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:

    “I think it’s the real world. The people we’re writing about in professional sports, they’re suffering and living and dying and loving and trying to make their way through life just as the brick layers and politicians are.”
    —Walter Wellesley (Red)

    “The genius of democracies is seen not only in the great number of new words introduced but even more in the new ideas they express.”
    —Alexis de Tocqueville (1805–1859)