## Glossary of Mathematics terms

Many terms used in mathematics have very specific definitions. This is an evolving glossary designed to address common questions specific to the new syllabus.

**Faces, vertices and edges**

Although it is common to define terms separately, the above three terms are used together when describing polyhedra (e.g. cubes, prisms).

Faces are always flat.

Edges are always straight. An edge is formed by the intersection of 2 faces.

Vertices are formed by the intersection of 3 or more faces.

A cube has 6 faces, 12 edges and 8 vertices.

A sphere has a surface but no faces, vertices or edges.

The terms faces, vertices and edges should only be used with polyhedra such as tetrahedrons, rectangular prisms or pyramids. A cone does not have faces, vertices and edges.

**Common problems with the definitions**

People often ask "how many edges are there on a cylinder?" Using the definition for polyhedra, edges are defined to be straight and so a cylinder has no edges. This often appears to run against common sense. If you allowed curved edges, a cylinder would have two.

However, edges, faces and vertices (corners) are only interesting on polyhedra because there is something the same about them on different polyhedra. Namely, the number of faces plus the number of vertices minus the number of edges is equal to two. This doesn't work for the rims (curved edges) of a cylinder.

**Perfect and amicable numbers**

The ancient Greeks were fascinated by whole numbers. They defined as perfect numbers those equal to the sum of their proper divisors, including 1.

For example, 6 is the smallest perfect number since it is the sum of its three proper divisors; 1, 2 and 3. The next perfect number is 28, which is the sum of 1, 2, 4, 7 and 14.

Amicable numbers are pairs of numbers where each number is the sum of the proper divisors of the other. The smallest such pair of amicable numbers is 220 and 284. The number 220 is evenly divisible by 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, which add up to 284. I leave you to check that the proper divisors of 284 add up to 220.

The Pythagorean brotherhood regarded 220 and 284 as numerical symbols of friendship.