If you evaluate
(Math.PI).ToString("G51")
the result is
"3,141592653589793115997963468544185161590576171875"
Math.Pi is correct up to the 15th decimal which is in accordance with the precision of a double. "G17" is used for round tripping, so I expected the ToString to stop after "G17", but instead it continued to produce numbers up to "G51".
Q Since the digits after 15th decimal are beyond the precision afforded by the 53 bits of mantissa, how are the remaining digits calculated?
A double has a 53 bit mantissa which corresponds to 15 decimals
A double has a 53 bit mantissa which corresponds to roughly 15 decimals.
Math.Pi is truncated at 53 bits in binary, which is not precisely equal to
3.1415926535897
but it's also not quite equal to
3.14159265358979
, etc.
Just like 1/3 is not quite 0.3, or 0.33, etc.
If you continue the decimal expansion long enough it will eventually terminate or start repeating, since Math.PI, unlike π is a rational number.
And as you can verify with a BigRational type
using System.Numerics;
var r = new BigRational(Math.PI);
BigRational dem;
BigRational.NumDen(r, out dem);
var num = r * dem;
Console.WriteLine(num);
Console.WriteLine("-----------------");
Console.WriteLine(dem);
it's
884279719003555
---------------
281474976710656
And since the denominator is a power of 2 (which it has to be since Math.PI has a binary mantissa and so a terminating representation in base-2). It also therefore has a terminating representation in base-10, which is exactly what was given by (Math.PI).ToString("G51"):
3.141592653589793115997963468544185161590576171875
as
Console.WriteLine(r == BigRational.Parse("3.141592653589793115997963468544185161590576171875"));
outputs
True