savio13
Savio Cabral
The conjecture is that 0.9 recurring (i.e. 0.999....,
with the digits going on forever) is actually equal to 1
(For this exercise I will use the notation 0.999... as notation for 0.9 recurring,
the correct way would be to put a little dot above the 9, or a line on top like this: 0.9)
•
Let X = 0.999...
•
Then 10X = 9.999...
Subtract X from each side to give us:
•
9X = 9.999... - X
but we know that X is 0.999..., so:
•
9X = 9.999... - 0.999...
or: •
9X = 9
Divide both sides by 9:
•
X = 1
But hang on a moment I thought we said X was equal to 0.999... ?
Yes, it does, but from our calculations X is also equal to 1. So:
•
X = 0.999... = 1
•
Therefore 0.999... = 1
with the digits going on forever) is actually equal to 1
(For this exercise I will use the notation 0.999... as notation for 0.9 recurring,
the correct way would be to put a little dot above the 9, or a line on top like this: 0.9)
•
Let X = 0.999...
•
Then 10X = 9.999...
Subtract X from each side to give us:
•
9X = 9.999... - X
but we know that X is 0.999..., so:
•
9X = 9.999... - 0.999...
or: •
9X = 9
Divide both sides by 9:
•
X = 1
But hang on a moment I thought we said X was equal to 0.999... ?
Yes, it does, but from our calculations X is also equal to 1. So:
•
X = 0.999... = 1
•
Therefore 0.999... = 1