- #1

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[itex]\sqrt {4.51} = ?[/itex]

I tried to use a linearization:

[itex]\sqrt x = \sqrt {n + \varepsilon } = \sqrt {\left[ {{\rm{integer}}} \right] + \left[ {{\rm{decimal part}}} \right]} = \sqrt n + L(\sqrt x ) \cdot \varepsilon [/itex]

…where you “linearly” count your way from [itex]\sqrt {n + \varepsilon } [/itex] to [itex]\sqrt x [/itex] by way of the linearization, [itex]L(\sqrt x )[/itex], namely, the derivative:

[itex]L(\sqrt x ) = {\left. {\frac{d}{{dx}}\sqrt x } \right|_{x = n}} = \frac{n}{{2\sqrt n }}[/itex]

But that’s another square root to calculate, which would further-roughen my approximation. Are there any other numerical-approximations you’d suggest?