Approximating Pi to Different Digits

[soroban]

Watch this . . .$$\pi \;=\;3.141592645$$

. . $$= \;3 + 0.141592654 \;=\; 3 + \dfrac{1}{7.062573306}$$ . [1]

. . $$=\;3 + \dfrac{1}{7 + 0.062573306} \;=\; 3 + \frac{1}{7+ \dfrac{1}{15.99659441}}$$ .[2]

. . $$=\;3 + \dfrac{1}{7 + \dfrac{1}{15 + 0.99659441}} \;=\; 3 + \dfrac{1}{7 + \dfrac{1}{15 + \dfrac{1}{1.003417228}}}$$

. . [$$=\;3 + \dfrac{1}{7 + \dfrac{1}{15 + \dfrac{1}{1 + 0.003417228}}} \;=\;3 + \dfrac{1}{7 + \dfrac{1}{15 + \dfrac{1}{1 + \dfrac{1}{292.6348491}}}}$$ .[3]If we stop at [1]: .$$\pi \;\approx\;3+\frac{1}{7} \;=\;\frac{22}{7} \;=\;3.142857...$$

If we stop at [2]: .$$\pi\;\approx\;3 + \frac{1}{7 + \dfrac{1}{16}} \;=\;\frac{355}{113} \;=\;3.14159292...$$

If we stop at [3]: .$$\pi \;\approx\;3 + \frac{1}{7+\dfrac{1}{15+\dfrac{1}{1 + \dfrac{1}{293}}}} \;=\;\frac{104,\!348}{33,\!215} \;=\;3.141592654...$$

 

[Prove It]

This was part of a series of lessons I did at a Harry Potter site, so disregard the first few sentences :P