Irregular Webcomic! #470

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< No. 470 >

Comic #470

For the non-mathematics geeks amongst you, an abelian group is a particular type of algebraic structure - being a set of objects with a binary operation defined on it such that:

the result of the operation on any two elements is a member of the set,
the operation is associative,
there exists an identity element,
for each element of the set there exists an inverse element (these conditions being sufficient to describe a group), and
the operation is commutative.

This is essentially a fancy way of saying "the integers, and addition" considered as a whole thing. There are other sorts of groups, some of which are not abelian, and some of which are, but they're not important for this comic. Maybe another time.

Anyway, if you take an abelian group and define a second binary operation on the elements of the set such that:

the result of the new operation on any two elements is a member of the set,
the new operation is associative,
the new operation is distributive over the original operation, and
there exists an identity element for the new operation,

then you produce what is known in mathematics terminology as a ring. Note that the second operation in a ring does not necessarily have to have inverse elements or be commutative.

This is essentially a fancy way of saying "the integers, and addition, and multiplication". Again, there are other examples, but they're not important right now.

So, if you know how to add and multiply integers, next time a mathematician asks you if you know anything about group theory and ring theory, you can say, "Why, yes!"

And for the non-Tolkien geeks amongst you, the One Ring is the fabled Ring of Power forged by the Dark Lord Sauron in the Second Age of Middle-earth, with the power to...

Oh sod it. Like anyone reading this far hasn't read the books or at least seen the movies by now.

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