Hero's Method/Examples/92
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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:
- $\sqrt {92} \approx 9.5917$
$\blacksquare$
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Hero's method (Heron's method)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Hero's method (Heron's method)