Short answers, pulled from the story.
An equation is a mathematical formula that expresses the equality of two expressions by connecting them with an equals sign. The expression on each side is called the left-hand side and the right-hand side. Solving an equation means determining which values of the variables make the equality true.
The equals sign was invented in 1557 by Robert Recorde. He chose two parallel straight lines of the same length because he considered that nothing could be more equal. The symbol appears in every equation.
An identity is an equation that is true for all values of its variables, while a conditional equation is true only for particular values. The difference of two squares is an example of an algebraic identity, and trigonometry contains many identities involving sine and cosine.
Polynomial equations of degree five or higher cannot always be solved algebraically, a limit proven by the Abel-Ruffini theorem. Equations of degree one, two, three, or four can be solved by a finite number of operations on their coefficients. Higher-degree equations often require computing approximations of their real or complex solutions instead.
A Diophantine equation is a polynomial equation in two or more unknowns for which only integer solutions are sought. It is named after Diophantus of Alexandria, a Hellenistic mathematician of the 3rd century who studied such equations. He was one of the first mathematicians to introduce symbolism into algebra.
René Descartes invented Cartesian coordinates in the 17th century, providing the first systematic link between Euclidean geometry and algebra. His Cartesian coordinate system transforms a geometric problem into an analysis problem, which is why the field is called analytic geometry. This enriched and modified the geometry conceived by the ancient Greek mathematicians.
An ordinary differential equation, or ODE, contains a function of one independent variable and its derivatives. A partial differential equation, or PDE, involves unknown multivariable functions and their partial derivatives. PDEs describe phenomena such as sound, heat, electrostatics, fluid flow, elasticity, and quantum mechanics.