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Approximations of pi

soroban

Well-known member
Feb 2, 2012
409

Watch this . . .


[tex]\pi \;=\;3.141592645[/tex]

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

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

. . [tex]=\;3 + \dfrac{1}{7 + \dfrac{1}{15 + 0.99659441}} \;=\; 3 + \dfrac{1}{7 + \dfrac{1}{15 + \dfrac{1}{1.003417228}}} [/tex]

. . [[tex]=\;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}}}} [/tex] .[3]


If we stop at [1]: .[tex]\pi \;\approx\;3+\frac{1}{7} \;=\;\frac{22}{7} \;=\;3.142857...[/tex]

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

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

Prove It

Well-known member
MHB Math Helper
Jan 26, 2012
1,403
This was part of a series of lessons I did at a Harry Potter site, so disregard the first few sentences :p