Short answers, pulled from the story.
Robert Recorde published the book The Whetstone of Witte in the year 1557. This publication introduced the symbol that eventually became the most recognized mark in all of mathematics.
A Diophantine equation is a polynomial equation in two or more unknowns for which the unknowns must take integer values. These equations were first studied by the Hellenistic mathematician Diophantus of Alexandria in the third century.
René Descartes invented Cartesian coordinates in the 17th century. This invention provided the first systematic link between Euclidean geometry and algebra, creating a new branch of mathematics known as analytic geometry.
The Abel-Ruffini theorem demonstrates that equations of degree five or more could not always be solved using a finite number of operations involving just the coefficients. This discovery showed that while solutions exist, they cannot always be expressed using elementary algebraic operations.
A differential equation is a mathematical equation that relates some function with its derivatives, where the functions usually represent physical quantities and the derivatives represent their rates of change. These equations are used to model processes that involve the rates of change of the variable in areas such as physics, chemistry, biology, and economics.