# Square Root/Examples/3

## Decimal Expansion

The decimal expansion of $\sqrt 3$ starts:

$\sqrt 3 \approx 1 \cdotp 73205 \, 08075 \, 68877 \, 2935 \ldots$

## Historical Note

The square root of $3$ was the second number, after the square root of $2$, to be identified as being irrational.

This was achieved by Theodorus of Cyrene.

Archimedes of Syracuse provided the approximation:

$\dfrac {1351} {780} < \sqrt 3 < \dfrac {265} {153}$

What he actually demonstrated was:

$26 - \dfrac 1 {52} < 15 \sqrt 3 < 26 - \dfrac 1 {51}$

These can be achieved by interpreting Pell's Equation, to obtain:

$1351^2 - 3 \times 780^2 = 1$
$265^2 - 3 \times 153^2 = -2$

Thus it appears that Archimedes was familiar with Pell's Equation.