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, measure and/or accuracy:
“What is life but the angle of vision? A man is measured by the angle at which he looks at objects. What is life but what a man is thinking all day? This is his fate and his employer. Knowing is the measure of the man. By how much we know, so much we are.”
—Ralph Waldo Emerson (1803–1882)
“I don’t measure America by its achievement, but by its potential.”
—Shirley Chisholm (b. 1924)
“In everything from athletic ability to popularity to looks, brains, and clothes, children rank themselves against others. At this age [7 and 8], children can tell you with amazing accuracy who has the coolest clothes, who tells the biggest lies, who is the best reader, who runs the fastest, and who is the most popular boy in the third grade.”
—Stanley I. Greenspan (20th century)