Lecture 0 - An Introduction to Numbers

Summary

We started today by getting to know the policies and expectations in the course. All of this is available already on the syllabus, but if you have any questions don't be shy about moc.liamg|ztluhcs.c.werdna#ydnA gniliame. We also spent some time introducing ourselves briefly; this will be continued as you post your own profiles for Homework 0. Afterwards, we started talking about the basics in number theory, starting with the axioms. We finished by introducing the notion of divisibility for the integers.

The Axioms of Number Theory

When trying to build a mathematical discipline from the ground up, one needs to describe the fundamental objects and operations in the discipline and then define the basic properties these objects will obey. These properties are called axioms, and they are the "ground rules" the objects and operations must satisfy. With axioms in place, one can then start proving theorems by manipulating the axioms.

In number theory, the basic objects of interest are integers. You might know these objects as whole numbers. In this class we'll denote the set of all integers as $\mathbb{Z}$:

(1)
\begin{align} \mathbb{Z} = \{\dots, -3, -2, -1, 0, 1, 2, 3, \dots \}. \end{align}

The basic operations we have on the integers are addition, subtraction and multiplication. We've avoided division since division doesn't behave very well on the integers: the quotient of two integers is frequently not another integer. We also have basic tools for comparing integers, namely equality and inequality.

With the fundamental objects, operations and comparisons in place, we can start writing down the basic rules they all satisfy. Here's the list that we were able to come up with in class:

  • for any $a \in \mathbb{Z}$$a = a$ (reflexivity of equality);
  • for any $a,b,c \in \mathbb{Z}$, if $a=b$ and $b=c$ then $a=c$ (transitivity of equality);
  • for any $a,b \in \mathbb{Z}$, $a=b$ implies $b=a$ (symmetry of equality);
  • for any $a,b \in \mathbb{Z}$, the sum $a+b$ is an integer (closure under addition) [note: we didn't mention this in class];
  • for any $a,b \in \mathbb{Z}$, the product $ab$ is an integer (closure under multiplication) [note: we didn't mention this in class];
  • for any $a,b,c \in \mathbb{Z}$, $(a+b)+c = a+(b+c)$ (associativity of addition);
  • for any $a,b,c \in \mathbb{Z}$, $(ab)c = a(bc)$ (associativity of multiplication);
  • for any $a,b,c \in \mathbb{Z}$, $(a+b)c = ac + bc$ (distributivity);
  • for any $a,b \in \mathbb{Z}$, $a + b = b+a$ (commutativity of addition);
  • for any $a,b \in \mathbb{Z}$, $ab = ba$ (commutativity of multiplication);
  • for any $a \in \mathbb{Z}$, $a+0 = a$ (additive identity) [note: we didn't mention this in class];
  • for any $a \in \mathbb{Z}$, $a\cdot 1 = a$ (multiplicative identity) [note: we didn't mention this in class];
  • for any $a \in \mathbb{Z}$ there exists $-a \in \mathbb{Z}$ so that $a + (-a) = 0$ (additive identity);
  • for any $a,b,c \in \mathbb{Z}$ with $a \neq 0$, then $ab = ac$ implies $b =c$ (cancellation of multiplication);
  • for any $a,b,c \in \mathbb{Z}$, $b=c$ implies $ab = ac$ (substitution for multiplication);
  • for any $a,b,c \in \mathbb{Z}$, $b=c$ if and only if $a+b = a+c$ (substitution for addition; cancellation of addition);
  • for any $a \in \mathbb{Z}$, exactly one of the following is true (1) $a<0$ (2) $a=0$ (3) $a>0$ (Trichotomy law);
  • for any $a,b,c \in \mathbb{Z}$, if $b<c$ then $a+b<a+c$;
  • for any $a,b,c \in \mathbb{Z}$, if $b<c$ and $a>0$, then $ab<ac$;
  • for any $a,b,c \in \mathbb{Z}$, if $b<c$ and $a<0$, then $ab>ac$; and
  • if $S$ is a nonempty set of positive integers, then $S$ has a least element (Well ordering principle).

The last axiom — the well ordering principle — probably sticks out as the ugly duckling of the bunch, and it certainly isn't one which most people think of when rattling off basic properties of the integers. It is, however, essential to what we'll be doing in class, as it is logically equivalent to mathematical induction — a tool that we'll be using with some frequency in this course.

Playing around with the axioms

Our list of axioms is a little redundant, meaning that we could probably prove some of the axioms we've listed in terms of the other axioms. In this sense, it doesn't pass the usual mathematical aesthetic. To see that this is true, you can try to use the other axioms listed above to prove

Theorem: For any $a \in \mathbb{Z}$, $a\cdot 0 = 0$

In class, we sketched a proof of the following result

Theorem: For any nonzero $n \in \mathbb{Z}$, if $a \in \mathbb{Z}$ satisfies $n\cdot a = 0$, then $a= 0$

Proof: We started by noting that we could assume a is either greater than or less than 0; by trichotomy we know that one of (1) $a<0$ (2) $a = 0$ or (3) $a>0$ is true, and if we had $a = 0$ then we'd be done with the theorem. By a similar token we know that $n>0$ or $n<0$, since $n =0$ is ruled out by assumption. So we broke things into 4 cases based on whether $a<0$ or $a>0$ and whether $n>0$ or $n<0$.

Case I: $a>0$ and $n>0$
In this case, one of our axioms on inequality tells us that $an > 0$. This contradicts the fact that $an = 0$, and so we know this case is impossible.

One could proceed with analyzing the other cases, each time you would find a contradiction to the given equality $an = 0$. At the end, one concludes that all the possibilities lead to a contradiction, and hence neither $a>0$ nor $a<0$ are possible. This leaves only $a = 0$, the desired result.
$\square$

Though working through axiomatic proofs is good exercise for building your proof-muscles, in practice we won't be quite so explicit in our use of these familiar axioms during class. This won't present any real problems since you are more accustomed to manipulating these axioms then perhaps you realize.

Putting the Elementary in Number Theory

With the basic ground rules set, we had a chance to talk about the most important property of integers in this whole class: divisibility. It is the study of this property which makes the number theory we'll study "elementary." One can think of divisibility as the attempt to carry division into the realm of the integers, made appropriately cautious to reflect the fact that the integers don't always behave so well under division.

An integer d is said to divide an integer a if there exists an integer q so that a = dq. If d divides a we write $d \mid a$, and if d does not divide a then we write $d \nmid a$.

An example

This definition should agree with your own intuitive notion of divisibility in the integers, so hopefully it isn't too surprising. To see an example in action, notice that $4 \mid 8$ since we can find an integer q to solve the equation

(2)
\begin{align} 8 = 4\cdot q; \end{align}

in this case, the integer q is simply 2.$\square$

A non-example

Let's try to prove that $2 \nmid 5$. For this, we need to show that we cannot find an integer q satisfying the equation

(3)
\begin{align} 5 = 2\cdot q. \end{align}

For this notice that $5 > 2\cdot 2$; for any integer q satisfying $2 \geq q$ we have $2\cdot 2 \geq 2\cdot q$ (a slight modification on one of our axioms), and hence we have

(4)
\begin{align} 5 > 2\cdot 2 \geq 2\cdot q. \end{align}


Likewise we know that $5 < 2 \cdot 3$, and for all integers q satisfying $q \geq 3$ we get

(5)
\begin{align} 5 < 2\cdot 3 \leq 2\cdot q. \end{align}

Since all integers fall into one of the categories we have described, we conlude that $5 \neq 2\cdot q$ for any integer q, and so $2 \nmid 5$.$\square$

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