You're wrong. They are EXACTLY equal BY DEFINITION. 9.99999999... is just another way of writing 1.
It is simple to prove this by contradiction.
Suppose it is not exactly equal to 1, that they're two separate numbers.
Let a=.9999..... b=1
Since the space of real numbers is compact is a compact set, there exists by neccessity a number c between 9.9999... and 1. a<c<b (strict inequality)
Yet, for any number c that you chose, c<a
Therefore, we have a contradiction... this implies that a=b.
Originally Posted by raiderfan19
Look dude, i understand ive already explained that for the purposes of equations yes you can subsitute the two because you have essentially no degree of error BUT in theoretical truth, they arent completely equivalent and I have had calc 2 and analysis classes.