Thursday, June 10, 2021

The Most Irrational Number

This is a fascinating discussion about the Golden Ratio by Jordan Ellenberg appearing in the online magazine Slate. 

Something totally new to me was the "barcode" representation of a number and how it can be used to determine some sense of the irrationality of an irrational number.


Wednesday, June 9, 2021

A Simple Way to Solve Any Quadratic Equation

I stumbled across this last December but had no time to investigate it until now. The technique is from Po-Shen Loh of the Department of Mathematical Sciences, Carnegie Mellon University. Relevant articles:

·        Math Genius Has Come Up with a Wildly Simple New Way to Solve Quadratic Equations (a bit of hyperbole, but what can you expect from the website

·        A Different Way to Solve Quadratic Equations

·        A Simple Proof of the Quadratic Formula

·        Quadratic Method: Detailed Explanation

·        Quadratic Method: Related Work

In Elementary and Intermediate Algebra classes we spend a huge amount of time on the factoring of polynomials, particularly second-degree polynomials with integer coefficients, as a way of solving certain quadratic equations; i.e., those with rational number solutions. Of course, in practical applications, the coefficients are not integers and the solutions are almost never rational numbers. So, we then talk about radicals and complex numbers, completing the square and eventually work our way up to the quadratic formula as the ultimate way to solve any quadratic equation. What Po-Shen has described is a single technique that can be taught at the beginning when the solutions are rational numbers, but will still work on quadratic equations with real number and even complex number solutions. In the above articles, he shows (including several videos) how to justify and teach the technique right after the student has learned how to multiply two binomials and has been introduced to the idea of a quadratic equation.

The basic idea in practice for solving any quadratic equation of the form Ax2 + Bx + C = 0 is:

1.     Take the quadratic equation and put it in the form x2 + Bx + C = 0.

2.      Let r = (-B/2 - u) and  s = (-B/2 + u) where u is any complex number. [note that r + s = – B]

3.      Now note that r x s = B^2/4 - u^2. Set B^2/4 - u^2 = C and solve for u. r and s are the solutions to the original quadratic equation.

Note that in addition to solving any quadratic equation, the technique also obviously leads to the factorization of the second-degree polynomial: P(x) = (xr)(xs).