Editing Pell number (section)

== Approximation to the square root of two ==

[[Image:Pell octagons.svg|thumb|300px|Rational approximations to [[regular polygon|regular]] [[octagon]]s, with coordinates derived from the Pell numbers.]]
Pell numbers arise historically and most notably in the [[diophantine approximation|rational approximation]] to {{sqrt|2}}. If two large integers ''x'' and ''y'' form a solution to the [[Pell equation]]
:<math>x^2-2y^2=\pm 1,</math>
then their ratio ''{{sfrac|x|y}}'' provides a close approximation to {{sqrt|2}}. The sequence of approximations of this form is
:<math>\frac11, \frac32, \frac75, \frac{17}{12}, \frac{41}{29}, \frac{99}{70}, \dots</math>
where the denominator of each [[fraction]] is a Pell number and the numerator is the sum of a Pell number and its predecessor in the sequence. That is, the solutions have the form
:<math>\frac{P_{n-1}+P_n}{P_n}.</math>
The approximation
:<math>\sqrt 2\approx\frac{577}{408}</math>
of this type was known to Indian mathematicians in the third or fourth century BCE.<ref>As recorded in the [[Shulba Sutras]]; see e.g. Dutka (1986), who cites Thibaut (1875) for this information.</ref> The Greek mathematicians of the fifth century BCE also knew of this sequence of approximations:<ref>See Knorr (1976) for the fifth century date, which matches [[Proclus]]' claim that the side and diameter numbers were discovered by the [[Pythagoreans]]. For more detailed exploration of later Greek knowledge of these numbers see Thompson (1929), Vedova (1951), Ridenhour (1986), Knorr (1998), and Filep (1999).</ref> Plato refers to the numerators as '''rational diameters'''.<ref>For instance, as several of the references from the previous note observe, in [[Plato's Republic]] there is a reference to the "rational diameter of 5", by which [[Plato]] means 7, the numerator of the approximation {{sfrac|7|5}} of which 5 is the denominator.</ref> In the second century CE [[Theon of Smyrna]] used the term the  '''side and diameter numbers''' to describe the denominators and numerators of this sequence.<ref>{{citation|title=History of Greek Mathematics: From Thales to Euclid|first=Sir Thomas Little|last=Heath|author-link=Thomas Little Heath|publisher=Courier Dover Publications|year=1921|isbn=9780486240732|page=112|url=https://books.google.com/books?id=drnY3Vjix3kC&pg=PA112}}.</ref>

These approximations can be derived from the [[simple continued fraction|continued fraction expansion]] of <math>\sqrt 2</math>:
:<math>\sqrt 2 = 1 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \ddots\,}}}}}.</math>
Truncating this expansion to any number of terms produces one of the Pell-number-based approximations in this sequence; for instance,
:<math>\frac{577}{408} = 1 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2}}}}}}}.</math>

As [[Donald Knuth|Knuth]] (1994) describes, the fact that Pell numbers approximate {{sqrt|2}} allows them to be used for accurate rational approximations to a regular octagon with vertex coordinates {{nowrap|(±''P<sub>i</sub>'', ±''P''<sub>''i''+1</sub>)}} and {{nowrap|(±''P''<sub>''i''+1</sub>, ±''P<sub>i</sub>'')}}. All vertices are equally distant from the [[origin (mathematics)|origin]], and form nearly uniform [[angle]]s around the origin. Alternatively, the points  <math>(\pm(P_i+P_{i-1}),0)</math>, <math>(0,\pm(P_i+P_{i-1}))</math>, and <math>(\pm P_i,\pm P_i)</math> form approximate octagons in which the vertices are nearly equally distant from the origin and form uniform angles.