New Proof Illuminates the Hidden Structure of Common Equations

In a recent paper, Manjul Bhargava of Princeton University has settled an 85-year-old conjecture about one of math’s most ancient obsessions: the solutions to polynomial equations such as x² – 3x + 2 = 0. “It’s a great problem, famous old question,” said Andrew Granville, a professor at the University of Montreal. “[Bhargava] had an interesting, somewhat different approach, which was very creative.”

To understand polynomials, mathematicians study their roots, the values of x that make the polynomial equal zero. If you plug the number 1 or 2 into x² – 3x + 2, you’ll get zero, making 1 and 2 the roots of that polynomial.

The equation x² – 5 = 0 is a bit trickier. The polynomial can’t be solved using a rational number — a fraction made by dividing two integers. So mathematicians define a new number that solves the equation and call it $\sqrt{5}$. But all we know about $\sqrt{5}$ is that its square is 5. Once you have $\sqrt{5}$, you can easily multiply it by –1 to get a second root: –$\sqrt{5}$.

These two equations differ in another critical way. The roots of x² – 5 = 0 help solve lots of other equations in our mathematical system, like x² – 20 = 0. (Note that here, our mathematical system is limited to polynomials and rational numbers.) But if we start using them this way, we’ll find that $\sqrt{5}$ and –$\sqrt{5}$ are completely interchangeable. Both $2\sqrt{5}$  and –$2\sqrt{5}$ work equally well as solutions of x² – 20 = 0 — and, more generally, in any context. Anywhere $\sqrt{5}$ is helpful, so is –$\sqrt{5}$.