Hero's Method/Examples
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Examples of Use of Hero's Method
Square Root of $5$
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:
- $\sqrt 5 \approx 2 \cdotp 23606 \, 79774 \, 99789 \, 6964 \ldots$
$\blacksquare$
Square Root of $92$
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:
- $\sqrt {92} \approx 9.5917$
$\blacksquare$