Question

Finding the better estimate of lt=" \sqrt{2} " align="absmiddle" class="latex-formula">
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Answer 1

There's no such thing as a "better" (or best) estimate of the square root of 2. This is an irrational number, i.e. it has infinitely many decimal digits, and they don't follow any patter.

So, the most you can do is approximate the number to an arbitrary number of decimal digits. For example, if you want 4 decimal digits, you have

$\sqrt{2}\approx 1.4142$