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

September 2005 Mathematics Learning Log

An Example of a "Learning Process" Journal

Thursday September 1, 2005
Learning Log Number 6

5:50 am Lethbridge, Alberta

It is early morning and I have a fresh cuppa in hand. I read the next two chapters of "Prime Obsession" last evening. In an optimistic vein, I wonder if I can maintain this pace of two chapters each day (i.e. note making in the morning and reading in the afternoon).


I have begun reading "Prime Obsession" by John Derbyshire. I hope to stay with this book until the end and then return to Michael Spivak's "Calculus".

My next step will be to make a few notes about chapters 3 & 4 [p. 32 - 62].


Prime Obsession. Chapter 3 The Prime Number Theorem [p. 32 - 47]

"The main idea of a function is that some number depends on some other number according to a fixed rule or procedure." Often this rule is expressed as a formula, but it is perfectly acceptable for it to be expressed verbally as a process.

The values that the function takes as inputs are referred to as the domain of the function, and the values it gives as outputs are called the range. Another common phrase is to say that a function is a way to map a number to another number.

Quite often there is a restriction that each value in the domain must map to a unique number in the range. Such functions are called single-valued, or if the context is clear, simply functions where it is taken for granted that they are single-valued. There is a special topic called multi-valued functions that deals with the more general case.

One function that is of special interest in number theory is called the Prime Counting Function. It is symbolized by p(N). For any value of N (a Natural number), it gives the number of prime numbers up to N (inclusive). This function is very interesting, in large part because there is no known formula for giving this number. There are a number of procedures for algorithmically going through the numbers and noting which ones are prime and then adding the number of such primes. The procedures themselves are a subtopic of investigation.

Two very important families of functions for Number Theory are the logarithmic and exponential functions. Therefore it is important to become very familiar with these two types of functions. This also involves the special number , named for Leonard Euler.

Another important idea is that of an inverse function. An inverse function is one that returns the argument to its original value. For example, division is the inverse of multiplication, subtraction is the inverse of addition, integration is the inverse of differentiation, and exponentiation is the inverse of logarithmization.

The Prime Number Theorem:

Note that the PNT is not an equation, but a statement about asymptotic approximation as N gets very large.

Using the fact that when dealing with all the numbers up to some large N, most of those numbers resemble N in size. This allows us to make a few more "approximate" statements:

The probability that N is prime is

The N-th prime number is .

The above statements are examples of the general topic of the distribution of the prime numbers. This is a rich and open-ended topic that has attracted the interest of many famous mathematicians, including Gauss, Euler and Riemann.

Two noteworthing features of the distribution of prime numbers is :

  • there is a thinning out of the prime numbers as N gets very large
  • there is a certain quality of randomness to the distrubution.

Prime Obsession. Chapter 4 On the Shoulders of Giants. [p. 48 - 62]

Leonhard Euler (1707 - 1783) (Swiss) is the only mathematician to have two numbers named after him, one of which is the number . Euler spent much of his adult life in St. Petersburg. Derbyshire calls one of Euler's discoveries "the Golden Key", although I am not sure at the moment what this is.

Carl Frederich Gauss (1777 - 1855) (German) is recognized at the first person to see the truth of the Prime Number Theorem. One can spend hours browsing the Web looking at sites that discuss some aspect of Gauss's work.


Here are a few quotes from "Prime Obsession" that I enjoyed:
"Is there a rule, a formula, to tell me how many primes there are less than a given number? " [p. 34]

No. There are only asymptotic approximations.

I wonder what would happen if one flipped the question to be, "How many composite numbers are there less than a given number?". Then approach this recursively using a variation of the sieve of Erastothenes. Hmmm.

" 'Function' is one of the most important concepts in all of math, the second or third most important, I should think, after 'number' and possibly 'set'. " [p. 35] I like this idea of trying to group ideas/concepts into a few categories ranked by importance. What is the most important idea in a course? Most math courses/textbooks are just a (large) collection of concepts with little sense of what is most valuable.
"Most interesting functions, however, have some limits on their domain. Either there are some arguments for which the rule doesn't work, usually because you would have to divide by zero, or else the rule only applies to certain kinds of numbers." [p. 37]  
"Mathematicians generally get a feel for a particular function by working intimately with it for a long time, obsering all its features and peculiarities. A table or a graph rarely encompasses the whole thing." [p. 37]  

These two chapters are still setting the foundation for Riemann's work. The next chapter (5) discusses Riemann's zeta function. This looks like a major step forward.

7:30 am

Total elapsed time: 1 hr 40 min.

The next step is to read the next two chapters, 5 & 6 [p. 63 - 98]

Total elapsed time for the day: 1 hr 40 min