English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Omar Khayyam's studies on cubic equations inspired the 12th century Persianmathematician Sharaf al-Din Tusi to investigate the number of positive roots.According to the noted mathematical historian Rashed, Tusi analyzed the problemfor five different types of equations. In fact all cubic equations arereducible to a form {\it Tusi form} $x^2-x^3=c$. Tusi determined that themaximum of $x^2-x^3$ on $(0,1)$ occurs at $\frac{2}{3}$ and concluded when$c=\frac{4}{27} \delta$, $\delta \in (0,1)$, there are roots in $(0,\frac{2}{3})$ and $(\frac{2}{3},1)$, ignoring the root in $(-\frac{1}{3},0)$.Given a {\it reduced form} $x^3+px+q=0$, when $p <0$, we show it is reducibleto a Tusi form with $\delta = \frac{1}{2} + {3\sqrt{3} q}/{4\sqrt{-p^3}}$. Itfollows there are three real roots if and only if $\Delta=-(\frac{q^2}{4}+\frac{p^3}{27})$ is positive. This gives an explicitconnection between $\delta$ in Tusi form and $\Delta$ in Cardano's formula.Thus when $\delta \in (0,1)$, rather than using Cardano's formula in complexnumbers one can approximate the roots iteratively. On the other hand, for areduced form with $p >0$ we give a novel proof of Cardono's formula. WhileRashed attributes Tusi's computation of the maximum to the use of derivatives,according to Hogendijk, Tusi was probably influenced by Euclid. Here we showthe maximizer in Tusi form is computable via elementary algebraicmanipulations. Indeed for a {\it quadratic Tusi form}, $x-x^2=\delta/4$, Tusi'sapproach results in a simple derivation of the quadratic formula, comparablewith the pedagogical approach of Po-Shen Loh. Moreover, we derive analogousresults for the {\it general Tusi form}. Finally, we present a novel derivationof Khayyam's geometric solution. The results complement previous findings onTusi's work and reveal further facts on history, mathematics and pedagogy insolving cubic equations.

On Tusi's Classification of Cubic Equations and its Connections to Cardano's Formula and Khayyam's Geometric Solution