Editing Diophantine approximation (section)

=== Equivalent real numbers ===

'''Definition''': Two real numbers <math>x,y</math> are called ''equivalent''<ref>{{harvnb|Hurwitz|1891|p=284}}</ref><ref>{{harvnb|Hardy|Wright|1979|loc=Chapter 10.11}}</ref> if there are integers <math>a,b,c,d\;</math> with <math>ad-bc = \pm 1\;</math> such that:
:<math>y = \frac{ax+b}{cx+d}\, .</math>

So equivalence is defined by an integer [[Möbius transformation]] on the real numbers, or by a member of the [[Modular group]] <math>\text{SL}_2^{\pm}(\Z)</math>, the set of invertible 2 &times; 2 matrices over the integers. Each rational number is equivalent to 0; thus the rational numbers are an [[equivalence class]] for this relation.

The equivalence may be read on the regular continued fraction representation, as shown by the following theorem of [[Joseph Alfred Serret|Serret]]:

'''Theorem''': Two irrational numbers ''x'' and ''y'' are equivalent if and only if there exist two positive integers ''h'' and ''k'' such that the regular [[Simple continued fraction|continued fraction]] representations of ''x'' and ''y''
:<math>\begin{align}
  x &= [u_0; u_1, u_2, \ldots]\, , \\
  y &= [v_0; v_1, v_2, \ldots]\, ,
\end{align}</math>

satisfy
:<math>u_{h+i} = v_{k+i}</math>
for every non negative integer ''i''.<ref>See {{harvnb|Perron|1929|loc=Chapter 2, Theorem 23, p. 63}}</ref>

Thus, except for a finite initial sequence, equivalent numbers have the same continued fraction representation.

Equivalent numbers are approximable to the same degree, in the sense that they have the same [[Markov constant]].