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

August 2005 Mathematics Learning Log

An Example of a "Learning Process" Journal

Friday August 26, 2005
Learning Log Number 2

6:50 am Lethbridge, Alberta

The sky is clear this morning. The early sunrise reminded me of many that I watched in Australia while looking over the ocean. The blending of many shades of orange was very peaceful, particularly since there was no wind. I now have a hot cuppa beside me and am about to make some notes about number and hopefully about mathematical proof and the relationship between mathematics and science. Early morning is prime time for me and the sound of our little fountain outside the window is soothing. Now, to begin ....


The Natural numbers (sometimes referred to as the counting numbers) consist of the positive integers. They begin with the number one and proceed in increments of one, creating two, three, four and so on. It is easy to show that there are an infinite number of such numbers. My content notes from the previous session cover material from the book "The Art of the Infinite: Our Lost Language of Numbers" by Robert & Ellen Kaplan (2003), pages 1 - 12.

My next step will be to make a few notes about the remainder of chapter one "Time and the Mind" [p. 12 - 28] and then to follow this with notes about chapter two "How Do We Hold These Truths?" [p. 29 - 56].



The Natural numbers consist of the positive integers beginning with the number one, and for each number there is a successor, created by adding one to that value.

However it is also natural to think of the result of taking something away from a collection. This might be sheep or sheckels. If I have four sheep and I give you one sheep, then how many sheep do I have left? At first the idea of taking a large number from a smaller number was simply forbidden as not making any sense, but in the world of commerce this was quickly incorporated into the idea of owing or of having a debt. This led, eventually, to the idea of negative numbers, and to the mathematical idea of subtraction. Further development involved the idea of the number zero, and the entire collection of positive integers, negative integers and zero were referred to as the Integers and were represented by the letter Z (from the German word for number, Zahl). At this point we have an infinite collection of numbers and two operations, addition and subtraction.

Although the exact history of the evolution of the concept of number to include the operations of multiplication and division will never be known with any certitude, we are able to see the consequences of such operations. Importantly, the operation of division gives rise to a new kind of number, now called a rational number. The collection of all rational numbers is represented by the letter Q (for quotient). A rational number is defined as the result of dividing one integer by another, where the special case of division by zero is not allowed. Such rational numbers have a number of interesting properties.

Theorem: Between any two rational numbers lies another.
Proof: Let and be any two rational numbers, where the first number is less than the second. That is, .

Create a new number which is the average of the two numbers.

Then , which is also a rational number since a,b,c, and d are all integers.

It remains to show that .

Multiplying all terms by 2bd gives

which is true since .

This obvious property of the rational numbers is stunning. It is saying that no matter how close together I have two rational numbers, there is always room for another that is between them. In fact it is now easy to see that there are an infinity of such numbers since there are an infinite number of ways of creating such an intermediate number. This is sometimes expressed by saying that the rational numbers are dense on the number line.

Theorem: There exist numbers that are not rational.
Proof: Consider a right triangle with sides of length 1. Then the length of the hypotenuse is not a rational number.

This may be proved my contradiction. That is, assume that the length of the hypotenuse is a rational number that has been reduced so there are no common factors: .

Then .

. That is is an even integer, and thus p is also an even integer. Let it be 2m.

Then , and

Therefore q is also an even integer, contradicting the original assumption. Thus this number (which we symbolize as ) is not a rational number. We call such a number irrational. It is relatively easy to show that there are an infinite number of such irrational numbers as well.

Notation: Rational numbers may also be expressed using a decimal notation. Any such rational number will then have an expression that is either finite (e.g. ) or which repeats in an endless cycle (e.g. ). This leads to the conclusion that if one knows that a number has a nonrepeating decimal expansion then it must be irrational.

There are an infinite number of easy ways to create an irrational number. Simply give a rule for writing the denominator that is an infinite nonrepeating number.

The collection of all the rational and irrational numbers are called the Real numbers, and are represented by the letter R.

The idea of taking the square root of a negative number gives rise to another kind of number, called imaginary and to the more general idea of a complex number, represented by the letter C. Such a number is not positive, since a positive times a positive is still positive, and it is not negative since a negative times a negative is also positive, and it is not zero. Therefore it does not lie on the Real number line. We currently represent this by means of another line that intersects the Real number line at zero.

It is difficult not to become excited by the richness of these number types and of their properties.

End of chapter 1. [p. 28]

Here are a few quotes from "The Art of the Infinite: Our Lost Language of Numbers" that I enjoyed:
"Exhilarated by its widened conception of number ... " [p. 14] Exhilaration is a word I rarely see in mathematics textbooks, particularly at the k-12 level. Sad.
"It is this artistic motivation and reckless commitment to whatever consequences follow that is the mathematician's real tetractys, the sign to kindred spirits across millenia; and that is what makes for the glories and despairs of mathematics." [p. 19]  
"The irrationals lay undiscovered in the body of mathematics as the system of tectonic plates lay undiscovered in the earth's until recently: both were there to be found, and who knows what other systems may still operate unknown?" [p. 25]  
"Then you think to yourself: with just a handful of digits - some before a decimal and some after - I can invent a number most likely never thought of before. Invent or discover, discover or invent - or do numbers evolve organically, like forms of life, when demands and conditions coincide?" [p. 25]  

The title of this first chapter is "Time and the Mind". The idea is that it takes an infinity of time to write an infinite number. Thus time is inextricably bound up in mathematics.

9:00 am

Total elapsed time: 2 hr 10 min.

Time for a break. This has been fun. I have been totally immersed in the math for just over two hours and time has stood still. But I have yet to make some notes about the second chapter.

Total elapsed time for the day: 2 hr 10 min