What is an Algebra? (Part III)

In our next few Algebras, we broaden our sense of elements. Until now, we’ve looked at different types of numbers as our elements.

Now, consider matrices as elements. Given any two 2×2 matrices, A and B, we can certainly computer A+B and A\cdot B. And crucially, this sum and this product are still 2×2 matrices. So, we see that the set of 2×2 matrices forms an Algebra. It shouldn’t be too surprising that the set of 3×3 matrices also forms an Algebra, and more generally the set of nxn matrices do as well.

We could also use functions (say, ones that turn a real number into another real number) as our elements. In that case, we have addition and multiplication of functions as operations. And we have another reasonable operation to look into–composition.

Do we have division in the world of matrices or the world of functions? If you multiply a matrix A by a matrix B, is there another matrix to multiply by that will get you back to A? What about something that undoes composition by f or multiplication by f in the same way that subtracting f undoes adding f?

 

 

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What is an Algebra? (Part II)

(This is Part II of a multi-part series. See Part I)

Consider the following statements about integer arithmetic.

even + even = even
even + odd = odd
odd + even = odd
odd + odd = even

even * even = even
even * odd = even
odd * even = even
odd * odd = odd

Speaking loosely, we can say that we have an Algebra consisting of the elements “even” and “odd.” We call this Algebra the Integers modulo 2. We write 0 as shorthand for even and 1 as shorthand for odd, and use the following notation to distinguish this modular arithmetic from our standard arithmetic.

0 + 0 = 0 (mod 2)
0 + 1 = 1 (mod 2)
1 + 0 = 1 (mod 2)
1 + 1 = 0 (mod 2)

0 * 0 =  0 (mod 2)
0 * 1 = 0 (mod 2)
1 * 0 = 0 (mod 2)
1 * 1 = 1 (mod 2)

A note on notation: think of the (mod 2) as describing the statement as a whole, not the element on the right-hand-side of the equals sign. What we are really saying is that in the Algebra of the Integers modulo 2, we have 1 + 1 = 0.

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The following statements are a bit more of a mouthful, but are analogous to the example above.

[A Multiple of 3] + [A Multiple of 3] = [A Multiple of 3]
[A Multiple of 3] + [A number that is one more than a multiple of 3] = [A number that is one more than a multiple of 3]

Some simple notation is necessary. We use (mod 3rather than (mod 2) here to indicate that we are thinking about multiples of 3 rather than multiples of 2. We use to mean [A Multiple of 3], to mean [A number one more than a multiple of 3], and to mean [A number that is two more than a multiple of 3].

Then, we can write the statements above as
0 + 0 = 0 (mod 3)
0 + 1 = 1 (mod 3)

We leave it to you to write out the remaining 7 statements of addition and the 9 statements of multiplication. These statements describe the Algebra of the Integers Modulo 3.

So far, we have seen the Integers Modulo 2 and the Integers Modulo 3. It’s not hard to imagine the notation or interpretation for the Integers Modulo 4 or the Integers Modulo 37.

So if you are asked to find 2+2, you better know what Algebra you are working in! In the Algebra of the Reals or the Integers, 2+2=4. But in the Integers Modulo 3, we have 2+2=1(mod 3).

Given that our motivation is to understand what “An Algebra” is rather than to understand the intricacies of any particular Algebra, we’ll close our discussion of this family of Algebras here. The curious reader may want to explore whether division makes sense in this Algebra.

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.