Commutative algebra and graph theory are two vibrant areas of mathematics that have grown increasingly interrelated. At this interface, algebraic methods are applied to study combinatorial structures, ...

Before reading this guide, you may find it helpful to read the guide on graphs from Module 7 (M7) and the guide on indices from Module 8 (M8). In Module 7 (M7) Graphs, you learned that simultaneous ...

👉 Learn how to graph linear inequalities written in standard form. Linear inequalities are graphed the same way as linear ...

David Conlon and Asaf Ferber have raised the lower bound for multicolor “Ramsey numbers,” which quantify how big graphs can get before patterns inevitably emerge. “There is no absolute randomness in ...

Keep Learning is available to stream on pbs.org and the free PBS App, available on iPhone, Apple TV, Android TV, Android smartphones, Amazon Fire TV, Amazon Fire Tablet, Roku, Samsung Smart TV, and ...

👉 Learn how to graph exponential functions involving horizontal shift. An exponential function is a function that increases ...

Apple improved the Math Notes feature in iOS 26 and iPadOS 26, allowing users to input equations to create graphs in three dimensions. Math Notes was added in iOS 18 and iPadOS 18. It works in the ...