Follow your curiosity

What discovery has been shared with you?

FACTREE gives you something interesting to discover every day — and something worth talking about together.

Start with one fact. Explore it, go deeper, then follow whichever branch catches your imagination.

Choose subjects for a surprise

Exploring any topic

Begin your discovery

Your next discovery is one click away.

Choose one or more subjects above, or leave Any Topic selected and let curiosity decide.

Mathematics

The Cauchy-Schwarz Inequality and Its Many Faces

Quick fact

Cauchy-Schwarz is the mathematical reason why the correlation coefficient always lies between -1 and 1, a fact that every statistician relies on.

Why this is interesting

You might have noticed that the dot product of two vectors is never larger than the product of their lengths—no matter how long they are. But why is that true, and what does it have to do with statistics, geometry, and signals?