2.3 Two Major Theorems

Objectives:

•

Discover two novel proof ideas that showcase the power of the tools we have learned so far
•

Rework those proof ideas to apply them to other similar problems

Example 2.3.1.

Define the set of real numbers.

2.3.1 What is an irrational number?

Like nearly everything we use in mathematics, we did not even know irrational numbers existed for hundreds and perhaps thousands of years, even when we were using them widely in construction.

If we are constructing a 45-degree diagonal wall, we are implicitly using the value $\sqrt{2}$ .

However, the fact that this number is irrational (cannot be written as a ratio of integers) is widely credited to Hippassus of Metapontum (flourished c. 500 BCE).

Some believe that the Pythagoreans were so horrified by the idea of irrational numbers that they threw Hippassus overboard on a sea voyage and vowed to keep the existence of irrational numbers a secret.

But we need the irrational numbers in order to define the real numbers. There is a potential definition of real numbers as, “the set of complex numbers whose square is non-negative”, but even this definition presupposes the existence of complex numbers. What are the complex numbers but the set of numbers $a+bi$ , where $a$ and $b$ are real numbers?

So what is another definition? The most natural one I can think of: “the union of the sets of rational numbers and irrational numbers”. We have a good definition of $\mathbb{Q}$ :

\left\{\frac{m}{n}:m\in\mathbb{Z},n\in\mathbb{N}\right\}.

But what is the definition of the set of irrational numbers? Certainly we don’t mean “the set of numbers which are not rational”, as that would involve complex numbers as well, and we don’t consider complex numbers to be irrational.

It turns out this definition is very difficult to make precise - it is typically covered halfway through MATH-409. Today we will tackle a slightly smaller problem that is certainly easy to prove, right? Right?

The number $\sqrt{2}$ is irrational.

Example 2.3.2.

The following “proofs” below are all missing crucial information. Can you determine why they are wrong?

Suppose BWOC that $\sqrt{2}=\frac{m}{n}$ for $m\in\mathbb{Z}$ and $n\in\mathbb{N}$ . Then multiplying both sides by $n$ yields $m=n\sqrt{2}$ . But the number on the left-hand side is an integer while the number on the right-hand side is not an integer.

This is a contradiction. The only thing we assumed was that $\sqrt{2}$ was rational, so $\sqrt{2}$ must be irrational.

Suppose BWOC that $\frac{m}{n}=\sqrt{2}$ for some $m\in\mathbb{Z}$ and $n\in\mathbb{N}$ . Then multiplying both sides by $n$ yields $m=n\sqrt{2}$ .

Square both sides; we get $m^{2}=2n^{2}$ . We conclude that $m^{2}$ is even.

But we never claimed that $m^{2}$ was even. Indeed, if $m$ is odd, then $m^{2}$ is odd (by Section 1-1 results). This is a contradiction.

The only thing we assumed was that $\sqrt{2}$ was rational, so $\sqrt{2}$ must be irrational.

Example 2.3.3.

After having reviewed the examples on the previous page, one student decides to work on the proof overnight. Below is what they come up with. Can you find anything wrong with it? Space is given between each line so you can ask questions or clarify steps that the student makes.

Suppose BWOC that $\sqrt{2}=\frac{m}{n}$ for some $m\in\mathbb{Z}$ and $n\in\mathbb{N}$ . Then multiplying both sides by $n$ yields $m=n\sqrt{2}$ .
Blank space for you to write your work.
Square both sides; we get $m^{2}=2n^{2}$ . Call this Equation (1); we will return to it later.
Blank space for you to write your work.
We conclude that $m^{2}$ is even.
Blank space for you to write your work.
We know from Section 1-1 that, if $m$ is odd, then $m^{2}$ is odd. The contrapositive of this statement must also be true: since $m^{2}$ is even, $m$ must be even.
Blank space for you to write your work.
By definition of an even number, then, $m=2k$ for some integer $k\in\mathbb{Z}$ .
Blank space for you to write your work.
We plug in $m=2k$ to Equation (1). So we have that $(2k)^{2}=2n^{2}$ , or $4k^{2}=2n^{2}$ , or $2k^{2}=n^{2}$ .
Blank space for you to write your work.
But this means that $n$ is even as well. This is a contradiction, as $m$ and $n$ can’t both be even! If they were, $\frac{m}{n}$ would be an unsimplified fraction, as both $m$ and $n$ are divisible by 2! We could have started the process by simplifying the fraction so that at least one of $m$ and $n$ is odd, and then we have a contradiction with the fact that both $m$ and $n$ must be even.
Blank space for you to write your work.
The only thing we assumed was that $\sqrt{2}$ was a rational number, so we are forced to conclude that $\sqrt{2}$ is irrational.
Blank space for you to write your work.

Perhaps we should change this proof slightly, as the contradiction seems a bit hidden. The student’s idea is very good, but it is a little hard to make out with all of the details of the contradiction set up at the very end.

Take the student’s idea and adapt it for the following theorem. The student relies on the fraction at the beginning of the process being simplified - go ahead and say that “WLOG” the fraction $\frac{m}{n}$ is given in simplest form.

Example 2.3.4.

Prove that $\sqrt{3}$ is irrational.

(More space for Example 2.3.4.)

This proof tells us something quite important: most statements worth proving in mathematics take a lot of thought to prove. What’s more, proving even slightly different statements can result in widely differing proof techniques! This is where we tell you the truth about mathematics:

There is no one right answer for how to do mathematics.

Example 2.3.5.

Interpret this statement in your own words.

2.3.2 How Many Primes Are There?

Now that we have shown that primes are the building blocks of the positive integers, it would be a good idea to figure out how many primes there are. Certainly there are infinitely many numbers, right? So are there infinitely many prime numbers?

We begin with a thought experiment. Recall that $a$ is a factor of $n$ if $a$ divides $n$ . Also recall that a number $n \in ℕ$ is prime if $n \geq 2$ and its only factors are 1 and itself. Finally, a number $n \in ℕ$ is composite if $n \geq 2$ and not prime. (The number 1 is neither prime nor composite.)

Example 2.3.6.

We say that a number $n$ has a prime factorization if $n=p_{1}p_{2}\cdot\cdots\cdot p_{n}$ for prime numbers $p_{1},p_{2},\dots,p_{n}$ where these prime numbers can repeat. What natural numbers (other than 1) do not have a prime factorization?

Example 2.3.7.

We say that a number $n$ has a primitive prime factorization if $n=p_{1}p_{2}\cdot\cdots\cdot p_{n}$ for prime numbers $p_{1},p_{2},\dots,p_{n}$ which do not repeat. What is the largest number that has a primitive prime factorization? Explain.

Example 2.3.8.

If your answer to the previous example were true, how many prime numbers would have to exist?

This claim that there are infinitely many prime numbers was proven about 2000 years ago by Euclid, whose texts on geometry established the criteria for rigorous proof that we still use today. Let us investigate proving this by contradiction, similar to what we did in the proof of $\sqrt{2}$ being irrational.

You might be noticing a pattern here… if we are showing something is irrational (meaning not rational) or infinite (meaning not finite), it is typically better to use a contradiction proof! We have a much better understanding of rational and finite things than we do of things that are not rational or not finite.

Example 2.3.9.

For the rest of this section, assume that there are only finitely many primes. We don’t know how many primes that would be, so say there are $n$ primes for some natural number $n$ . If there were infinitely many primes, we would not be able to choose this $n$ . Explain why.

Example 2.3.10.

We also don’t know what all the primes would be (save for a few). Let’s call them $p_{1},p_{2},\dots,p_{n}$ , then. Given our supposition, what is the largest number with a primitive prime factorization?

Example 2.3.11.

Continuing our assumptions from before, investigate whether the number

(p_{1}p_{2}\cdot\cdots\cdot p_{n})+1

has a prime factorization (primitive or not!). Hint: first decide whether $p_{1}$ can divide this number. Then decide whether $p_{2}$ can divide this number. What primes can divide this number?

Example 2.3.12.

Write a formal proof of the following statement: “There are infinitely many distinct prime numbers.”