Which fraction is the clostest to the actual value of pi?

Answer 1

No fraction is truly the closest to the "actual value" of pi. Pi is a nonstop number, thus it has no true value. The only way you could ask, "What's closest to pi?" is if you wanted a number rounded to a certain place. Sorry if I've disappointed you, but it's impossible.

But as a reference, the American education system wants you to use 3.14. If the choice is yours to make (i.e., in your job), then I would recommend using 3.14159, which is pi rounded to the hundred-thousandth place. +++

A common approximation, the integer fraction closest to pi and sufficient in practical problems not needing high accuracy, or as a quick estimate of likely answer, is 22/7. I'm surprised US schools stop at only 2 decimal places, when 3 (i.e. pi = 3.142) is more accurate but just as easy to use; but scientific calculators normally have a pi key to umpteen decimal places anyway.

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π is a transcendental number (if you sit and think about π are you doing transcendental meditation?) which is an irrational number which is not the solution to a polynomial with rational coefficients.

Depending how accurate you want your rational approximation there are various fractions that can be used:

22/7 is 3.142857... which is accurate when rounded to 2 decimal places: 3.14

355/113 is 3.1415929... which is accurate when rounded to 6 decimal places: 3.141593

314159265.../100000000... is as accurate as you want it - for each extra digit of π that you append to the numerator you append an extra zero to the denominator.

For most purposes taking π rounded to 2 decimal places (3.14) is sufficient. If you need greater accuracy, taking it rounded to 6 decimal places (3.141593 ≈ 355/113) will do - it all depends upon the accuracy of the other numbers (measurements) in the calculation.

Computer programming languages with in-built values for π are still limited in their accuracy due to the number of bits used to store π. A single precision floating point value can only store π to about 7 digits, a double-precision to about 11 digits; electronic (hand held) calculators are accurate to about 11 digits.

When working with (irrational) numbers like π, it is often better to see if you can simplify the calculation before reaching for a calculator - as the calculation is done each intermediate result can be rounded with the accumulated rounding affecting the final result - I have seen such a calculation in a test paper where the mark scheme allows for errors in the result which come about through the intermediate rounding, but the actual answer is an exact whole number (if the calculation is simplified before being done the exact answer will be calculated).