Homotopy Type Theory Sandbox (Rev #30)

Comments about school mathematics

A replacement for the quadratic formula

Very few people use the quadratic formula for real quadratic functions in applications of mathematics. (The quadratic formula also doesn’t hold for most types of real numbers in constructive mathematics.) Instead, they use numerical solvers in a computer or on a calculator, and it would be better for students to understand the numerical analysis algorithms behind the numerical solvers:

Given a real quadratic function $f:\mathbb{R} \to \mathbb{R}$, $f(x) \coloneqq a x^2 + b x + c$ for real numbers $a:\mathbb{R}$, $b:\mathbb{R}$, $c:\mathbb{R}$, where $\vert a \vert \gt 0$, suppose we want to find an approximate zero of $f$. The real quadratic function has discriminant $\Delta = b^2 - 4 a c$ and has 2 zeroes if $\Delta \gt 0$, 1 zero of multiplicity 2 at $2 a x_i + b = 0$ if $\Delta = 0$, or no zeroes of $\Delta \lt 0$. (In constructive mathematics, there are real quadratic functions where it isn’t possible to determine the number of zeroes the real quadratic function has.) Assume that $\Delta \gt 0$. Select a real number $x_0:\mathbb{R}$ as the initial guess, where $2 a x_0 + b \gt 0$ or $2 a x_0 + b \lt 0$, and the next guess would be defined as

The algorithm isn’t valid when $2 a x_0 + b = 0$ because of division by zero. When $2 a x_0 + b \gt 0$, the algorithm converges towards the zero at $x \gt -\frac{b}{2a}$, and when $2 a x_0 + b \lt 0$, the algorithm converges towards the zero $x \lt -\frac{b}{2a}$, where $-\frac{b}{2a}$ is where the extremum (minimum/maximum) of the parabola occurs.

This is known as Newton’s method for real quadratic functions, and could be introduced in any secondary school algebra course. The proof would have to wait until a calculus course, but the proof of the fundamental theorem of algebra requires some form of real or complex analysis and the fundamental theorem of algebra is still taught in secondary school algebra.