Approximations of square roots/10

This is a list of all fractional approximations of the square root of 10 (√10). The first 50 decimal digits of √10 are 3.16227766016837933199889354443271853371955513932521.

2-5 digits

 * 19/6 = 3.167... (convergent of √10's continued fraction; first approximation by denominator to be accurate to 2 decimal digits)
 * 79/25 = 3.16 (2 decimal digits)
 * 117/37 = 3.16216... (convergent of √10's continued fraction; first approximation by denominator to be accurate to 3 decimal digits)
 * 370/117 = 3.1624... (3 decimal digits)
 * 721/228 = 3.16228... (convergent of √10's continued fraction; first approximation by denominator to be accurate to 4 decimal digits)
 * 990/313 = 3.1629... (3 decimal digits)
 * 1,249/395 = 3.16203... (3 decimal digits)
 * 2,280/721 = 3.162275... (5 decimal digits)
 * 3,950/1,249 = 3.1625... (3 decimal digits)

6-10 digits

 * 4,443/1,405 = 3.16227758... (convergent of √10's continued fraction; first approximation by denominator to be accurate to 6 decimal digits)
 * 14,050/4,443 = 3.1622777... (6 decimal digits)
 * 27,379/8,658 = 3.162277662... (convergent of √10's continued fraction; first approximation by denominator to be accurate to 8 decimal digits)
 * 168,717/53,353 = 3.16227766011... (convergent of √10's continued fraction; first approximation by denominator to be accurate to 10 decimal digits)

11-15 digits

 * 1,039,681/328,776 = 3.1622776601698... (convergent of √10's continued fraction; first approximation by denominator to be accurate to 11 decimal digits)
 * 6,406,803/2,026,009 = 3.16227766016834... (convergent of √10's continued fraction; first approximation by denominator to be accurate to 13 decimal digits)
 * 39,480,499/12,484,830 = 3.16227766016838... (convergent of √10's continued fraction; first approximation by denominator to be accurate to 14 decimal digits)

16-20 digits

 * 243,289,797/76,934,989 = 3.16227766016837931... (convergent of √10's continued fraction; first approximation by denominator to be accurate to 16 decimal digits)
 * 1,499,219,281/474,094,764 = 3.162277660168379333... (convergent of √10's continued fraction; first approximation by denominator to be accurate to 17 decimal digits)
 * 9,238,605,483/2,921,503,573 = 3.16227766016837933198... (convergent of √10's continued fraction; first approximation by denominator to be accurate to 19 decimal digits)