# How do you approximate 2/sqrt3?

Jul 6, 2015

Use an iterative method to get a good rational approximation for $\sqrt{3}$ then use that to calculate $\frac{2}{\sqrt{3}}$ to get (say)

$\frac{2}{\sqrt{3}} \cong \frac{2}{\frac{97}{56}} = \frac{112}{97} \cong 1.155$

#### Explanation:

Start with a reasonable approximation ${a}_{0} = 2$ for $\sqrt{3}$.

Then iterate using the formula:

${a}_{i + 1} = \frac{{a}_{i}^{2} + 3}{2 {a}_{i}}$

${a}_{1} = \frac{{a}_{0}^{2} + 3}{2 {a}_{0}}$

$= \frac{{2}^{2} + 3}{2 \cdot 2}$

$= \frac{7}{4}$

${a}_{2} = \frac{{a}_{1}^{2} + 3}{2 {a}_{1}}$

$= \frac{{\left(\frac{7}{4}\right)}^{2} + 3}{2 \cdot \frac{7}{4}}$

$= \frac{\frac{49}{16} + \frac{48}{16}}{\frac{7}{2}}$

$= \frac{97}{56}$

We will stop here, but if you want more accuracy, just iterate again.

In general, to find the square root of $n$, pick a reasonable first guess as ${a}_{0}$, then iterate using:

${a}_{i + 1} = \frac{{a}_{i}^{2} + n}{2 {a}_{i}}$