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# explainers

## Latest Articles

### What Is Machine Learning?

Neural networks and other forms of machine learning ultimately learn by trial and error, one improvement at a time.

### How the Square Root of 2 Became a Number

Useful mathematical concepts, like the number line, can linger for millennia before they are rigorously defined.

### The Enduring Mystery of How Water Freezes

Making ice requires more than subzero temperatures. The unpredictable process takes microscopic scaffolding, random jiggling and often a little bit of bacteria.

### The S-Matrix Is the Oracle Physicists Turn To in Times of Crisis

Particle physicists in search of the next theory of reality are consulting a mathematical structure that they know will never fail: a table of possibilities known as the S-matrix.

### Computer Scientists Invent an Efficient New Way to Count

By making use of randomness, a team has created a simple algorithm for estimating large numbers of distinct objects in a stream of data.

### How Do Machines ‘Grok’ Data?

By apparently overtraining them, researchers have seen neural networks discover novel solutions to problems.

### The Hidden Connection That Changed Number Theory

Quadratic reciprocity lurks around many corners in mathematics. By proving it, number theorists reimagined their whole field.

### The Deep Link Equating Math Proofs and Computer Programs

Mathematical logic and the code of computer programs are, in an exact way, mirror images of each other.

### How Many Microbes Does It Take to Make You Sick?

Exposure to a virus isn’t an all-or-nothing proposition. The concept of “infectious dose” suggests ways to keep ourselves safer from harm.