Monday, September 28, 2015

What are Amicable Numbers?

Two numbers, k and j, make are amicable if the sum of the proper divisors of k is equal to j and if the sum of the proper divisors of j is equal to k.

History of the Amicable Numbers:

Although the equation that derives pairs of amicable numbers was popularized by Fermat and Descartes in the early 16th century, the concept and derivation of these unique numbers was first investigated by an Iranian mathematician named Thābit ibn Qurra (826, 901). Qurra was also one of the first mathematicians to describe the phenomenon that we now know as the Pythagorean Theorem. 

After Fermat and Descartes studied these unique numbers and rediscovered the pairs, Euler extended their work in the 18th century and discovered dozens of new pairs. After years of extension on these pairs of numbers, as of 2007, there are 12,000,000 known amicable pairs.

Interpretation of the Amicable Numbers:

Let's try to better our understanding of this definition by looking at an example. Firstly, the "proper divisors" of a number n are all of its divisors excluding n itself. Normally, we would list the divisors of the number 6 as

1, 2, 3, 6

However, the proper divisors of 6 (for this example we will denote this by p(6) ) would be listed as
p(6)= {1, 2, 3}

So, now that we know what a proper divisor is, we can continue interpreting the definition of an amicable pair.  Using our proper divisor notion, it follows that two numbers, k and j, are amicable if
Σp(k)=j and Σp(j)=k

The numbers 220 and 284 are  amicable, let's see why that is.

p(220)= {1, 2, 4, 71, 142}
p(284)= {1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110}

From the definition, it should follow that Σp(220)=284 and Σp(284)= 220, let's double check:

1 + 2 + 4 +71 + 142 = 284
1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 220.

So, we can see that 284 and 220 are amicable numbers.

Here is a great NumberPhile video for more explanantion on amicable numbers:

Finding Amicable Numbers:

So many great mathematicians studied amicable numbers because of the difficulty of finding these numbers. Part of this difficulty lies within the relationship between amicable numbers and prime numbers. This can be seen in Euler's rule for finding amicable numbers as follows:

"Euler's rule is a generalization of the Thâbit ibn Qurra theorem. It states that if

p = (2(n - m)+1) × 2m − 1,
q = (2(n - m)+1) × 2n − 1,
r = (2(n - m)+1)2 × 2m + n − 1,
where n > m > 0 are integers and p, q, and r are prime numbers, then 2n×p×q and 2n×r are a pair of amicable numbers."

Where Do We See Amicable Numbers?:

 Although there is no direct application of these pairs of integers, amicable numbers have a significant historical impact. Not only did the search for these pairs attract so many recognizable mathematicians over time, but they were representations of "friendship." In Genesis 32:14, Jacob gifted Esau with 220 goats, a number chosen to exemplify his love. 

Another instance in which amicable pairs were used to portray positive relationships was in Medieval horoscope projections. If two people were given a horoscope containing the numbers 220 and 284, then this indicated a future romance between them. These whimsical applications of amicable numbers further emphasizes the natural attraction that people have for these pairs.

Sunday, September 13, 2015

The Influence of Greek Mathematics on Modern Proof Methods

As mathematics progressed from simply being a field of computation and quantitative observation, the techniques of proof that we use today began to develop. The first historical observation of what we today consider a mathematical proof was during the time of ancient Greek mathematicians. The contribution of proof was incredibly vital in the progression of the field of mathematics.

Editor of the Ancient History Encyclopedia, Cristian Violatti, claimed that there were three proof techniques that arose during the development of ancient Greek mathematics:

  1. The technique of abstraction
  2. The technique of generalization
  3. The art of deductive reasoning
The first technique contributed by the Greeks, abstraction, is a key quality of communicating in modern mathematics. Before the concept of abstraction was introduced to mathematics, each problem had to be approached as a new, unique situation. Problems like the length of rope needed to create a 4-sided fence or the angle to which the peak of a roof had to be constructed were seen as unique problems that had to be solved in context. However, with the development of abstraction, these every day problems could be taken out of their real world situation and be solved using visual and mathematical representation.

Another technique that was contributed was generalization. The concept of generalization allowed mathematicians to make general claims about situations that were true in all cases. An example given by Violatti was the Pythagorean Theorem. For example, if an applicable real-world situation could be abstracted as a 3:4:5 right triangle, the theorem would apply to this abstraction. However, with the use of generalization, mathematicians can now claim and prove that theorems, such as the Pythagorean Theorem,  can be applied to all situations. As math students, we see the use of generalizations every day in our studies. We have the tools to prove statements with universal quantifiers and for multiple cases thanks to the progress that the Greeks made towards modern proof technique.

The last technique that was mentioned by Violatti was deductive reasoning. Since the technique of generalization allows us to prove claims for general cases of mathematical situations, we are then able to use those proven claims to produce desired conclusions. This method of deductive reasoning that was established by the Greeks has been used to perpetuate every field of mathematics since it allows us to use what we already know in order to create new conclusions and mathematical relationships.

Without the contribution of new proof techniques by the Greeks, modern proof techniques would not have been established and the field of mathematics could not have progressed as far as it has.
Cristian Violatti. “Greek Mathematics,” Ancient History Encyclopedia. Last modified September 24, 2013. /article/606/.

Thursday, September 3, 2015

What is Math?

As an aspiring mathematician, I often get asked, "Lindsay, why do you like math? Isn't math just equations and numbers that you have to memorize to pass an exam?" I am then forced into defense mode in which I must defend the subject that is so near and dear to my heart. However, this is such a difficult thing to do. Sure, the quadratic equation and triangle congruence rules that we had to memorize in high school fit under the umbrella of "Mathematics," but that's not all that it includes.

It is my understanding that mathematics is a collection of tools that we use to quantify and describe the world around us. We use mathematics very similarly to how we use language. Using language, we can identify objects, convey ideas, and argue. Math can be used in the exact same way when communicating scientific ideas, defining mathematical objects, and proving theorems. The most interesting relationship between language and mathematics is that both can be utilized to describe events and objects that do not exist in the physical universe. Language can be used to create poetry which describes abstract emotions or fictional events. Similarly, mathematics can be used to describe things like how a 2-dimensional circle would look if it were projected on a 1-sided Klein bottle (like what Dr. William Dickinson at GVSU researches.) I think that, at its very root, mathematics is a form of communication.

A very large turning point for math was when mathematical concepts could be argued and verified through what we all now recognize as a proof. This began in the field of geometry and this method of logical argument showed that math concepts were concrete and could be verified. The conceptual topics that were being proved early in mathematical history were rooted in applications. Another key turning point for math was when observable events and objects were abstracted into general cases. This pushed the field of mathematics from simply being computation and quantification into a field that could describe both concrete and abstract situations.