Approximations of e

This is a list of all fractional approximations of the mathematical constant e, the base of the natural logarithm. The first 50 decimal digits of e are 2.71828182845904523536028747135266249775724709369995.

This list is not exhaustive by any means; ultimately, there are infinite numbers, so this list can always be expanded.

Organization
This list will be organized the same way as the list of approximations of π.

2-5 digits

 * 19/7 = 2.714... (convergent of e's continued fraction; first approximation by denominator to be accurate to 2 decimal digits)
 * 87/32 = 2.71875 (convergent of e's continued fraction; first approximation by denominator to be accurate to 3 decimal digits)
 * 106/39 = 2.7179... (convergent of e's continued fraction; 2 decimal digits)
 * 193/71 = 2.7183... (convergent of e's continued fraction; 3 decimal digits)
 * 269/99 = 2.717... (2 decimal digits)
 * 299/110 = 2.71818... (3 decimal digits)
 * 878/323 = 2.71827... (first approximation by denominator to be accurate to 4 decimal digits)
 * 1,264/465 = 2.7182796... (convergent of e's continued fraction; 4 decimal digits)
 * 1,457/536 = 2.718284... (convergent of e's continued fraction; first approximation by denominator to be accurate to 5 decimal digits)

6-10 digits

 * 2,721/1,001 = 2.7182817... (convergent of e's continued fraction; first approximation by denominator to be accurate to 6 decimal digits)
 * 23,225/8,544 = 2.71828184... (convergent of e's continued fraction; first approximation by denominator to be accurate to 7 decimal digits)
 * 25,946/9,545 = 2.718281823... (convergent of e's continued fraction; first approximation by denominator to be accurate to 8 decimal digits)
 * 49,171/18,089 = 2.7182818287... (convergent of e's continued fraction; first approximation by denominator to be accurate to 9 decimal digits)
 * 96,499/35,500 = 2.71828169... (6 decimal digits)
 * 138,472/50,941 = 2.7182819... (6 decimal digits)
 * 271,801/99,990 = 2.7182818282... (9 decimal digits)
 * 517,656/190,435 = 2.718281828445... (convergent of e's continued fraction; first approximation by denominator to be accurate to 10 decimal digits)
 * 566,827/208,524 = 2.71828182847... (convergent of e's continued fraction; 10 decimal digits)
 * 1,599,418/588,393 = 2.7182818286... (9 decimal digits)

11-15 digits

 * 1,084,483/398,959 = 2.7182818284586... (convergent of e's continued fraction; first approximation by denominator to be accurate to 11 decimal digits)
 * 13,580,623/4,996,032 = 2.71828182845907... (convergent of e's continued fraction; first approximation by denominator to be accurate to 13 decimal digits)
 * 14,665,106/5,394,991 = 2.71828182845903... (convergent of e's continued fraction; 13 decimal digits)
 * 28,245,729/10,391,023 = 2.7182818284590459... (convergent of e's continued fraction; first approximation by denominator to be accurate to 15 decimal digits)

16-20 digits

 * 410,105,312/150,869,313 = 2.71828182845904521... (convergent of e's continued fraction; first approximation by denominator to be accurate to 16 decimal digits)
 * 438,351,041/161,260,336 = 2.71828182845904525... (convergent of e's continued fraction; 16 decimal digits)
 * 848,456,353/312,129,649 = 2.7182818284590452348... (convergent of e's continued fraction; first approximation by denominator to be accurate to 17 decimal digits)