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

February 2006 Mathematics Notebook

An Example of a "Learning Process" Journal

The intent is to continue making notes for Coincidences, Chaos, and All That Math Jazz. Part II "Embracing Figures" is about numbers and will have some bearing on my interest in Number Theory. The entire book is riddled with puns and word play.

Chapter 4 Secrets Held, Secrets Revealed.

• Idea: Although it is easy to multiply two numbers together it is, in practice, impossible to determine the two prime factors of a large number. In principle this could be done by trial and error, but simple calculation shows that this would take longer using any known method than there are seconds in the history of the universe. Although this idea has been well known for centuries, it is only recently that we had the insight that this fact could be used to create unbreakable cipher codes. One needs to know these two prime factors in order to decode the message.
  
  
• Here is the overall coding/decoding process (known as public key cryptography).
  • Convert each character in the message to a number.
  • String all of these numbers together into one huge number.
  • Raise this huge number to a very large power giving an enormous number
  • Divide this enormous number by another large number N which only the sender and receiver know its two prime factors
  • The remainder from this division is sent as the message.
  • The receiver takes this remainder and raises it a power involving the two prime factors.
  • This large number is then divided by N.
  • The resulting number is then converted back to characters using the same coding system used in the first step.
    
    
• The important point is understanding WHY this works. I do not have this understanding at the moment.
  
  
• The next important point is trying to understand why finding the factors of a large number is so difficult.
  
  
• "Schools often present mathematics in dull and distorted ways. In particular, many students are left with the impression that all of mathematics is understood and all mathematical questions have been answered." [p. 73 - 74]
  
  
• "For thousands of years, people have loved numbers and found patterns and structures among them." [p. 74]
  
  
• "THE GOLDBACH CONJECTURE: Every even number greater than 2 is the sum of two primes." [p. 75]
  
  
• "TWIN PRIME CONJECTURE: There are infinitely many twin primes." [p. 76]
  
  
• Think of any whole number. Perform the following easy steps:
  • If the number is even, divide it by 2
  • If the number is odd, multiply it by 3 and then add 1
  • Repeat the process unless the answer is 1.

Is there a number where this process fails to eventually reach 1? No one knows. [p. 76 - 77]

Chapter 5 Sizing Up Numbers

• "... the process of counting has a potency far beyond its seeming simplicity. Few abstract ideas in human history have given us greater power to see the world in a nuanced way. [p. 79]

• Most natural numbers (i.e. the counting numbers, the positive integers) "are really big and thus have absolutely no true meaning to us." [p. 80]
  
  
• "Isn't it amazing that the entire age of the universe in trillionths of a second can be measured with a number less than 30 digits long? A 500-digit number, for all practical purposes, doesn't mean anything. It's just too big." [p. 82]
  
  
• Here are some examples to provide a frame of reference for large numbers:

Thousands	1,000	The number of stars visible to the naked eye in the night sky.
Millions	1,000,000	A high-resolution digital picture has a few mega-pixels. A million dollars would weigh about 1600 pounds.
Billions	1,000,000,000	There are about 6.4 billion people on earth. The age of the universe is about 13.7 billion years. A gigabyte means roughly a billion bytes.
Trillions	1,000,000,000,000	The US national debt is 7 trillion dollars.
Quadrillions		50 fold of paper are 128 million miles thick (greater than distance to sun). Repeated doubling is an explosive process.
		Number of atoms in the universe.

• But these are all relatively small numbers!

Chapter 6 A Synergy Between Nature and Number

• "... counting often opens our eyes to new insights ... we might make the effort to move from the qualitative ('There are many spirals') to the quantitative ('The exact number of spirals is ...')." [p. 103]
  
  
• "When we open our eyes to spirals, we find them everywhere." (pineapples, sunflowers, pine cones) [p. 104]
  
  
• Fibonacci numbers: Beginning with 1, 1 then each successive number is the sum of the two preceding numbers.
  
• "Why do so many of nature's spirals conform to this sequence of numbers? Actually neither mathematicians nor biologists have a complete explanation." [p. 108]
• The ratio of consecutive Fibonacci numbers approaches a limit: 1.61803... which is also known as the Golden Ratio.
• If one begins with the two numbers 2, 1 and add the two preceding numbers to get the next, we end up with a different sequence that is called the Lucas sequence after a French mathematician.
• It can be shown that the ratio of consecutive Lucas numbers also approaches the Golden Ratio. It can even be shown that if you begin with any two numbers and use the same principle the ratio of consecutive numbers will approach the Golden Ratio.
  
  

  Beginning with any two numbers as seeds is one way to extend the idea. Another way is to try playing with triads of numbers. Lets look at the first few numbers. 1, 1, 1, 3, 5, 9, 17, 31, 57, 105, ... Using Excel: 1 1 1 3 1 1 3 5 1.666667 1 3 5 9 1.8 3 5 9 17 1.888889 5 9 17 31 1.823529 9 17 31 57 1.83871 17 31 57 105 1.842105 31 57 105 193 1.838095 57 105 193 355 1.839378 105 193 355 653 1.839437 193 355 653 1201 1.839204 355 653 1201 2209 1.839301 653 1201 2209 4063 1.839294 1201 2209 4063 7473 1.839281 2209 4063 7473 13745 1.839288 4063 7473 13745 25281 1.839287 7473 13745 25281 46499 1.839286 13745 25281 46499 85525 1.839287 25281 46499 85525 157305 1.839287 46499 85525 157305 289329 1.839287 85525 157305 289329 532159 1.839287 157305 289329 532159 978793 1.839287 289329 532159 978793 1800281 1.839287 532159 978793 1800281 3311233 1.839287 978793 1800281 3311233 6090307 1.839287 1800281 3311233 6090307 11201821 1.839287 This also appears to approach a limit, but it does not look like it is the Golden Ratio. Let's try triads again, but starting with 2, 1, 1. 2 1 1 4 1 1 4 6 1.5 1 4 6 11 1.833333 4 6 11 21 1.909091 6 11 21 38 1.809524 11 21 38 70 1.842105 21 38 70 129 1.842857 38 70 129 237 1.837209 70 129 237 436 1.839662 129 237 436 802 1.83945 237 436 802 1475 1.839152 436 802 1475 2713 1.839322 802 1475 2713 4990 1.839292 1475 2713 4990 9178 1.839279 2713 4990 9178 16881 1.83929 4990 9178 16881 31049 1.839287 9178 16881 31049 57108 1.839286 16881 31049 57108 105038 1.839287 31049 57108 105038 193195 1.839287 57108 105038 193195 355341 1.839287 105038 193195 355341 653574 1.839287 193195 355341 653574 1202110 1.839287 355341 653574 1202110 2211025 1.839287 653574 1202110 2211025 4066709 1.839287 1202110 2211025 4066709 7479844 1.839287 2211025 4066709 7479844 13757578 1.839287 This also seems to approach the same limit, even though it is not the Golden Ratio. One last extension. Let's try working with quads. 1 1 1 1 4 1 1 1 4 7 1.75 1 1 4 7 13 1.857143 1 4 7 13 25 1.923077 4 7 13 25 49 1.96 7 13 25 49 94 1.918367 13 25 49 94 181 1.925532 25 49 94 181 349 1.928177 49 94 181 349 673 1.928367 94 181 349 673 1297 1.927192 181 349 673 1297 2500 1.927525 349 673 1297 2500 4819 1.9276 673 1297 2500 4819 9289 1.927578 1297 2500 4819 9289 17905 1.927549 2500 4819 9289 17905 34513 1.927562 4819 9289 17905 34513 66526 1.927564 9289 17905 34513 66526 128233 1.927562 17905 34513 66526 128233 247177 1.927562 34513 66526 128233 247177 476449 1.927562 66526 128233 247177 476449 918385 1.927562 128233 247177 476449 918385 1770244 1.927562 247177 476449 918385 1770244 3412255 1.927562 476449 918385 1770244 3412255 6577333 1.927562 918385 1770244 3412255 6577333 12678217 1.927562 1770244 3412255 6577333 12678217 24438049 1.927562 3412255 6577333 12678217 24438049 47105854 1.927562 Convergence yes, but also to a new value. I wonder what would happen if I extended the number of terms to a large value? Rather than read about mathematics, the last few minutes have been spent doing mathematics. We need to find ways to encourage this type of activity in schools.