Skimpression

After reading my first book, I switched books with Morgan Sparber and read my second book titled Love and Math: The Heart of Hidden Reality by Edward Frenke. While discussing first books in class, Morgan's account of this book piqued my interest by depicting the book to be the journey of a young man going through his studied and his love of mathematics that drives him. Obviously being a math lover myself, I related well to this aspect. What I did not expect, however, was to find a part of math that I had not discovered myself; Frenke helped me fall in love with this part. He depicts math as more of an art and more creative rather than black-and-white as those starting off in mathematics sees it as.

Newton and Mathematics

Sir Issac Newton (1642-1727), though is largely attributed towards the advancement in physics, was a key contributor to the advancement of mathematics as well. It is said that the beginning of his education focused on law and it was not until he picked up an astrology textbook in which he did not understand the mathematics contained in it, that he became interested in math. Newton - though proclaimed to be predominantly self-taught - studied underneath mathematicians such as William Oughtred and John Wallis, and Descartes. He is greatly known for contributions such as differentiation and integration, at the time known as method of fluxions and inverse method of fluxions (later on Leibniz would ultimately define these methods as differentiation and integration).

The famous Leibniz vs. Newton dispute concerned who exactly came up with these fundamental calculus ideas. Followers of Newton claimed that Leibniz had glimpsed Newton's notes of his fluxions in 1676 during a visit which then lead to Lebniz publishing this pleigerized work in 1684, stealing the thunder from Newton.

Newton also came up with the Binomial Theorem, which states,

General Form: 

He discovered such a way to expand any binomial, fractions, complex, etc., when he exiled himself to keep from contracting the plague. His general form contains works of Pascal in which (n  / k) relates to numbers in Pascal's Triangle. The general Binomial Theorem form is present on Newton's tombstone in Westminster Abbey.

e : the Story of a Number (a review)

     Unlike a typical stiff, boring, Mathematics textbook one might come across in a classroom setting, e: the Story of a Number by Eli Maor (1994) humanizes the number e in way of depicting its long history in which it was "invented" (or discovered, whichever you believe). It balances equations and - as the book puts - pure mathematics with applied mathematics and the history of the mathematicians that have formed mathematics as we know (and love) today.

     I was (pleasantly) surprised that the book not only covered the number e and its brother ln (which I was expecting) but also covered various trigonometric topics, differentiation, anti-differentiation,  and other famous numbers such as pi and i. This book is very well rounded and very well written: most ideas written in layman's terms.

     Since most (including myself) have learned math through a series of building blocks, it is refreshing to explore along with Maor how those long ago learned mathematics. Also, reading of the lives of mathematicians was rewarding as well; it allowed the static "characters" in a typical math textbook to be more dynamic and relate-able. I would easily recommend this book even to those with average math background and skills, especially if one were newly starting their journey in the subject; the book does a fantastic job of giving a deeper explanation of how different mathematics links together.

     To conclude, I cannot say with confidence that I was eager to start this book, but come the end of it I was in full math-nerd mode, happy that I did.

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Extensions/Aspects I Found interesting:

-The "log base 10" being the normal base came from the mathematician Henry Briggs who suggested 

 

-“The logarithm function remains central to almost every branch of mathematics” (Maor 16). I have noticed this substantially while studying mathematics myself throughout these past few years. There have been many-an-equation in which log was included.

- That e can be written as

- logarithmic spiral: First studied by Descartes in 1638 and then by Jakob Bernoulli, the logarithmic spiral is quite interesting. For example, I had not known before that every straight line through the origin intersects the spiral at the same angle. This is the only curve that does this.

Also, that inverting the logarithmic spiral causes a mirror reflection upon itself. Though I have studied The Golden Ratio and Fibonacci's numbers before, I had not realized that the logarithmic spiral and the formers were the same. This got me to explore the logarithmic spiral in nature:

Credit

- Euler's Formula and Identity fascinated me. We discussed the topic in class however the book divulged much more information. Chapter Thirteen is titled most appropriately:

The talk about how amazing Euler's Identity was had me searching the web for their reactions to the famous identity:

(1) I like this video for it shows a way to describe the famous identity to those without much mathematical background

(2)This Video shows all of the components of such a nice equation.

Hippo takes a plunge

An in-class activity summarized and analyzed:

Our task was to calculate how many rubber bands were needed to allow a plastic hippo to bungee jump the high of a flight of stairs. The goal was to minimize error and optimize the distance of the fall without exceeding it (ie. not cashing to the floor).

We first attached one rubber band to the foot of the hippo and tested its maximum fall distance. We then added a second rubber band and did the same. We repeated this for the first five rubber bands and then the seventh. We noted that each rubber band averaged an additional 22 cm. to the fall distance. I, myself, speculated that with each rubber band added, the additional distance to the fall was slowly decreasing.

Since the height of the jumping platform to the ground was about 505.46 cm. (199 in.) we calculated that 22 rubber bands would suffice. We found that with seven rubber bands, the hippo dropped a total of 156.5 cm. By calculations, we determined that with 21 rubber bands, (156.5(3)) the hippo would drop a total of 469.5 cm. With an additional 22 cm an extra rubber band would allow - increasing our rubber band count to 22 - we speculated that the hippo would drop a total of 491.5 cm.. This leaves the hippo just 13.96 cm. off of the ground. We determined that with averaging the cm/rubber band, 13.96 cm. from the ground leaves room for unaccounted error.

Below is a recorded account of the bungee jump:

Above is an account of our second jump. It should be added that the first jump was roughly 40 cm. away from the floor however when dropped, the rubber bands bunched up and did not allow for a clean drop. Thus, we dropped again. On the second drop, we ended up a mere17 cm. from the floor. This leads to an error of 3.04 cm. away from the floor based on our calculations.

 This activity made me think of limits. When it came to the distance each rubber band added allowed the hippo to drop, the number, ultimately, slowly shrank. However I cannot imagine that if one were to add an obscenely large amount of rubber bands (x rubber bands tending towards infinity) that the last rubber band added would allow a distance of 0 cm more distance dropped. Thus, the limit of what each rubber band would contribute to the drop would have to be greater than zero, but less than the last distance recorded (being 18cm). I should add that I do not know this number and that more testing would be involved to calculate it but this is just a speculation.

Also, it had me wondering about elasticity. According to Anthony Coy, "When the rubber bands were heated, the particles stretched out, making them more elastic and able to withstand greater force" (2005). This has me curious, wondering what would happen to the limit a rubber band might add to the drop when they are heated: would the limit be greater than that of the room-temperature bands or would the limits be equal?

Below is an account for our rough data:

Al Kharaji Summation

Given the summation

find an equation that defines it.
I start by constructing the sequence xn.

 That is,

Using exploration and the series, I derived an equation that defined the above summation:

 See scratch work below:

First I tried different values of n. That resulted in the circled numbers: 1, 9, 36, 100, and 225. I noticed that all these numbers were perfect squares with the corresponding numbers 1, 3, 6, 10, 15, 21 as their square roots. These numbers form a sequence defined as xn above. I used this sequence to derive a way to find the summation of any n.

Tessellations

Above is the tessellation I completed in class. I will continue to expand on this project.
From the looks of it, there are two complete figures above:

Figure 1: Yellow Hexagon               
               Red Trapezoids
                Orange Squares
               

Figure 2: Yellow Hexagon
                Blue Parallelograms
                Green Equal-lateral Triangles
                Tan Elongated Parallelograms

Note: Though the figures can be determined in multiple different ways because of connecting pieces, to make things easier I have divided things as such.

Reasons for the tessellation being incomplete is because of time constraints and limited material. If I had enough time and pieces, however, I would construct three more of Figure 1 and attach them to the bottom of Figure 2. This would complete the circle of Figure 1's around Figure 2. From there, I would construct multiple Figure 2's and fill in such spaces around the six Figure 1's. This process could continue on and on without spaces between Figures.

I researched other tessellations and included them below:

                                                  
 Credit                                                 Credit                                        Credit                     

The Nature of Mathematics

What is an axiom? 

 According to the Merriam-Webster online dictionary, an axiom is,

In mathematics terms, an axiom is typically a proposition or rule that does not require a proof for supporting evidence of its truth. Axioms can take different forms and can also be used in other areas of study other than mathematics such as law, the sciences, etc..

What is the role of axioms in mathematics?

Axioms in mathematics play a large role since mathematics focuses on proving conjectures. These conjectures are proved in a similar way of building blocks: you must lay a foundation and build upon those blocks. These foundations typically contain axioms. When considering axioms as building blocks, these do not need to be proven every use but simply recalled when used. It reminds me of the idea of infinity and how if we had to prove each axiom with its use would also then call for a proof of the axiom's axiom, and this could go on for an unfortunate long period of time. This causes too much unnecessary work. Thus, axioms are very convenient for mathematicians.

What is Math?

What exactly is mathematics, you ask? If you were to ask me that question even two years ago, my answer would be extremely different. At that time, I was still completing the Calculus sequences (Calc 1, Calc 2, and Calc 3). At that time, math meant strict calculations: whether it be addition, deriving, or integrating. At that moment in my life, math was very literal and very black and white.

My answer now includes those original thoughts, but goes much, much deeper. Now, I believe Mathematics to be a subject that encompasses everything logical. Mathematics is an tool to problem solve, whether that problem be dealt with numbers or patterns excluding numbers (such that Discrete has taught me). It is a subject that can account for so much of our world and our existence. Mathematics is a tool that, if used correctly, can explain many natural and man-made occurrences that surround us.