Some musings on π

3.1415926535897932384626433832795028841971693993751058209749445923078164062
8620899862803482534211706798214808651328230664709384460955058223172535940812848
1117450284102701938521105559644622948954930381964428810975665933446128475648233
7867831652712019091456485669234603486104543266482133936072602491412737245870066
0631558817488152092096282925409171536436789259036001133053054882046652138414695
1941511609433057270365759…

 
Let’s write the closest value to π by using all digits   0 – 9   once each

(1 – 23 + 45 * 67) / 890 = 2993/890 = 3.3629213483146067415730337078651685393258426966292134…

2993/890   –   π   =   0.2213286947248135031103903245856656551286732972541076…
 
Better approximations:

1/(234 + 56 – 7) * 890   =   890/283  
=   3.1448763250883392226148409893992932862190812720848056…

890/283   –   π   =   0.0032836714985459841521976061197904020219118727096998…
 

0   +   1   +   2   +   3/(4*5)   –   6/(78*9)   =   7351/2340
=   3.1414529914529914529914529914529914529914529914529914…
Repeating decimal:   3.14145299
 

With 10 digits accuracy:

3   +   4/28   –   1/(790 + 5/6)   =   104348/33215
=   3.1415926539214210447087159415926539214210447087159415…

Repeating decimal:   3.1415926539214210447087159