Verify that \( 1=0 . \overline{9} \)

Question

Verify that \( 1=0 . \overline

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Answer 1

Let n = 0.999999999999999999...

Then 10n = 9.999999999999999...    by multiplying both sides by 10

And 10n = 9 + 0.9999999999999...    by splitting off the integer part

Or 10n = 9 + n by the definition of n as shown in the first line above

So 9n = 9 by subtracting n from both sides

Finally, n = 1 by dividing both sides by 9

Therefore n = 0.99999999999999... = 1