What is an Algebra? (Part I)

We’ll answer this question in this series of posts by giving a hands-on introduction to a variety of different Algebras (yes, there are lots!). Loosely speaking, an Algebra is a system that consists of elements and one or more operations. For now, you can think of the elements as numbers and the operations as the basic arithmetic you’re familiar with: addition or multiplication, for example.

So, our first example of an Algebra is the one we generally study in high school: the Real Numbers. And our operations are the basic four we know from elementary school.

Our second example of an Algebra is the set of integers. Here, our Algebra does not include all of our basic operations: division is problematic in that the quotient of two integers is not necessarily another integer. So, the Algebra of the integers includes addition, subtraction, and multiplication; but not division.

We’ll close this post with one more Algebra that is familiar from high school: the Algebra of the Complex Numbers. Addition, subtraction, and multiplication all behave nicely. Division is less clear to the novice eye: Given two complex numbers z=a+bi and y=c+di, does the notion of \frac{z}{y} make sense? Is it still a complex number? If the answers to these questions are yes, the Algebra of the Complex Numbers seems to be more like the Reals. And if not, the Algebra of the Complex Numbers seems to be more like the Integers. We’ll leave these questions about complex division for you to consider.

In the next post, we’ll consider an extremely useful family of Algebras: the Integers modulo n.


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