Title
What is .999...?

Problem Statement
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?

 

Investigation/Exploration of the Problem

To solve this problem, I entered the the given information on a spreadsheet to see if I could detect any patterns. See the table below or click here for the complete spreadsheet.

1 0.111111
2 0.222222
3 0.333333
4 0.444444
5 0.555556
6 0.666667
7 0.777778
8 0.888889
9 1

 

As we view the information in total, we can see another pattern beginning with 5. The last number of the repeating decimal is 6. If we round the next number to the left of the 6 , the next number will round to six and the next will round to 6 until the final rounded decimal is .6666666. The pattern continues with 6,7,and 8.

The question is.... why is 9/9 equal to one?

Since we can also write 9/9 as .999999...(repeating)
Let's  let x = 0.999... and  10x = 9.999...
If we subtract 10x - x = 9.999... -.999.. we get 9x = 9 or x = 1
 We see that x = 1 but we stated that x = 0.999... so 1 = 0.999...(repeating)

Author & Contact

Laura Thomas

Memorial Middle School

Conyers, GA
LThomas@rockdale.k12.ga.us