Hero's Method/Examples/5

Examples of Use of Hero's Method

The calculation of the square root of $5$ by Hero's Method proceeds as follows:

$\ds x_0$

$=$

$\ds 2$

as the initial approximation

$\ds \leadsto \ \ $

$\ds x_1$

$=$

$\ds \dfrac {2 + \frac 5 2} 2$

$\ds $

$=$

$\ds 2 \cdotp 25$

$\ds \leadsto \ \ $

$\ds x_2$

$=$

$\ds \dfrac {2.25 + \frac 5 {2.25} } 2$

$\ds $

$=$

$\ds 2 \cdotp 236 \, 111 \, 111 \dots$

$\ds \leadsto \ \ $

$\ds x_3$

$=$

$\ds \dfrac {2 \cdotp 236 \, 111 \, 111 \dots + \frac 5 {2 \cdotp 236 \, 111 \, 111 \dots} } 2$

$\ds $

$=$

$\ds 2 \cdotp 236 \, 067 \, 978 \dots$

$\ds \leadsto \ \ $

$\ds x_4$

$=$

$\ds 2 \cdotp 236 \, 067 \, 978 \dots$

Comparing with Square Root of $5$, we have:

$\sqrt 5 \approx 2 \cdotp 23606 \, 79774 \, 99789 \, 6964 \ldots$

$\blacksquare$