Square Root/Examples/3
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Decimal Expansion
The decimal expansion of $\sqrt 3$ starts:
- $\sqrt 3 \approx 1 \cdotp 73205 \, 08075 \, 68877 \, 2935 \ldots$
This sequence is A002194 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Historical Note
The square root of $3$ was the second number, after the square root of $2$, to be identified as being irrational.
This was achieved by Theodorus of Cyrene.
Archimedes of Syracuse provided the approximation:
- $\dfrac {1351} {780} < \sqrt 3 < \dfrac {265} {153}$
What he actually demonstrated was:
- $26 - \dfrac 1 {52} < 15 \sqrt 3 < 26 - \dfrac 1 {51}$
These can be achieved by interpreting Pell's Equation, to obtain:
- $1351^2 - 3 \times 780^2 = 1$
- $265^2 - 3 \times 153^2 = -2$
Thus it appears that Archimedes was familiar with Pell's Equation.
Sources
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions ... (previous) ... (next): Table $1.1$. Mathematical Constants
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 1$: Special Constants: $1.4$
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $1 \cdotp 732 \, 050 \, 807 \, 568 \, 877 \, 293 \, 527 \, 446 \, 341 \, 505 \, 872 \, 366 \, 942 \ldots$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $1 \cdotp 73205 \, 08075 \, 68877 \, 29352 \, 74463 \, 41505 \, 87236 \, 69428 \ldots$