Communicating Math: Amicable Numbers

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Amicable (which means friendly) numbers were discovered by Pythagoreans a really, really, really long time ago (see, even old guys liked finding “cutesy” math things). It is said that their properties stem from Pythagoras’ response to the question, “What is a friend?” He responded, “A friend is one who is the other I.” Now most of us here in the 21st century hear that response and say, “Huh?,” but when we dissect two amicable numbers, it becomes much clearer. There are many ways to define or describe Amicable numbers, but the clearest definition I’ve found so far is this: Two numbers are called Amicable (or friendly) if each equals to the sum of the proper divisors of the other (proper divisors are all the divisors excluding the number itself).   

The first pair of amicable numbers ever discovered are 220 and 284. Let’s take a look:

220: 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110, 220

284: 1, 2, 4, 71, 142, 284

I put the final factor of each number in italics to show that these are not proper factors of each number, so when summing the factors we will not be adding these. If you plug the factors 1-110 into your calculator and add, you should get 284 as the answer. Similarly, if you plug the factors 1-142 into your calculator and add, you should get 220 as the answer. And there we have it, Amicable numbers.

Thabit ibn Qurra, braniac he was, decided to take this a step further–he proved that numbers of the form T(n)=3*(2^n)-1 (now called 321 numbers) are also very special. If the 321 number for n and n-1 is prime and 9*2^(2n-1)-1, then the numbers (2^n)*T(n)*T(n-1) and (2^n)*(9*2^(2n-1)-1) are amicable. Don’t ask me how he managed to figure this all out–he must have had a lot (and I mean a lot) of time on his hands! And, even after all this work that Thabit did, the only known examples of this are for n=2, 4 and 7. That’s it. Check ’em out and see for yourself! (Hint: n=2 is our 220 and 284 example :))

For many years, this was the only known pair (220 and 284) of amicable numbers. It wasn’t until 1636 that Pierre de Fermat discovered another: 17296 and 18416. Descartes contributed the third pair: 9363584 and 9437056, and from there Euler took over, finding 63 pairs. Today, roughly 11994387 pairs of amicable numbers are known. 

 

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Weekly 2: History of Math, The House of Wisdom

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For my weekly work I did some research into the House of Wisdom and its influence on mathematics. To begin, the House of Wisdom was a library, translation institute and school established in Baghdad, Iraq. Works on agriculture, mathematics, philosophy and medicine were translated into Arabic here.  The House of Wisdom helped transform Baghdad into a hub for the study of humanities and for sciences, including astronomy, chemistry, zoology and geography, as well as alchemy and astrology. It is said that the House of Wisdom was the key institution in the Translation Movement and was considered to have been a major intellectual center of the Islamic Golden Age (Institute Mohamed Ali). This is simply a brief background on the House of Wisdom–next I’ll get into the role it played in mathematics.

The Abbasid Caliph, al-Ma’mun, established the House of Wisdom and invited Al-Khawarizmi (who will later come to be known as the Father of Algebra) to come and help him with something. He wanted to prove Allah’s existence through the “complexity and beauty of” mathematics.  Al-Khawarizmi began by translating ancient Greek and Indian texts. From the great Indian book on math, The Opening of the Universe, al-Khawarizmi adopts the idea of the zero as a number. This opened up a whole new world of mathematical possibilities and complexities unknown to the world at this time. The old Roman numeral system made more complex math almost impossible, but with a number system that includes 0, al-Khawarizmi introduces new ideas. Areas of mathematics such as algebra and geometry of the Greeks are developed, which eventually lead to math such as trigonometry and calculus. 

However, he still has a problem. Zero cannot be proven to exist using math, because even though the Indian texts he’s translated insist that zero divided by zero equals zero, al-Khawarizmi knows that dividing anything by zero is impossible. So, he decides that zero must simply be accepted without being proven–much like the existence in Allah. al-Khawarizmi is later cited by European mathematicians by a name that much more closely resembles ‘algorithm’ than it does his actual name. The mathematical word ‘algorithm’ is derived from his name, and, as many know, means complex, mathematical formula. Not only is he remembered by this, but also for his book titled Kitab al-Jabr wa-l-Muqabala, in which he lays out the principles of Algebra.

The House of Wisdom hosted scholars and researchers of all fields and studies, not just mathematics. However, the discoveries that were made by al-Khawarizmi truly provided the foundation of mathematics as we know it today. His discovery of the number zero and the concept of algebra laid the basis for almost every other type of mathematics that is taught in schools across the world. Therefore, the role of the House of Wisdom in the history of mathematics is more fundamental and crucial than I (and I’m guessing most others) ever realized. 

 

 

 

Daily 2: LCM & GCF

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For my daily work I took a look at the LCM (least common multiple) and GCF (greatest common factor) to see if I could come up with any connections or observations. Here are some of the notes I took while working on this task…Image

So, to begin with, I just started choosing random integers (ones that I knew were not prime, because then their greatest common factor would be one and their least common multiple would be their product) and determined their GCF and LCM by simply listing out all of the factors of each integer and some of the multiples and seeing where they overlapped. I was then thinking about it and thought there had to be a way that made more sense, or a way that would better illustrate a connection between the two. I decided to try prime factorization on the two integers to see what those factors presented.

First I determined how to find the GCF using the prime factorization results. I noticed that it had to do with the factors that both integers had in common. For example, 10 and 12 have 2 in common (10 has 2 as a prime factor once, while 12 has 2 as a prime factor twice). Since they only overlap once, 2 is the GCF. However, for the integers 12 and 18, they both have at least one 2 and one 3, so the GCF is 2*3=6. I tried this for the remainder of my integers and found it to be true, so I moved onto LCM. Since I had some idea of how to approach this at this point, I began by trying to just multiply the common factors (from my prime factorization), but realized this wasn’t quite right. So, after a bit of tinkering, I finally came up with this: you multiply all of the prime factors together, unless there are any overlapping ones. Then you just use that number once. For example: 9, 12 (9: 3, 3; 12: 2, 2, 3) 2*2*3*3=36. All of the prime factors are multiplied except for one of the overlapping threes.

I found this website, which does a great job (in my opinion) representing this relationship visually.

http://www.lamath.org/journal/vol5no1/Venn_Diagrams.pdf

So how does this tie in with axioms? Let me tell you…

An axiom is a statement or proposition that is regarded as being established, accepted, or self-evidently true. So, they are guiding lights to doing mathematics. Euclid proved entire postulates using only the few axioms he had predetermined. Likewise, using the prime factorization in the LCM/GCF exploration, I used basic rules (axioms) to determine what constituted as a prime number and what I could therefore do when dividing the integers further and further. Basic statements about what are prime numbers and division allow us to expand our explorations and practice to new things, like prime factorization, GCF and LCM, for example. Then, once I had finished coming up with a relationship, I wrote down the basic rules I had created for doing so. Thus, creating more axioms.

 

What is Math?

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When I was younger I used to think that math was all about formulas, equations and numbers–and that it always had a “right” answer. I had a hard time coming up with those “right” answers, and often felt discouraged and dumb based on the accusations and actions of my teachers. So, I decided to write math classes off as “Cs get degrees” courses (just kidding, I was never someone who could settle with a C, so I just cried about the fact that I would never understand :)). Who needed a subject that just made you memorize equations and plug-and-chug data with no real goal in sight?

This attitude carried through my elementary and middle school years, so I saw no reason why it shouldn’t continue right through high school, too. That was until my freshman year geometry class. I was expecting day after day of boring notes and homework problems, but what we got instead were group projects, collaborative assignments and lessons in which we were supposed to prove theorems. My teacher did the impossible–he made math fun. I loved the challenge of being expected to work through difficult problems with classmates and see what we could discover on our own. It was so much cooler to take what I knew and shuffle it around until something new emerged. And working with classmates made everything possible. There were times when I had no idea what to do next (these would have been the times in the past where I would have given up and called it good enough), but thankfully had two or three other minds to bounce ideas off of and gain new insight from. I learned that math was more than I had ever thought. Math literally helped explain the world around me, and applied to nearly every facet of my life. Sure, it included things like numbers, variables, operations and solutions like I mentioned before, but it was more about thinking critically, problem solving and working collaboratively with others than I had ever thought before. And so began my love affair with mathematics.

I recently had another “aha” moment during my semester of teacher assisting in a sixth grade math classroom. I realized the power of justifying one’s response or reasoning when it comes to math. My favorite questions (and likely my students’ least favorite question) quickly became “why?” Why does that work? Why is that the solution? Why are we able to do that in mathematics? It soon became habit for them to have their explanation prepared to share with myself and the rest of the class. It was wonderful to see them explain their thoughts to their peers using pictures, words and gestures–it was like they were now the teachers!

So, I guess I would conclude that “math” is a slightly challenging word to define. It’s almost just as much data and numbers as it is teamwork and thought processes.