A while back I saw someone claiming that .99̅ is equal to 1, due to this:

0/9 = .00

1/9 = .11̅

2/9 = .22̅

3/9 = .33̅

4/9 = .44̅

5/9 = .55̅

6/9 = .66̅

7/9 = .77̅

8/9 = .88̅

Supposedly any positive integer less than 10, when divided by 9, is written as a decimal point followed by that integer, repeating forever. So it makes sense that 9/9 should be .99̅ (which does approach 1 but is still not equal to 1), except that's obviously not the case since it is 1 (due to mathematical law... any real non-zero number divided by itself is 1).

And that is the basis that someone was using to claim that those two values are equal.

However, I am certain that .99̅ is not equal to 1. But I can't really figure out what's going on in this case...

 

The other case would be something like 1/3 + 2/3, which is 3/3, which is 1... of which the problem could be written as .33̅ + .66̅... however, if you do write the answer as .99̅ instead of 1 then that's more of a rounding error, since you cannot add an infinitely long string of numbers with complete accuracy. In other words, it'd clearly be incorrect.

You can't say the same about the 9/9 thing though... since there is no rounding or maths at all even involved in that case...

Edited June 2, 201213 yr by Xenidal

I think it's because if you were to do the problem "1 - .99̅ = ", since the .99̅ goes on for infinity, you'll never find a value in the answer. It'll just go on as .000000000000 for infinity. So I can see how you can say it's basically one. The "difference" is so substantially small that it cannot even be factored in.

I'm not going to lie. I came here with every intention to read this and attempt to break out some new math skillz.

I did not even bother reading past the first sentance. I did however read Oishii's post since it was right above the quick reply, and I see the logic there. ..

Yeah that makes sense, however, 1-.9̅ is still a non-zero number (even though it would be impossible to calculate). If you ever could reach the end of .9̅ (which you can't since it never ends) then theoretically 1-.9̅ should end in a 1. Meaning it is not 0 but simply an infinitely small non-zero number. Meaning that 1/(1-.9̅) should be an unsolvable infinitely large number but still distinguished from 1/0... and (1-.9̅)/(1-.9̅) should be 1 rather than undefined... and .9̅-1 should be negative (since it is less than 0, even though it is unmeasurably less than 0).

In all practicality, the difference between 1 and .9̅ is so small that it's as if it doesn't even exist, but mathematically there should still be a difference.

I'm not going to lie. I came here with every intention to read this and attempt to break out some new math skillz.

I did not even bother reading past the first sentance. I did however read Oishii's post since it was right above the quick reply, and I see the logic there. ..

 

I did the exact same thing lmao