Binomial Theorem Approximations/Examples/Arbitrary Example 2

Example of Binomial Theorem Approximation

$\sqrt {25 \cdotp 1} \approx 5 \cdotp 0100$

to $4$ decimal places.

Proof

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\(\ds \sqrt {25 \cdotp 1}\)

\(=\)

\(\ds 5 \times \paren {1 + 0 \cdotp 004}^{1/2}\)

\(\ds \)

\(\approx\)

\(\ds 5 \times \paren {1 + \dfrac 1 2 \paren {0 \cdotp 004} + \dfrac {\paren {\frac 1 2} \paren {-\frac 1 2} } {2!} \paren {0 \cdotp 004}^2 + \cdots}\)

\(\ds \)

\(\approx\)

\(\ds 5 \times \paren {1 + 0 \cdotp 002 - 0 \cdotp 000002}\)

\(\ds \)

\(=\)

\(\ds 5 \cdotp 00999\)

\(\ds \)

\(\approx\)

\(\ds 5 \cdotp 0100\)

$\blacksquare$

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text I$. Algebra: The Binomial Theorem: Approximations: Example $2$.