Dan Tests Layouts: Numbers

For the next three entries, I want to talk about numbers. Now, us theoretical mathematicians don't like numbers all that much, but as they are a necessary part of the world, I'd like to show how they relate to our theoretical adventures. Actually, its not really true that we don't like numbers - we like the complex numbers quite a bit because they're a nice (algebraically closed) field and we do like to count things and look at the sizes of things - we just don't really like the primary focus to be on numbers. At any rate, below is an overview of the systems of numbers that we're going to talk about.

Natural Numbers: $\mathbf{N} = \left \{ 0 , 1 , 2 , \cdots \right \}$
Integers: $\mathbf{Z} = \left \{ \cdots , -2 , -1 , 0 , 1 , 2 , \cdots \right \}$
Rational Numbers: $\mathbf{Q} = \left \{ \frac{p}{q} : p,q\in \mathbf{Z} \;\text{and}\; q\neq 0 \right \}$
Real Numbers: $\mathbf{R}$ is the set of all decimal numbers - even non-terminating and non-repeating decimals
Complex Numbers: $\mathbf{C} = \left \{ a + b i : a,b\in \mathbf{R} \;\text{and}\; -1=i^2 \right \}$

Number System Advantages

This covers most of your basic number systems and whether you know it or not, every time you work on a problem you have to choose a number system to live in. Induction problems use the natural numbers, calculus problems probably use the reals, and solving polynomials usually requires the complex numbers. You'll notice the nice relationship where

\[ \mathbf{N} \subset \mathbf{Z} \subset \mathbf{Q} \subset \mathbf{R} \subset \mathbf{C} \]

and the way it works is that every time you move to a "larger" number system you get a new property that you get to work with.

Natural Numbers: We're going to start our journey in the natural numbers, $\mathbf{N}$. First what we need to know is what we're allowed to do in $\mathbf{N}$. You can multiply numbers in $\mathbf{N}$ and you can add numbers in $\mathbf{N}$, but what you can't do is subtract or divide and still stay in $\mathbf{N}$. Clearly neither $6-8$ nor $\frac{5}{4}$ are natural numbers.
Integers: So then we move to $\mathbf{Z}$ and, as promised, we gain a property. In $\mathbf{Z}$ we get negative numbers, which means we get subtraction and, in the language of abstract algebra, we get additive inverses. However, you still don't get to divide integers.
Rational Numbers: You don't get division until you move to the rationals, $\mathbf{Q}$, which gives us fractions and multiplicative inverses. It may seem like all we need to do, we can do in $\mathbf{Q}$, but that's not true yet.
Real Numbers: The sequence $3, 3.1, 3.14, 3.141, \cdots$ contains only rational numbers and will converge to $\pi$, but $\pi \notin \mathbf{Q}$. What we gain in the real numbers, $\mathbf{R}$, is the limits of sequences. Every convergent sequence of real numbers converges inside $\mathbf{R}$ (we call $\mathbf{R}$ "complete" in topological terms).
Complex Numbers: The largest number system we've got (for now) is the complex numbers, $\mathbf{C}$, and the benefits that come from the complex numbers make it very useful for the purposes of abstract algebra. The particular benifit is that its "algebraically closed." If we're just living in $\mathbf{R}$, then the polynomial $p(x)=x^2 + 1$ has no roots - that is it has no real roots. In $\mathbf{C}$, every polynomial of degree greater than 1 with complex coefficients has a complex root.

Number System Disadvantages

Now, we just learned that we have all these number systems, and as the number systems gets larger we gain more and more properties. As it turns out, though, as the number systems get larger we also lose properties from some added complications.

Natural Numbers: Again, we're going to start in the natural numbers and then show how at each step we lose something. There are a lot of restrictions on $\mathbf{N}$, as we already learned, but it turns out that $\mathbf{N}$ has a lot of nice properties, too. The first of which is called the Well Ordering Principle which says that any subset of the natural numbers has a smallest element.
Integers: It is not difficult to see that the integers do not have the Well Ordering Principle. Indeed the set $\left \{ k \in \mathbf{Z} : k < 0 \right \}$ does not have a smallest element.
Rational Numbers: The cost in moving from the integers to the rationals is a little more complicated. The integers have something called a "discrete topology." What that really means is that there is some space between the integers and there's not that space between rationals. In $\mathbf{Z}$, you can pick two numbers (say 2 and 3) such that there's no integer between them, but this can't be done in $\mathbf{Q}$ (for any $a,b \in \mathbf{Q}$ with $a<b$, $\frac{a+b}{2} \in \mathbf{Q}$ and $a < \frac{a+b}{2} < b$). Also, bounded subsets of $\mathbf{Z}$ have a maximum and a minimum and are finite, but on the other hand, bounded subsets of $\mathbf{Q}$ don't have to have maximums or minimums and can be infinite (such as the interval $(a,b) \subset \mathbf{Q}$).
Real Numbers: The rational numbers are countable (meaning that there's a bijection between $\mathbf{N}$ and $\mathbf{Q}$) and when you move to the reals you lose this. In fact, this is one of the defining properties of the reals.
Complex Numbers: Finally, when you leave the comforts of the reals and move to the wide world of algebraic completeness afforded by the complex numbers you lose order. If $x,y \in \mathbf{R}$ then either $a < b$, $a > b$, or $a = b$. If $a,b \in \mathbf{C}$ then we can talk about what it means for them to be equal, but the statement $a < b$ doesn't make any sense.

Most of the time the set of numbers that we use is implied or is obvious, but at times, its important to be careful which set you're using or at least to be aware of the properties at your disposal in each of them. Next time we'll take a look at where these numbers fall in the world of abstract algebra. It turns out that although most of these form groups in one way or another, none of them are only groups.

References

Wikipedia: Natural Numbers

Wikipedia: Integers

Wikipedia: Rational Numbers

Wikipedia: Real Numbers

Wikipedia: Complex Numbers