advertisement

What is .999…? Tina Roberts Use a calculator to find the decimal representation of 1/ 9. Based on the result, predict the decimal representations of 2/ 9, 3/ 9, 4/ 9, ..., and 8/ 9. Use a calculator to check your predictions. Continuing with the pattern, what do you think the decimal representation of 9/ 9 will be? How can that be? This problem was asking you to represent proper fractions for 9ths. Based on these fractions, you can predict certain outcomes for other fractions with 9 in the denominator. You can use a calculator to check your answer. Continuing with the pattern, you can predict other representations of a fraction with 9 in the denominator, including improper fractions. I used Excel to solve this problem. I created 3 columns, one for the numerator, one for the denominator, and one for the answer (=a2/b2). I put consecutive numbers in the numerator column starting with the numbers 1, 2, 3,…. I used only the number 9 in the denominator column because we were only working with fractions that had 9 in the denominator. The answer column automatically filled in once I entered the numbers for the numerator and denominator columns. I highlighted and drug the mouse down to about 20 so I could find the pattern and some sort of correlation to 9 being the only denominator. I found that when the numerator was less than nine, there was no whole number in the decimal answer. The decimal answer did equal the numerator (1/9=0.1111111, 2/9=0.22222) and so on until 9/9. Just looking at the initial pattern, I would have assumed 9/9 would equal 0.99999, but it just equaled 1 whole. I continued to look at the pattern that continued down the Excel chart I created. I noticed that when I went beyond 9 in the numerator (an improper fraction), there was a whole number on the left side of the decimal. These whole numbers increased by one as you continue down the answer column for every eight numbers. The numbers on the right side of the decimal I figured out represented the proper fractions I had done prior to the improper fractions. For example, 12/9=1.333333, 13/9=1.4444444, which means 12 and 13 can only be divided by 9 and even 1 amount of times and the numbers to the right of the decimal represent the proper fractions 3/9 and 4/9. I also noticed that the numbers at the right of the decimal were rounded and the very last number in each decimal is one more than its preceded number. For example, when I looked at 14/9=1.555556, 6 is one more than 5, so I knew the numbers had been rounded. This pattern continues down the answer column. Also, once you get to a multiple of 9, like 9, 18, or 24, there are whole numbers only represented in the answer column. This is because these are all divisible by 9 an even amount of times. An extension to this problem would be to see if you could use other numbers for the denominator to find patterns for fractions and decimal representations.