Diophantine Approximation - Best Diophantine Approximations of A Real Number

Best Diophantine Approximations of A Real Number

Given a real number α, there are two ways to define a best Diophantine approximation of α. For the first definition, the rational number p/q is a best Diophantine approximation of α if

for every rational number p'/q' such that 0< q' ≤ q.

For the second definition, the above inequality is replaced by

A best approximation for the second definition is also a best approximation for the first one, but the converse is false.

The theory of continued fractions allows us to compute the best approximations of a real number: for the second definition, they are the convergents of its expression as a regular continued fraction. For the first definition, one has to consider also the semiconvergents.

For example, the constant e = 2.718281828459045235... has the (regular) continued fraction representation

Its best approximations for the second definition are

while, for the first definition, they are

3, \tfrac{5}{2}, \tfrac{8}{3}, \tfrac{11}{4}, \tfrac{19}{7}, \tfrac{30}{11}, \tfrac{49}{18}, \tfrac{68}{25}, \tfrac{87}{32}, \tfrac{106}{39}, \ldots\, .

Read more about this topic:  Diophantine Approximation

Famous quotes containing the words real and/or number:

    “The mother must teach her son how to respect and follow the rules. She must teach him how to compete successfully with the other boys. And she must teach him how to find a woman to take care of him and finish the job she began of training him how to live in a family. But no matter how good a job a woman does in teaching a boy how to be a man, he knows that she is not the real thing, and so he tends to exaggerate the differences between men and women that she embodies.”
    —Frank Pittman (20th century)

    “I think, for the rest of my life, I shall refrain from looking up things. It is the most ravenous time-snatcher I know. You pull one book from the shelf, which carries a hint or a reference that sends you posthaste to another book, and that to successive others. It is incredible, the number of books you hopefully open and disappointedly close, only to take down another with the same result.”
    —Carolyn Wells (1862–1942)