Thursday 3 November class

Topics:
• Proving that the square root of 2 is irrational: here are my notes, which I handed out in class. We also imitated that proof in the case of the square root of 3.

A similar proof will work in any case of a square root of a prime number. All square roots of prime numbers are irrational. But even more is true: If n is any positive integer (natural number) which is not a perfect square, then the square root of n is irrational. It is just a little more work to prove this if n is not a prime number, as you will see in the exercises.

• Density of rational numbers:
Theorem: between any two rational numbers, there is another rational number.
Proof: Take the average of the two rational numbers. The result is another rational number which is halfway between them.

Example: between $\frac{1}{2}$ and $\frac{3}{4}$ we can find the number $\left(\frac{1}{2} + \frac{3}{4}\right)\frac{1}{2}$ (their average, where instead of dividing by 2 we have multiplied by 2, which is the same thing. This comes out to be $\frac{5}{8}$, which is in between $\frac{1}{2}$ and $\frac{3}{4}$ and is a rational number.

Note: this is not the only rational number in between $\frac{1}{2}$ and $\frac{3}{4}$: but we only needed to show that there was at least one. So this one is convenient to find.

• A consequence of this theorem is that between any two rational numbers there is an infinite number of rational numbers. You can find rational numbers as close together as you like on the real number line. We say that the rational numbers are dense in the real numbers.

But even though the rational numbers are so close together and "thick" on the real line, there is still room enough for an infinite number of irrational numbers! In fact, as we will see next time, there are (in a very concrete sense) more irrational numbers than rational numbers.
• It follows from this that in between every two rational numbers there are infinitely many rational numbers. (Just keep taking averages of averages...)

• Comment: in fact, between every two real numbers (whether or not they are rational) there is a rational number, and also between every two real numbers there is an irrational number. The proofs of those facts are harder so we do not discuss them in this course. Ask Dr. Math gives a proof of the first statement here.

Homework:
• Journal assignment:
- The proof that the square root of 2 is irrational uses at one point the fact that if  a² is divisible by 2, then a must itself be divisible by 2. This happens because of the Fundamental Theorem of Arithmetic (Unique Prime Factorization). Explain why this is so. (Give an example, it may help. Factor both a and a² into primes and see how their prime factorizations are related to each other.)
- What goes wrong if you try to imitate this proof to prove that the square root of 4 is irrational?


• Problems to do:
Related to 0.999999,,, = 1:
1) Write 0.09999999... as a fraction in lowest terms.
2) Write 0.29999999... as a fraction in lowest terms
3) Write 0.45999999... as a fraction in lowest terms

4) Prove that the square root of 5 is irrational

5) Prove that the square root of 6 is irrational. Since 6 is not a prime number, you will have to think about how to handle the step where we say "since a² is divisible by 6, then a must itself be divisible by 6."

6) Challenge! This is extra credit. Prove that the square root of 12 is irrational. You must be extra-careful at the above-mentioned step here!

Next up: Hilbert's Hotel, which always has room for more: up to a point, that is. Measuring the size of infinity. Yes, it makes sense.
Please watch this video and this video (for the Cantor proof) , and here is another video as well. (all three of them are worth watching.)