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:

    “What difference is there, do you think, between those in Plato’s cave who can only marvel at the shadows and images of various objects, provided they are content and don’t know what they miss, and the philosopher who has emerged from the cave and sees the real things?”
    —Desiderius Erasmus (c. 1466–1536)

    “While I do not suggest that humanity will ever be able to dispense with its martyrs, I cannot avoid the suspicion that with a little more thought and a little less belief their number may be substantially reduced.”
    —J.B.S. (John Burdon Sanderson)