$10^9 \times \sqrt{3}$ what are first two digits after the decimal point? Because of floating point error, my computer basically says $10^9 \sqrt{3} \approx 1.73205 \times 10^9$ so that if we ignore the numbers before the decimal point, the fractional part is:
$$\{ 10^9 \sqrt{3} \} = 0$$
That's obviously wrong.  As the algorithm progresses, the program keeps losing accuracy, effectively getting the error I just described.  if I type it directly we get the correct answer:
$$ 10^9 \sqrt{3} = 1732050807.\color{#00D000}{56}88772$$
but the computer doesn't store all these digits forever.   I could like to know how to get the first two decimal places without a calculator.

You want the two digits before the decimal point of 
$10^{11} \sqrt{3} = \sqrt{3 \times 10^{22}}$.  Let's see if we can 
get a good rational approximation of that.  Let $A = 3 \times 10^{22}$.
The Newton's method iteration for $\sqrt{A}$ is
$x \to (x + A/x)/2$.  If we start with $x_0 = 2 \times 10^{11}$ and apply
this iteration, rounding each result to the nearest integer, we get
$$ \eqalign{x_1 &=  175000000000\cr
            x_2 &= 173214285714\cr
            x_3 &= 173205081001\cr
            x_4 &= 173205080757\cr
            x_5 = x_4 &= 173205080757\cr}$$
Check that $x_4^2 > A$, so what we want is $x_4 - 1 = 173205080756$, and its lowest two digits are the answer: $56$.
