Class #7

Erdös: The Man Who Loved Only Numbers


Elliptical Stairway

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NOTE: Please see the last Resources entries below for material promised in Class #6 about Andrew Wiles and his proof of Fermat's Last Theorem.

Assignments for Class #7

• Read Chapter 5, "God Made the Integers".

• Read about the harmonic series, on page 217, and then read the first few paragraphs of these two Wikipedia entries:
1) Harmonic series (mathematics), and 
2) Harmonic series (music).

• Watch the first 5 minutes of this video on the harmonic series in music:

• Read the first paragraph (and any more that interests you) of the Wikipedia entry, Set (mathematics).

• Learn the basics of one-to-one correspondence by watching the first two minutes of this video:

• Even Galileo had difficulty understanding infinite sets.  Watch this short video:

• Progress in understanding has been made, but it is still unintuitive.  Watch this video: 

Questions to Think About

• Why is the series (1/1) + (1/2) + (1/3) + (1/4) ... + (1/n) called the harmonic series ?

• What is likely to be the first one-to-one correspondence that you ever made?
Hint: it was very handy.

• You are cleaning up after a party.  You started the evening with the same number of forks and spoons but now you think you may have lost a spoon.  How can you tell, without actually counting either the forks or the spoons?

•  Look at this picture


Step 1 starts with the interval consisting of all points on the number line between 0 and 1.  


Step 2 removes the middle third of that interval, leaving 2 disconnected shorter intervals.  


Step 3 removes the middle third of each of those 2 intervals, leaving 4 disconnected even-shorter intervals. 


Step 4 removes the middle third of each of those 4 intervals, leaving 8 disconnected even-shorter intervals. 


Subsequent steps carry on similarly, removing the middle third of each existing interval to produce more and more  - but shorter and shorter - disconnected intervals 


Question: Are there any points remaining after doing an infinite number of steps?  If yes, how many points are left?


Other Resources

• More about the Golden Ratio at Wikipedia

• HOW INFINITE HOTEL RAN OUT OF ROOM

• We've talked about all sorts of numbers: integers, rationals, reals, imaginaries, complex ... . Confused? Overwhelmed? 

Click to enlarge.

Here's Matt Parker of Numberphile with an overview of "All The Numbers".

Note that other lists of types of numbers might vary in the number and contents of categories. Most Numberphile entries probably are consistent with this one. But all categorizations of types (of anything) involve personal interests and preferences to some extent.

• Here's Leonard Bernstein to demonstrate how music changes as new harmonics come to be included:


• If you are interested in learning more about music, you might like this these resources:
-- Book: What to Listen For in Music, by Aaron Copland, 

• More about Pope Leo XIII, mentioned at the bottom of page 223 of our text, from Wikipedia: "As soon as he was elected to the papacy, Leo XIII [pope from 1878 until his death in 1903], worked to encourage understanding between the Church and the modern world. When he firmly reasserted the scholastic doctrine that science and religion coexist, he required the study of Thomas Aquinas[20] and opened the Vatican Secret Archives to qualified researchers, among whom was the noted historian of the Papacy Ludwig von Pastor. He also refounded the Vatican Observatory "so that everyone might see clearly that the Church and her Pastors are not opposed to true and solid science, whether human or divine, but that they embrace it, encourage it, and promote it with the fullest possible devotion."[21]

Resources promised at Class #6: 

• BBC documentary about Fermat’s Last Theorem:   https://www.dailymotion.com/video/x223gx8

• Andrew Wiles's proof of Fermat’s Last Theorem from the Annals of Mathematics Journal (1995). The first 3 pages are a brief synopsis as introduction; the next 10 pages are a somewhat more detailed synopsis.  The next 90 or so pages is the actual proof.   I (Steve) cannot understand even the first 3 pages. 

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