For most of human history, there was no zero. The ancient Egyptians counted without it. The Greeks, who invented geometry and proved the irrationality of the square root of two, had no symbol for nothing. The Romans built an empire with a numeral system (I, V, X, L, C, D,…

The smartphone in your pocket, the laptop on your desk, the servers that run the internet, the supercomputers that simulate climate models, and the chips inside your car’s engine all share a common ancestor. They are all, at their core, implementations of an architecture described in a 101-page document written…

Probability, as it was understood in the 18th century, worked in one direction. If you know the composition of a deck of cards, you can calculate the probability of drawing an ace. If you know the bias of a coin, you can calculate the probability of getting heads ten times…

On April 10, 2019, the Event Horizon Telescope (EHT) collaboration released the first image of a black hole: a glowing orange ring surrounding a dark central shadow. The object was the supermassive black hole at the center of the galaxy Messier 87 (M87), located 55 million light-years from Earth. It…

By the late 19th century, mathematics had grown so vast that no single person could master all of it. The era of the universal mathematician, the scholar who could make original contributions to every branch of the discipline, was ending. The last person widely acknowledged to have achieved this was…

The history of physics can be told through a handful of books. Not textbooks or popular summaries, but the original publications in which physicists first presented the ideas that changed everything. These are the works that introduced new frameworks for understanding the universe, frameworks so powerful that they replaced everything…

Around 300 BCE, the Greek mathematician Euclid compiled the Elements, a systematic presentation of geometry built on a small number of definitions, common notions, and five postulates (axioms). The first four postulates are simple and self-evident: you can draw a straight line between any two points; you can extend a…

Every student of mathematics encounters, at some point, the statement that the square root of negative one is called i, and that numbers involving i are called “complex” or “imaginary.” The terminology is unfortunate. It suggests that these numbers are artificial, mysterious, or somehow less real than ordinary numbers. Mathematicians…

On July 20, 1969, the Apollo 11 Lunar Module Eagle was descending toward the Sea of Tranquility. Neil Armstrong and Buzz Aldrin were three minutes from becoming the first humans to land on the Moon. Mission control in Houston was monitoring every parameter. Everything was nominal. Then the alarms started.…

In the winter of 1961, the meteorologist Edward Lorenz at MIT was running a simple computer simulation of weather patterns. The model used twelve equations to simulate atmospheric convection, and Lorenz had been running it to study long-term weather behavior. One day, wanting to re-examine a particular sequence, he restarted…