Hero's Method/Examples/92

Examples of Use of Hero's Method

The calculation of the square root of $92$ by Hero's Method proceeds as follows:

$\ds x_0$

$=$

$\ds 8 \cdotp 5$

as the initial approximation

$\ds \leadsto \ \ $

$\ds x_1$

$=$

$\ds \dfrac {8.5 + \frac {92} {8 \cdotp 5} } 2$

$\ds $

$=$

$\ds 10 \cdotp 8235$

$\ds \leadsto \ \ $

$\ds x_2$

$=$

$\ds \dfrac {10 \cdotp 8235 + \frac {92} {10 \cdotp 8235} } 2$

$\ds $

$=$

$\ds 9 \cdotp 66176 \dots$

$\ds \leadsto \ \ $

$\ds x_3$

$=$

$\ds \dfrac {9 \cdotp 66176 \dots + \frac {92} {9 \cdotp 66176 \dots} } 2$

$\ds $

$=$

$\ds 9 \cdotp 9519 \dots$

and so on.

The actual value of $\sqrt {92}$ is:

$\sqrt {92} \approx 9.5917$

$\blacksquare$