What is .999?

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?

Using a calculator I found that 1/9 = .1111....

Following this example I predict that 2/9, 3/9, 4/9, ...and 8/9  will also be repeating decimals that follow the numerator.

By using the calculator I find that my prediction is correct. Below is a spreadsheet that shows this.  This table was developed by using a formula that divides each whole number by 9.

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

But what happens when we get to 9/9?  The spreadsheet shows that 9/9 = 1.

 9 1

Is this logical?  We are taught that when we have a numerator = denominator that the quotient equals 1.  Seeing the pattern from above is this a true statement?

To get some answers I went to Math Forum and found some similar questions.

Solution 1

Solution 2

Solution 3

My personal thoughts...

First, I questioned what is the difference between .9999... and 9/9=1?  I really liked the explanation that posed the question what is the number between the two (solution3)? Only .9999.... repeating right?  In my opinion .9999... and 1 are not equal, but 9/9 equals 1.  The infinite .9999... will never reach 1, so it cannot be equal to 1.  However, if 9 is a whole number and it is divided by the same whole number, 9, then that equals 1.

Another thing to think about is 9 × 1/9 = 1.

9 × 0.1111... = .99999.

Examining these two expressions demonstrates that .9999... and 1 are not equal.