Math is a language written primarily using equations – little sentences of letters and numbers that entirely govern our understanding of the universe.

They can have a scary reputation, but really an equation is just a statement: it’s a math-y way of saying “These two things are equal to each other.” And each has a story behind it – a realization, a reaction, and consequences that literally change the world.

Here are the stories of three equations that you may not have heard before.

## Pythagoras’s Theorem: *a*^{2 }+* b*^{2} =* c*^{2}

^{2 }

^{2}

Describing the relationship between the lengths of sides in a right-angled triangle, the Pythagorean theorem was probably the first “proper” theorem most of us ever learned. And that’s fitting, actually – because it’s also one of the earliest formal theorems known to have been discovered at all, turning up as Proposition 47 of Book I of Euclid’s *Elements* – aka the oldest surviving and continuously used math textbook in the world.

And when we say it’s an old theorem, we mean *old*. It had already been kicking about for around 1,500 years when Euclid put out his famous tome – which, yes, means it predates the guy it was named after by a good millennium.

“Many thousands of clay tablets have been retrieved from the lost cities of ancient Babylon, in present-day Iraq,” explained Daniel Mansfield, a senior lecturer in mathematics at UNSW Sydney, in a 2021 article for The Conversation.

“One example is the approximately 3,700-year-old cadastral survey Si.427, which […] is in fact one of the oldest examples of applied geometry from the ancient world,” he wrote. Why? Because it contains sets of Pythagorean triples: groups of three numbers that satisfy the Pythagorean theorem and therefore can form the sides of a right-angled triangle.

Then there’s Plimpton 322: another ancient Babylonian tablet containing a list of Pythagorean triples, used for land surveying. But as impressive as this all is, it took the Greeks to turn “wow, neat and useful observation!” into a “formal theorem, with an actual proof.”

That said, its ancient formulation might not look too familiar to most of us. “For the purposes of higher mathematics, the Greeks worked with lines and areas instead of numbers,” explained Ian Stewart, Emeritus Professor of Mathematics at the University of Warwick, UK, in his 2012 book *Seventeen Equations that Changed the World*.

“So Pythagoras and his Greek successors would decode the theorem as an equality of areas: ‘The area of a square constructed using the longest side of a right-angled triangle is the sum of the areas of the squares formed from the other two sides.’”

Pythagoras’s proof of the Pythagorean theorem.

When did this wordy geometry problem become the elegant equation we know and love? Well, this is where the Pythagorean theorem really shines: not only does it symbolize a break with trial-and-error numeracy and the beginning of deductive math, but it also represents a bridge between two major areas of the subject: geometry and algebra.

“Pythagoras’s theorem was […] vital to the invention of coordinate geometry,” Stewart explained, since it allows mathematicians to represent a circle in the form of an algebraic equation. It directly led to the development of trigonometry, and as math branched out past Euclidean space, it also inspired another big equation on the list: the theory of relativity.

“Pythagoras’s equation first came into being around 3500 years ago to measure a farmer’s land,” Stewart wrote. “Its extension to triangles without right angles, and triangles on a sphere, allowed us to map our continents and measure our planet. And a remarkable generalization lets us measure the shape of the universe.”

“Big ideas have small beginnings.”

## Euler’s Identity: *e*^{i π }+ 1 = 0

Often known as the “most beautiful equation in math”, Euler’s identity brings together five of the most important constants in math: the additive constant zero, the multiplicative constant one, the imaginary unit *i*, and the two best-known transcendental numbers pi (π) and *e*. It includes addition, multiplication, and exponentiation. And it can send mathematicians into raptures.

“[It’s] like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep,” Stanford University mathematics professor Keith Devlin wrote in Wabash Magazine in 2002. “Euler’s equation reaches down into the very depths of existence.”

Dramatic? Perhaps. But brain scans have shown that mathematicians really do react to Euler’s identity the same way most of us respond to great works of art or music.

“It is simple to look at and yet incredibly profound,” mathematician David Percy told the BBC in 2014. “At first you don’t realize the implications […] then suddenly it becomes amazing as you realize its full potential.”

“Given that *e*, pi and *i* are incredibly complicated and seemingly unrelated numbers, it is amazing that they are linked by this concise formula,” he said.

So how exactly *are* they linked? And who figured it out? Well, as abstract as it may seem, in fact the equation is just a very specific example of a more general rule, known as Euler’s equation (mathematicians aren’t always the most inventive at naming things). And *that* has a fairly boring application, all things considered: it “describes two equivalent ways to move in a circle,” explained Kalid Azad in an article for his Better Explained blog.

“We can understand it by building on a few analogies,” he wrote. “Starting at the number 1, see multiplication as a transformation that changes the number: 1∙*e ^{i}*

^{π}.”

Now, the base *e* means we’re dealing with exponential growth, but how do we deal with that rogue *i*? Well, Azad explained, while “regular exponential growth continuously increases 1 by some rate for some time period[,] imaginary exponential growth continuously rotates 1 for some time period.”

In this case, π is the time period – and that’s just enough time to rotate halfway around a circle centered at the origin. In other words, you end up at -1.

If all this looks a little familiar, by the way, that’s not a coincidence – this is very similar to what we were talking about with the Pythagorean theorem. But there’s a big conceptual difference: “Sine and cosine describe motion in terms of a grid, plotting out horizontal and vertical coordinates,” Azad notes, while “Euler’s formula uses polar coordinates – what’s your angle and distance?”

Of course, even if none of that makes sense to you, it’s fine to just sit back and appreciate the beauty of a good equation.

“It is a real classic,” Percy said. “You can do no better than that.”

## Einstein’s Special Theorem: *E *= *mc*^{2}

Speaking of classics – few equations roll off the tongue quite as nicely as *E* = *mc*^{2}. It’s nothing short of iconic: neat, compact, and incredibly profound. But it probably doesn’t mean exactly what you think it does – and in fact, like Euler’s identity, it wasn’t originally formulated the way we know it today at all.

“When Einstein first derived the physical idea that it represents, he didn’t write it in the familiar way,” Stewart wrote. “It is perhaps typical of human culture that our most iconic equation is not, and was not, what it seems to be, and neither is the theory that gave birth to it.”

Most of us have some idea of why this equation is important: it describes the equivalence between energy and mass. What you might not realize is that it’s incomplete – and while we often refer to it as “the” theory of relativity, it’s actually one of two equations that share that title.

“‘Relativity’ covers two distinct but related theories: special relativity and general relativity,” explained Stewart. “Special relativity is about space, time, and matter in the absence of gravity; general relativity takes gravity into account as well.”

*E* = *mc*^{2} describes *special* relativity – and it’s the one Einstein figured out first. “[It] is known as special relativity because it applies only to frames of reference that are moving uniformly with respect to each other,” Stewart wrote. “Among its consequences are the Lorenz-FitzGerald contraction, now interpreted as a necessary feature of space-time.”

It’s because of special relativity that we know what happens at velocities close to the speed of light: that time runs slower, masses increase, and lengths contract. It set lightspeed as a universal upper limit – a theoretical point at which time stops running, mass becomes infinite, and size becomes zero – and, on the more scary and apocalyptic side, it allowed us to figure out the atom bomb.

So it’s probably not totally surprising that the theorem proved controversial when it started gaining publicity – and not only from cranks such as Philipp Lenard and Johannes Stark, whose main gripe with the discovery was that Einstein was Jewish.

It “profoundly shocked scientists and the general public,” wrote historian of science and author Jimena Canales in a 2016 article for Nautilus, with its proponents arguing that some of the most basic, common sense concepts – things like “what does ‘simultaneous’ mean?” or “time and space are different things” – needed to be overhauled.

And yet despite all this, Einstein was still unhappy with his equation. “It provided a unified theory of space, time, matter, and electromagnetism,” Stewart explained, “but it missed out one vital ingredient. Gravity.”

It would take him another decade to figure out the missing piece of the puzzle, and the final result – a set of ten tensor equations linking such mind-melting quantities as the shape of spacetime, curvature, and the relationship between energy and momentum – is definitely not as snappy as our old friend *E* = *mc*^{2}. But its consequences are massive (no pun intended): it rewrote the rules of gravity; explained phenomena that had previously seemed inexplicable; it even predicts the existence of black holes.

And perhaps the most incredible aspect of all? “It all comes from… Pythagoras’s Theorem,” Stewart wrote.

“One of the oldest equations in science leads to one of the newest.”