Originally Posted by
Flying Mullet
Alright, I'm going to start over on this proof and look at it two ways:
The first says that we haven't proved that 0.9 = 1
So 0.9 = X
We don't prove that X = 1 until after these series of statements so we have to use 0.9 for all instances of X
10X = 9.9
9X - 0.9 = 9.9 - 0.9
This step's incorrect, so this counter proof is invalid. YOu've taken 0.
9 from the right, but you've taken 0.
9 and X from the left.
Now, I will take a second approach:
Let's assume that 0.9 = 1 from a previous proof.
So 0.
9 = 1 = X (we can use these interchangably.
10X = 9.
9
This step's also invalid, though I think it's a typo judging from your next line, so I'll continue on this one
9X - 1 = 9.
9 - 1
9X = 8.
9
X = 8.
9 / 9
And we're back to where we started, dividing into an infinite number.
Yes, we're back to dividing by an infinite number, but you've stopped midway. You can then see 8.
9 = 8 + 0.
9 = 8 + 1 = 9, so that resolves the dividing by an infinite number, if you don't like dividing such things. Stopping somewhere just because you can't see a way to continue doesn't provide a counter-proof.
Let's try writing this proof out even longer so it's easier to see what's going on
We want to proove 0.
9 = 1.
Let's start with 0.
9.
Let's define X = 0.
9.
Now 10X = 9.
9, because we've shifted the decimal place as we're multiplying by the base.
Next we subtract X from both sides.
10X - X = 9.
9 - X
For the LHS, 10X - X = 9X, as you should know from algebra.
For the RHS, the only value of X we know is 0.
9 since that's what we defined earlier, so the RHS becomes 9.
9 - 0.
9, so we now have
9X = 9.
9 - 0.
9
You should be able to see the RHS simplifies to 9, giving us
9X = 9
Divide both sides by 9, and we get
X = 1.
Nowhere earlier have I used X = 1 in obtaining this value.
