Why 10/9 = 1.111… ; 9×1.111… = 9.999… ; but 9×10/9 = 10. Why is 10 equal to 9.999…?
In: Mathematics
We’ve got a proof in the expressions you’ve given.
We have:
10/9 = 1.1…
9*1.1… = 9.9…
and 9×10/9=10
So lets rewrite the last one in terms of the first two:
9×10/9 = 10/9 * 9 = 1.1… * (9.9…/1.1…)
We can cancel out the two 1.1… terms as one is a multiplication and one is a division.
So we *must* have 9×10/9 = 9.9…
This means we have both 9×10/9 = 10 _and_ 9×10/9=9.9…
Therefore we’ve proven that 10 = 9.9…
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