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Farey sequence
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==== Equivalent-area interpretation ==== Every consecutive pair of Farey rationals have an equivalent area of 1.<ref name=Austin2008>{{cite web |last1=Austin |first1=David |date=December 2008 |title=Trees, Teeth, and Time: The mathematics of clock making |website=[[American Mathematical Society]] |location=Rhode Island |language=en |url=http://www.ams.org/publicoutreach/feature-column/fcarc-stern-brocot |access-date=28 September 2020 |url-status=live |archive-url=https://web.archive.org/web/20200204014725/http://www.ams.org/publicoutreach/feature-column/fcarc-stern-brocot |archive-date=4 February 2020}}</ref> See this by interpreting consecutive rationals <math display=block>r_1 = \frac{p}{q} \qquad r_2 = \frac{p'}{q'}</math> as vectors {{math|(''p'', ''q'')}} in the xy-plane. The area is given by <math display=block>A \left(\frac{p}{q}, \frac{p'}{q'} \right) = qp' - q'p.</math> As any added fraction in between two previous consecutive Farey sequence fractions is calculated as the [[Mediant (mathematics)|mediant]] (β), then <math display=block>\begin{align} A(r_1, r_1 \oplus r_2) &= A(r_1, r_1) + A(r_1, r_2) \\ &= A(r_1, r_2) \\ &= 1 \end{align}</math> (since {{math|1=''r''<sub>1</sub> = {{sfrac|1|0}}}} and {{math|1=''r''<sub>2</sub> = {{sfrac|0|1}}}}, its area must be 1).
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