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.