Algebra

Mathematicians have long grappled with the reality that some problems just don’t have solutions.

Representation theory was initially dismissed. Today, it’s central to much of mathematics.

Explore our surprisingly simple, absurdly ambitious and necessarily incomplete guide to the boundless mathematical universe.

New work on the problem of “scissors congruence” explains when it’s possible to slice up one shape and reassemble it as another.

Odd enough to potentially model the strangeness of the physical world, complex numbers with “imaginary” components are rooted in the familiar.

In math, sometimes the most common things are the hardest to find.

The 19th-century discovery of numbers called “quaternions” gave mathematicians a way to describe rotations in space, forever changing physics and math.

An upstart field that simplifies complex shapes is letting mathematicians understand how those shapes depend on the space in which you visualize them.

To begin to understand what mathematicians and physicists see in the abstract structures of symmetries, let’s start with a familiar shape.