Unsymmetrical Approximation of Irrational Numbers
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چکیده
1 A 1 < £ < , 5B B SB so that we have the classical theorem of Hurwitz. For other values of r, approximations from both sides are permitted, but the errors allowed on the two sides are different; hence the term unsymmetrical approximation. The result here was new, and is so related to Hurwitz's inequality that one side is strengthened and the other weakened. Notice that the result for r > l is weaker than the result for r < l . For suppose tha t r > 1, and apply the theorem with r replaced by 1/r to the irrational number — £. In this way, the permissible errors on the right and left are interchanged, and we see that £ has infinitely many approximations A/B satisfying
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