Math 130 Ch 4.2

4a Find the decimal representation of 1/37 to at least 6 places (or as many as your calculator shows). Notice the repeating pattern

1/37= .027027027...

5a Find the decimal representation of 1 41 to at least 10 decimal places. Notice the repeating pattern.

1/41= .0243902439

4c Write 10* 1/37 and 100*1/37= 100/37 as mixed numbers.

10*1/37= 0 10/37 100* 1/37= 100/37= 2 26/37

5c. Write the numbers 10* 1/41 = 10/41 100* 1/41= 100/41 1000* 1/41= 1000/41 10,000* 1/41= 10,000/41 100,000* 1/41= 100,000/41 as mixed numbers

10*1/41= 0 10/41 100*1/41= 100/41= 2 18/41 1000*1/41=24 16/41 10,000*1/41=243 37/41 100,000*1/41=2439 1/41

4b Now find the decimal representation of 10/37 and of 26/37 to at least 6 places. Compare the repeating patterns with each other and to the decimal representation of 1/37. What do you notice?

10/37= 10* 1/37= .270270270..... 26/37= .0702702702....notice that 26*27= 702. Also notice that the digits of this base ten are the same as the digits in 1/37 and the same sequence, but out of sync--each digit (except the first 2) is shifted two places to the left

3 Now that you understand why multiplying a number by 10 shifts the digits 1 place to the left, explain how we can deduce that multiplying by 10,000 shifts the digits 4 places to the left and multiplying by 100,000 shifts the digits 5 places to the left. How should we think about the numbers 10,000 and 100,000 to make these deductions

If you think of 10,000 as 10*10*10*10 and of 100,000 as 10*10*10*10*10, it makes sense that multiplying by 10,000 shifts the digits four places to the left--- you are simply multiplying by ten four times. Likewise, multiplying by 100,000 is multiplying by ten five times, so the digits shift five places to the left.

2 Mary says that 10 * 3.7 = 3.70. Why might Mary think this? Explain to Mary why her answer is not correct and why the correct answer is right. If you tell Mary a procedure, be sure to tell Mary why the procedure makes se

Many people say that when multiplying a whole number by ten you add a zero to the end of the number. This rule does indeed produce the correct answer for whole numbers. Mary probably tried to apply this rule to the base-ten number3.7, but adding a zero to the end actually doesn't change the number! It would be better to help Mary learn to interpret multiplying by ten as shifting the digits to the left

1 Using the example 10*47 to illustrate, explain in your own words why we move the digits in a number 1 place to the left when we multiply by 10

Multiplying 10*47 can be interpreted as 10 sets of 47 units in each set. See for example figure 4.10 in the regular text. Where before we had four tens, we now have ten times that, or forty tens, which is four hundreds. Where before we had seven units, we now have ten times that, or seventy units, which is seven tens. In any number, when it is multiplied by ten each digit shifts one place to the left, since the multiplication gives you ten times the original value of that digit.

4d What happens to the decimal representation of a number when it is multiplied by 10? By 100? Use your answer, and part (c), to explain the relationships you noticed in part (b

Multiplying any number by ten shifts the digits of its base-ten representation one place to the left, as we see in the base-ten representation of 10/37. Multiplying any number by 100 shifts the digits of its base-ten representation two places to the left combining the answers to part b and c., we see that the full base-ten representation of 2 26/37+2.70270270...Every digit has been shifted two places to the left

5d Use your answers in part (b) to find the deci-mal representations of the fractional parts of the mixed numbers you found in part (c). Do not use your calculator or do long division; use part (b). Explain your reasoning.

The left side of each equation in part b. is the same as the left side of each corresponding in part c. Thus we know that the corresponding right sides are equal. Taking 100* 1/41 as an example: from the right sides of the corresponding equations in each part, we have 2.4390243902...=2 18/41 -> 18/41= .4390243902... So this problem simply asks students to match the fractional parts with the corresponding base-ten parts in each equation

5b Use your answer in part (a) to find the decimal representations of the numbers 10* 1/ 41 , 100 *1/4, 1000 * 1/41, 10,000 *1/41, 100,000* 1/41 without a calculator.

To get the base-ten representations this problem ask for, we simply shift digits one place to the left, shifting an additional time for each example: 10* 1/41= .2439024390 100*1/41= 2.4390243902 1000*1/41=24.3902439024 10,000*1/41=243.9024390243 100,000*1/41=2439.0243902439