Why did the babylonians approximated the square root of two the way they did?

Question

I was researching about the Newton-Raphson method and came across http://www.sosmath.com/calculus/diff/der07/der07.html. On the third line of the page near the end of the line it told us to consider $2/x_1$:

Consequently $3/2>\sqrt{2}$. If we now consider $2/x_1=4/3$, its square $16/9$ is of course smaller than $2$, so $2/x_1<\sqrt{2}$.

Why is that? Why did they divide $2$ by $x_1$ and not divide $1$ by $x_1$?

Answer 1

If $x > \sqrt{2}$ then $\dfrac1x < \dfrac{1}{\sqrt{2}}$. See the problem? $\dfrac{1}{\sqrt{2}}$ is not $\sqrt{2}$ but $\dfrac{2}{\sqrt{2}}$ is.

Answer 2

Since $x_1$ is close to $\sqrt2$, taking $2\over x_1$ will get you another number close to $\sqrt2$ but on the other side from $x_1$. This way you can take the average of $x_1$ and the new number to approximate $\sqrt2$.