Describe how you would estimate the square root of a number that is not a perfect square without using a calculator.
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If 
is an integer, you can use induction. First show the inequality holds for 
. You have 
, which is true.
Now assume this holds in general for
, i.e. that 
. We want to prove the statement then must hold for 
.
Because, you have
and this must be greater than
for the statement to be true, so we require
for
. Well this is obviously true, because solving the inequality gives 
. So you're done.
If you is any real number, you can use derivatives to show that 
increases monotonically and faster than .
Now assume this holds in general for
Because
and this must be greater than
for
If you