The tricky part is that there is no 0.999…9 because there is no last digit 9. It just keeps going forever.
If you are interested in the proof of why 0.999999999… = 1:
0.9999999… / 10 = 0.09999999…
You can divide the number by 10 by adding a 0 to the first decimal place.
0.9999999… - 0.09999999… = 0.9
because the digit 9 in the second, third, fourth, … decimal places cancel each other out.
Let’s pretend there is a finite way to write 0.9999999…, but we do not know what it is yet. Let’s call it x. According to the above calculations x - x/10 = 0.9 must be true. That means 0.9x = 0.9. dividing both sides by 0.9, the answer is x = 1.
The reason you can’t abuse this to prove 0=1 as you suggested, is because this proof relies on an infinite number of 9 digits cancelling each other out. The number you mentioned is 0.9999…8. That could be a number with lots of lots of decimal places, but there has to be a last digit 8 eventually, so by definition it is not an infinite amount of 9 digits before. A number with infinite digits and then another digit in the end can not exist, because infinity does not end.
Your way of thinking makes sense but you’re interpreting it wrong.
If you can round up and say “0,9_ = 1” , then why can’t you round down and repeat until “0 = 1”? The thing is, there’s no rounding up, the 0,0…1 that you’re adding is infinitely small (inexistent).
It looks a lot less unintuitive if you use fractions:
No, because that would imply that infinity has an end. 0.999… = 1 because there are an infinite number of 9s. There isn’t a last 9, and therefore the decimal is equal to 1. Because there are an infinite number of 9s, you can’t put an 8 or 7 at the end, because there is literally no end. The principle of 0.999… = 1 cannot extend to the point point where 0 = 1 because that’s not infinity works.
If .99…9=1, then 0.999…8=0.999…9, 0.99…7=0.999…8, and so forth to where 0=1?
The tricky part is that there is no 0.999…9 because there is no last digit 9. It just keeps going forever.
If you are interested in the proof of why 0.999999999… = 1:
0.9999999… / 10 = 0.09999999… You can divide the number by 10 by adding a 0 to the first decimal place.
0.9999999… - 0.09999999… = 0.9 because the digit 9 in the second, third, fourth, … decimal places cancel each other out.
Let’s pretend there is a finite way to write 0.9999999…, but we do not know what it is yet. Let’s call it x. According to the above calculations x - x/10 = 0.9 must be true. That means 0.9x = 0.9. dividing both sides by 0.9, the answer is x = 1.
The reason you can’t abuse this to prove 0=1 as you suggested, is because this proof relies on an infinite number of 9 digits cancelling each other out. The number you mentioned is 0.9999…8. That could be a number with lots of lots of decimal places, but there has to be a last digit 8 eventually, so by definition it is not an infinite amount of 9 digits before. A number with infinite digits and then another digit in the end can not exist, because infinity does not end.
Wonderful explanation. It got the point across.
That is the best way to describe this problem I’ve ever heard, this is beautiful
0.999…8 does not equal 0.999…9 so no
Your way of thinking makes sense but you’re interpreting it wrong.
If you can round up and say “0,9_ = 1” , then why can’t you round down and repeat until “0 = 1”? The thing is, there’s no rounding up, the 0,0…1 that you’re adding is infinitely small (inexistent).
It looks a lot less unintuitive if you use fractions:
1/3 = 0.3_
0.3_ * 3 = 0.9_
0.9_ = 3/3 = 1
No, because that would imply that infinity has an end. 0.999… = 1 because there are an infinite number of 9s. There isn’t a last 9, and therefore the decimal is equal to 1. Because there are an infinite number of 9s, you can’t put an 8 or 7 at the end, because there is literally no end. The principle of 0.999… = 1 cannot extend to the point point where 0 = 1 because that’s not infinity works.
If you really wants to understand the concept , you need to learn about limits
There is no .99…8.
The … implies continuing to infinity, but even if it didn’t, the “8” would be the end, so not an infinitely repeating decimal.