Chapter 1 in Stillwell.
Pythagorean musical scale and $\sqrt{2}$.
Preview
Response What are numbers and what are they for? Where do numbers exist and what allows them to exist? What kinds of things do you count as numbers? Fractions, decimals, infinite decimals, negative numbers, irrational numbers, complex numbers, quaternions, hyperreals, matrices? If you can argue for the inclusion/exclusion of one species, who might disagree and what might their reasons be?
Response What does Stillwell mean when he claims that “this rational world is impossible”?
Follow up We learn that not all the things the Greeks wanted to be numbers to be as represented by Geometric construction, namely all the possible lengths of lines, could be derived from basic operations of arithmetic from whole numbers. Numbers like $\sqrt 2$ were said to be non-commensurable with whole numbers. What can we learn about Greek mathematics from the idea that non-commensurable numbers may have bothered them? Have they ever bothered you or have you even ever herd of this concept before? Would you be surprised to learn that other advanced mathematical traditions of antiquity (Mesopotamia, China, India) did not seem to care about this?
Research Assuming that all the points on a line are numbers, a basic assumption we are all encouraged to make implicitly by our school math instruction and also made explicitly by the Greeks as well as in our imagined initial discoveries of math by humans, turns out to have deep consequences. The existence of irrational numbers like $\sqrt 2$ is just the beginning. We will see a further consequence in our module on infinity—a big surprise—when we try to count the points on a line. But this too is just the beginning. If you’d like to dive into the mysteries of the real line, you will discover much.
Problem Show that $\sqrt 2$ is irrational. Since there are at least dozens of such proofs on the web and we’ve already talked about it in class, yours should do something useful that those do not. For example:
- Do this in a test-like environment. Study. Wait at least one day. Give yourself 10 minutes, a piece of paper with this problem on it, and a pen/pencil. Nothing else. See how you do. Repeat until you get it.
- Typeset it and explain it well.
- Describe the dependencies of the proof. What are you assuming you already know about numbers and arithmetic? How does the proof change if you allow yourself better assumptions.
- Generalize this argument.
1.1 The Pythagorean Dream
Reflection Stillwell notes that for the ancient Greeks, learning was divided into seven disciplines. Where do you see the legacy of this classification today? Does it remain useful and how might it be revised? Where does it a align with a division of academic content area (subjects) and where does its desire to map out kinds of thought that are important to be capable of yield different divisions? Where does our current math education fit in?
Recall Keep track of the historical stories Stillwell tells about the Greeks and how he phrases these. Note names, places, works and deeds. Do you notice anything strange about them? Do you believe them? Why?
Research Take one of the people or works mentioned and dig up some more details. Wikipedia might be a good place to start, but you want to find more authoritative material eventually. Having learned a lot, what more would you want other students reading Stillwell to know about the historical material he brings in?
Try it Create your own experiment regarding harmony. Either using an existing instrument, creating a vibrating string, or using software that can digitally reproduce various frequencies of sound, experiment with different intervals. Is, as Stillwell claims, the octave the most harmonious? Can you make a fifth? After you try it yourself, try testing others.
Recall Why is a fifth called a fifth?
Recall What is a mode of vibration?
What is the frequency of a vibration?
Response What might be meant by the Pythagorean credo “All is number”? Where do you see this in place today and how might you further refine it? Where are there numbers behind everything, or people who think so? Where does this seem to be a good thing and where does it lead to trouble?
Practice Try out this basic lesson about sounds and vibrations.
Starting with a specific note, create a Pythagorean scale from fifths. Why is it called a fifth? Explain what goes wrong when you try to build a scale this way.
Problem Can a sum of fifths equal an octave? Why or why not and why does this matter?
Research Find (an)other chapter(s) in the history of musical scales and temperaments. What were those innovations comprised of and what did they aim to improve? What was the response to their creation?
Pythagorean Theorem
Research Read up on Plimpton 322
Problem Use Plimpton 322 to make a scale of triangles. What goes wrong?
Warmup Try to figure out the volume of a tetrahedron
Euclid
Not very many people make it through much of Euclid.
Research What’s so great about book 5?
Reflection What does it mean for a book to be influential? What are our expectations for people and math in relation to classical knowledge?
Irrational Numbers
Research Check out Simon Stevin’s l’Arithmetique of 1585 or his treatise on the use of decimal numbers La Thiende.
Recall What do we know and what do we not know about the digits in the infinite decimal expansion of $\sqrt{2}$?
Does every quadratic number have a periodic continued fraction representation? Why is the converse true?
Research Check out some of the other scales and tempers.
Help us to hear the differences in temperings or otherwise appreciate them in the application of them to music.
Reflection Interpret the Pythagorean credo “All is number” today. How is its applicability changed by the fact that number means something different now than then? Are there senses where it might be said to be more true than others in its original sense?
Follow up What exactly is the relationship between pitch and frequency of vibration and what is the essence of Beeckman’s contribution?
More Problem Suggestions
Show that $\sqrt 2 + 7$ is irrational. Here it is really important to tell the difference between mathematical argumentation and hand waving.
Show that $\sqrt 2 + \sqrt 3$ is irrational.
In the proof for the irrationality of $\sqrt 2$ it sometimes comes up that $p^2$ is odd implies $p$ is odd. This kind of reasoning can go further. As an introduction,
show that $p^2$ cannot have remainder 3 when divided by 4 for any whole number $p$.
The fancy way of writing this is $p \not \equiv 3 \mod 4$.
