When referring to a planar object, "free" means that the object is regarded as capable of being picked up out of the plane and flipped over. As a result, mirror images are equivalent for free objects.

The word "free" is also used in technical senses to refer to a free group, free semigroup, free tree, free variable, etc.

In algebraic topology, a free abstract mathematical object is generated by elements in a "free manner" ("freely"), i.e., such that the elements satisfy no nontrivial relations among themselves. To make this more formal, an algebraic gadget is freely generated by a subset if, for any function where is any other algebraic gadget, there exists a unique homomorphism (which has different meanings depending on what kind of gadgets you're dealing with) such that restricted to is .

If the algebraic gadgets are vector spaces, then freely generates iff is a basis for . If the algebraic gadgets are Abelian groups, then freely generates iff is a direct sum of the integers, with consisting of the standard basis.