Measure of The Accuracy of Approximations
The obvious measure of the accuracy of a Diophantine approximation of a real number α by a rational number p/q is However, this quantity may always be made arbitrarily small by increasing the absolute values of p and q; thus the accuracy of the approximation is usually estimated by comparing this quantity to some function φ of the denominator q, typically a negative power of it.
For such a comparison, one may want upper bounds or lower bounds of the accuracy. A lower bound is typically described by a theorem like "for every element α of some subset of the real numbers and every rational number p/q, we have ". In some case, "every rational number" may be replaced by "all rational numbers except a finite number of them", which amounts to multiplying φ by some constant depending on α.
For upper bounds, one has to take into accounts that not all the "best" Diophantine approximations provided by the convergents may have the desired accuracy. Therefore the theorems take the form "for every element α of some subset of the real numbers, there are infinitely many rational numbers p/q such that ".
Read more about this topic: Diophantine Approximation
Famous quotes containing the words measure of the, measure of, measure and/or accuracy:
“He gave man speech, and speech created thought,
Which is the measure of the universe;”
—Percy Bysshe Shelley (1792–1822)
“One might imagine that a movement which is so preoccupied with the fulfillment of human potential would have a measure of respect for those who nourish its source. But politics make strange bedfellows, and liberated women have elected to become part of a long tradition of hostility to mothers.”
—Elaine Heffner (20th century)
“Letters are above all useful as a means of expressing the ideal self; and no other method of communication is quite so good for this purpose.... In letters we can reform without practice, beg without humiliation, snip and shape embarrassing experiences to the measure of our own desires....”
—Elizabeth Hardwick (b. 1916)
“Such is the never-failing beauty and accuracy of language, the most perfect art in the world; the chisel of a thousand years retouches it.”
—Henry David Thoreau (1817–1862)