Descartes’ PB&J (or, the Clever Idea at the Heart of High School Math)

In cooking, the biggest breakthroughs come from surprising but natural combinations. Shrimp… and walnuts. Chicken… with peanut sauce. Cookie dough… in ice cream. Mouthwatering as the separate ingredients may be, the real magic happens when you throw them together. If you’re striving for complexity, excitement, and overriding deliciousness, then the unexpected harmony of disparate flavors is your holy grail.

And what’s true for the kitchen goes double for the math classroom. The best math explores deep connections between seemingly unrelated objects.

Take the coordinate plane. Since prehistory, humans have explored geometry—the relationships among shapes. And for over a millennium, mathematicians have investigated algebra—abstract patterns among numbers. But it took the philosopher Renee Descartes to unite the two, to realize that shapes and equations share the same DNA.

Descartes’ breakthrough was to take a pair of numbers—say, x and y—and represent it as a point on a piece of paper. Every pair of numbers yields a point, and every point signifies a pair. Though not entirely new—Descartes built on similar ideas from other mathematicians—the Cartesian plane was a striking step forward. What made it so radically inventive, so uniquely powerful?

It allowed algebraic questions to be reinterpreted as geometric ones—and vice versa.

Take this problem: If x + y = 10, and 2x + 3y = 24, then what could x and y be? We can solve this problem using the standard algebraic techniques; or we can turn it into a geometry problem:

Or take this challenge: Prove that the diagonals of a rectangle are congruent. We could respond in the Euclidean geometric style, using congruent triangles; or we could convert it into an algebra problem:

Descartes had achieved a peanut-butter-and-jelly-caliber breakthrough. The two seemingly disparate fields of geometry and algebra were no mere strangers. They were identical twins—dressed in different clothes, but spawned from the same logical genome.

The hallmark of powerful mathematics is its ability to translate smoothly between different types of problems. Performing those acts of translation requires a theoretical breakthrough like Descartes’, a realization that two seemingly separate universes are actually one.

Virtually our entire high school math curriculum now consists of playing out the consequences of Descartes’ innovation. Flip through any textbook, and you’ll see Descartes’ sticky, peanut-butter-and-jelly-coated fingerprints on almost every page.

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14 thoughts on “Descartes’ PB&J (or, the Clever Idea at the Heart of High School Math)

  1. This is why I love math! Surprising and deep connections between seemingly disparate facets of mathematics, such as this connection between equations and shapes, give us a vague glimpse of the true nature of mathematics.

    I didn’t enjoy mathematics until I realized that there was something profound, indescribable, and seemingly unreachable, hidden deep beneath the ideas of numbers and equations, which somehow gives birth to those numbers and equations. Math began to excite me when I realized, in the words of Leonhard Adleman, that mathematics “is less related to accounting than it is to philosophy.”

    Please write more about surprising and deep connections like this!

    • Thanks for reading! That’s a nice quote about accounting vs. philosophy. Accountants may use some math, but I’d say mathematicians use lots of philosophy.

  2. Always funny and interesting. I don’t know how you do to have always more interesting things to say every week. It’s truly impressive.

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