Question 645485
The fractions below


22/7, 355/113, 103993/33102, 2508429787/798458000


convert to the following numbers in decimal form


3.14285714285714, 3.14159292035398, 3.1415926530119, 3.141592653589794


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Now compare this with the given value of pi, which is 3.141592653589793...



First Rational Approximation (22/7)


3.141592653589793
3.14285714285714


Note: first number is the given value of pi and second number is the rational approximation


Notice how 22/7 only matches up to 3.14 and then it's off for the rest of the expansion.

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Second Rational Approximation (103993/33102)


3.141592653589793
3.14159292035398


Note: first number is the given value of pi and second number is the rational approximation


We got a better approximation this time since it matches 3.141592


So far, the second approximation is better

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Third Rational Approximation (355/113)


3.141592653589793
3.1415926530119


Note: first number is the given value of pi and second number is the rational approximation


We got a better approximation this time since it matches 3.141592653


So far, the third approximation is best

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Fourth Rational Approximation (2508429787/798458000)


3.141592653589793
3.141592653589794

Note: first number is the given value of pi and second number is the rational approximation


We get the most accurate match this time since it matches 3.14159265358979


The only difference between this rational approximation and the given decimal expansion is that the '3' at the end doesn't match with the 4.

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Answer:


So the fraction 2508429787/798458000 gives the best approximation of pi



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