Connecting Algebra and Numbers

In the last two entries, we've seen some of the most common number systems and the rules that each of them get us, and then some of the more common algebraic structures. Now we'll talk a little about how they fit together and along the way we'll learn a little bit more about algebra.

Before this mathematical adventure, there is something worth mentioning, and that is that we're looking for the exact algebraic structure to describe each number system. What I mean is that, as an example, the integers are both a group and a ring, but neither of those structures fully describe all of the algebraic properties that we get from the integers. Ideally we could find the exact list of properties that describe everything that we get to do in each number system, which is what I will now venture to do.

Natural Numbers

Our smallest number system that we had was the natural numbers. Now the natural numbers have both addition and multiplication which suggests the structure of a ring, but they don't have any inverses, so they're not even a group. We do have a structure for groups without inverses, and its called a monoid.

Definition: Monoid
Let $M$ be a set and let $\cdot$ be a binary operation defined on the elements of $M$. Then $M$ is a monoid if it satisfies the following three axioms:

Closure

$a \cdot b \in M \; \forall a,b \in M$

Associativity

$\left ( a \cdot b \right ) \cdot c = a \cdot \left ( b \cdot c \right ) \; \forall a,b,c \in M$

Identity

$\exists e \in M \; \text{such that} \; a \cdot e = a = e \cdot a \; \forall a \in M$

A monoid with the additional condition of commutivity is called a commutative monoid.

The natural numbers are indeed a monoid, but it turns out that they're a monoid under both addition and multiplication and we still have the distributive laws in the natural numbers. In fact, the natural numbers look just like a ring except they have a multiplicative identity and lack additive inverses. This algebraic concept is called a semiring.

Definition: Semiring
Let $S$ be a set and let $+$ and $\cdot$ be different binary operations on $S$ (called addition and multiplication) such that $S$ is a monoid under both addition and multiplication. Then $S$ is a semiring if it satisfies the following axioms in addition to the additive and multiplicative monoid axioms:

Additive Commutivity

$a + b = b + a \; \forall a,b \in S$

Left Distribution

$a \cdot \left ( b + c \right ) = \left ( a \cdot b \right ) + \left ( a \cdot c \right ) \; \forall a,b,c \in S$

Right Distribution

$\left ( a + b \right ) \cdot c = \left ( a \cdot c \right ) + \left ( b \cdot c \right ) \; \forall a,b,c \in S$

Annihilation

$0 \cdot a = 0 = a \cdot 0 \; \forall a \in S$

A semiring with the additional condition of multiplicative commutivity is called a commutative semiring.

The natural numbers form the stereotypical commutative semiring. This is not a common algebraic structure. You find group theorists in the world, you find people that spend their lives studying rings, and there are plenty of people that love fields because they need them for vector spaces or modules or algebras or plenty of other things. To my knowledge, there are not hoardes of people who dedicate their time to the mathematical advances of the semiring. We just don't get nearly as much usefulness out of semirings as we do from some of the other algebraic structures.

Integers

The integers form a ring. No long suspensful introduction or lengthy suspense this time, the integers form a ring. But wait, there's more! The integers aren't just a ring, they get some bonus properties. I mention first that they're a ring because they are actually very close to a ring and they are one of the first and best examples that you get of a ring. However, the integers are also commutative, they have a multiplicative identity, and they get one extra unexpected property. This structure is called an integral domain.

Definition: Integral Domain
Let $R$ be a commutative ring. Then $R$ is an integral domain if it follows the following axioms in addition to the ring axioms:

Multiplicative Identity (Unity)

$\exists 1 \in R \; \text{such that} \; 1 \neq 0 \; \text{and} \; a \cdot 1 = a = 1 \cdot a \; \forall a \in R$

No Zero Divisors

$\forall a,b \in R \; a \cdot b = 0 \; \text{implies that either} \; a = 0 \; \text{or} \; b = 0$

Integral domains are the structure that get us the exact properties of the integers. It is, in fact, that last property regarding zero divisors that generally defines integral domains - without it we would describe $R$ as "a commutative ring with unity." In fact, in many books that need rings but don't explicitly study rings, there is a sentence in the beginning that says something to the effect of "for the remainder of this text every ring will be assumed to be a commutative ring with unity." My point is to greatly emphasise here the importance of this final property. It may seem obvious, and it is obvious when regarding the numbers that we've used since we were five years old, but in arbitrary rings it is not obvious. As an example, suppose we're considering $\mathbf{Z}_p$ and we want to define multiplication similarly to the way we define addition - that is $a \cdot b = a \cdot p \; \text{mod p}$. Then in $\mathbf{Z}_8$ we have that $2 \cdot 4 = 0$ but there are no two non-zero numbers in $\mathbf{Z}_7$ that multiply to zero. That is the property that is really of interest in an integral domain.

Rational, Real, and Complex Numbers

Believe it or not, from an algebra standpoint there's no difference between the rational and real numbers. Topologists know the difference and analysts definitely know the difference, but even the most well-trained algebraist can only barely tell the difference between the real numbers and the rational numbers. If you remember, the bonus that we get from the reals is that all sequences in $\mathbf{R}$ converge in $\mathbf{R}$ but sequences and convergence is not really a concern in the world of algebra. The reals are "larger" than the rationals, as we've already seen, in that the rationals are countable and the reals are uncountable and there are times when this fact concerns a diligent algebraist, but in fact they are both identical in algebraic structure. That structure is the structure of a field. Both the real numbers and the rational numbers have the exact algebraic structure of a field.

The complex numbers also form a field, but as you probably guessed, they have exactly one extra property that turns out to be extremely important.

Definition: Algebraically Closed Field
A field, $F$, is algebraically closed if every polynomial with one variable of degree at least 1 with coefficients in $F$ has a root in $F$.

Consider the polynomial $p \left ( x \right ) = x^2 + 1$. The coefficients of $p$ live in any of the number systems that we've discussed so far, but as we may remember from calculus, there is no real number that is a root of $p$, but there are two complex numbers - namely $i$ and $-i$ - that are roots of $p$. As it turns out any polynomial

\[ p \left ( x \right ) = \sum_{k=0}^{n} \alpha_k x^k \]

of degree $n \geq 1$ with $\alpha_k \in \mathbf{C}$ has some root in $\mathbf{C}$. This is not true, though, in fields that are not algebraically closed like $\mathbf{R}$ and $\mathbf{Q}$.

Other Thoughts

There's a lesson to be learned here. We have all sorts of obvious uses for all of the various number systems that we have in calculus and analysis and topology and all sorts of other places and we have developed some algebraic language and ideas with which to talk about the most common numeric universes. It is obvious, though, that the most common algebraic structures don't really represent our number systems very well, which further justifies the study of algebra because the things that are accurately represented by algebraic structures cannot accurately be represented by the numbers that we've studied since the dawn of mathematics.

I should also mention that, as strange as it seems, all these definitions aren't necessarily concrete. Authors sometimes define things slightly differently depending on what they plan on doing with their structures. For example, semirings aren't always defined with either identity elements because that makes the analogy betweem ring:semiring and group:semigroup more accurate (even though we didn't define a semigroup). The definition of a ring can also sometime include an identity element. I know its confusing and sort of counter-intuitive to the rigidity of mathematics, but usually the context makes it clear.

This is a blog about algebra (for now at least), so I don't expect that my readers are worrying that I'll start a rigorous study calculus after glossing over the real numbers. On the other hand, I introduced rings, fields, monoids, semirings, integral domains, and algebraically closed fields without really explaining them or giving examples or anything. Don't feel too guilty if you don't understand how any of those things work - I don't plan on delving into the inner workings of any of these structures any time soon and if I do, I promise to reintroduce them later.

References

Previous Related Post: Algebraic Structures

Wikipedia: Monoid

Wikipedia: Semiring

Wikipedia: Integral Domain

Wikipedia: Algebraically Closed Field