced into 10 equally sized indivisible pieces. In this case, 0.857142 is more than 8 of these 10 pieces but less than 9 of these 10 pieces. Hence, the maximum error in this approximation is less than 1 piece of these 10 pieces.

26. © Math As A Second Language All Rights Reserved next In a similar way, if our measuring instrument can only give us guaranteed accuracy to the nearest hundredth, it is equivalent to the case in which our corn bread is presliced into 100 equally sized indivisible pieces. In this case, 0.857142 is more than 85 of these 100 pieces but less than 86 of these 100 pieces. Hence, the maximum error in this approximation is less than 1 piece of these 100 pieces. CornBreadehT

27. © Math As A Second Language All Rights Reserved next In a similar way, if our measuring instrument can only give us guaranteed accuracy to the nearest thousandth, it is equivalent to the case in which our corn bread is presliced into 1,000 equally sized indivisible pieces. In this case, 0.857142 is more than 857 of these 1,000 pieces but less than 858 of these 1,000 pieces. Hence, the maximum error in this approximation is less than 1 piece of these 1,000 pieces. CornBreadehT

28. © Math As A Second Language All Rights Reserved next Notice that in each case, the maximum error is less than 1 piece, but as the power of 10 increases, the size of the indivisible pieces get smaller and smaller and eventually the “thickness” of each piece will become less than the size of the error in our measuring instrument. CornBreadehT

29. © Math As A Second Language All Rights Reserved next As technology continues to improve, the error in our measuring instrument will most likely continue to improve. However, any line we draw must have some thickness (otherwise the line would be invisible). Therefore, the error will never completely disappear, and the error in our measurement will always be 1 of the indivisible pieces. However, eventually the thickness of each of the pieces will be less than the accuracy of our measuring device, and therefore, for all practical purposes, the error will become completely negligible. CornBreadehT

30. © Math As A Second Language All Rights Reserved next Enrichment Note #3 Our previous two enrichment notes (hopefully) explain quite clearly that when it comes to making measurements in the “real world” there is no need to “invent” any numbers beyond the rational numbers (that is, common fractions).

31. © Math As A Second Language All Rights Reserved next Enrichment Note #3 Yet in the study of pure mathematics much is made of endless decimals and irrational numbers (which will be discussed in detail later in our discussion of decimals). Examples such as this show that mathematics belongs as much to the field of liberal arts, as well as to the fields of science and engineering.

32. © Math As A Second Language All Rights Reserved next Enrichment Note #3 We feel it is important for you to try to help your students see mathematics in this light. In short, there are people who study mathematics for the same reasons that people study poetry and other liberal arts; that is, not to build better bombs but rather to better engage the human mind.

33. next This completes our present discussion. In our next presentation we shall discuss how we compare decimals by size. In particular we show the rationale by which we know that 0.6173 is greater than 0.6173 but less than 0.6174. © Math As A Second Language All Rights Reserved Comparing Decimals