Learning:
The Journey of a Lifetime
A Cloud Chamber of the Mind

March 2006 Mathematics Notebook

An Example of a "Learning Process" Journal

2:30 PM Ballina NSW Australia

This is a continuation of my notes for "Dr. Riemann's Zeros".

Chapter 3 New numbers for old

• "But for mathematicians nothing is unimaginable, and if the correct steps in reasoning lead to an unfamiliar or counter-intuitive answer, they sometimes see that as the starting point for a new journey." [p. 39]
• "The Greek mathematician Diophantus, who flourished around AD250, devised a type of equation which is now called Diophantine, with solutions in whole numbers.

• "Meanwhile the curriculum kept rolling on, and I could see that I couldn't stay behind ... I would have to pay attention to the next topic." [p. 40]
  
  
• "When mathematicians create new types of object, they like to devise or discover the rules that govern relationships between those objects. ... any new numbers, such as complex numbers, which are outside the real numbers should have their own set of rules." [p. 46]
  
  
• "The numbers we have all grown up with are just one of many number systems in mathematics. In fact, even within the familiar real numbers there are subsets such as the integers and the fractions (also called rationals, because they are depicted as ratios)." [p. 47]
  
• "The next step is crucial for understanding the Riemann zeta function. It is possible to use complex numbers (a + bi) in series. ... We can use z, as a complex number, in an expression such as . Furthermore, we can put it in a series, as . ... There's no need to worry for now about understanding what this series actually means. ... This series is like the Euler zeta function, but with complex numbers instead of s. And this is the important change that Riemann made, to come up with the Riemann zeta function." [p. 48]
  
  
• "It's because Riemann used complex numbers instead of real numbers, with s being a number of the form a + ib , that the behavior of the Riemann zeta function is far more mysterious than the behavior of the Euler zeta function." [p. 48]
  
  
• "Clearly, there was a time - now long gone - when educated men would see no shame in incorporating mathematics into their cultural lives. They had discovered that studying mathematics can be its own reward." [p. 51]
  
  
• "Artie Shaw, a giant among jazz composers and performers, suddenly abandoned public performance and plunged into an intensive course of self-education. Among the subjects he explored was mathematics." [p. 51]

• "There was never any question of passing or not passing - for you either knew the subject (in which case there was no doubt about it at all and you got a perfect score of 100%) or you didn't (in which case you had no business taking the exam until you had finished learning waht you had to in order to know it) ..." [p. 51]