I recently read the book How to Bake Pi by Eugenia Cheng. Cheng discusses what math really is and in an easy to read context that keeps you intrigued. Her goal isn’t to discuss math in terms of what is learned at the high school level, but rather seen through category theory, something she feels very passionate about. Through category theory, math becomes less about numbers and formulas and more about how we see and understand things. She turns math, something that many people find difficult, into something tangible and describes how math is actually here to make the hard things easier. Her perspective on what math is is the focus of the book. She relates the field of math to cooking, something she is also very knowledgeable in.
Math relates to cooking in that it is made up of ingredients and methods. It is defined by the techniques it uses to study things, and that the things it studies are determined by those techniques.
I really liked this book because she helped me to understand what math is in a different perspective from my own and was able to bring in real-life applications that made the book more interesting and relatable. It was an easy book to read because her personality stood out through the book and her humor made the read more intriguing. When I met Eugenia and listened to her discussion on the book, it made me appreciate the book even more because I could put a voice to the things she was discussing and I could tell how passionate she was about math, in particular category theory but also cooking. I would recommend this book to many people, even those who don’t study the field of mathematics. I think this book appeals to diverse readers and could clear up common misconceptions in mathematics, like what math really is and what its purpose is. I am glad that I had the background in math that I did before reading though because I had a deeper understanding in some of the material she was presenting. Reading this during my Capstone, I was able to understand topics she was discussing from the math classes at Grand Valley I have taken: hypothesis testing from Stats 312, proofs by contradiction and transitivity from Math 210, Fermat’s Last Theorem from Math 495, binary operations and identities from Math 310, axiomatized geometry from Euclidean Geometry, matrices from Linear Algebra….(the list goes on). This book was a great way for me to reflect on the many things I have learned in the past few years and see how they all connect and how they can be applied. Oh, and I also have a great collection of recipes now since there was one at the beginning of each chapter.
Some of my favorite quotes from the book:
“Mathematics is the study of anything that obeys the rules of logic, using the rules of logic.”
“Mathematics is like a marathon, the point is the journey itself, not the arrival at the destination.”
“Math works like Legos. You start with some basic building blocks and some ways of sticking them together, and then you see what you can build.”
“Category Theory seeks to illuminate math.”
Recently, our class discussed Galileo's findings and contributions to the field of mathematics. Although Galileo is most commonly known for revolutionizing physics and astronomy, it has been misunderstood how much he also contributed to the field of mathematics. But that leads us to an important question, Is math a science? If not, then what is it?
I feel that math is a science, specifically it's a language used to communicate and understand science. Science applies mathematics and is used to help explain scientific properties.
Let's look at an application that applies both mathematical and scientific principles: the Barbie Bungee!
The Barbie Bungee activity is a STEM activity that allows students to conduct an experiment, collect data, and then use the data to predict the maximum number of rubber bands that should be used to give Barbie a safe jump from a high, specific height. She should get as close to the ground as possible without hitting the ground; she's a daredevil.
Many different mathematical and scientific principles are used during this experiment. The distance to which the doll will fall is directly proportional to the number of rubber bands, so this context is used to examine linear functions. Depending on how you want students to expand on this activity, many different math and science standards apply to this activity such as creating a linear model or scatter plot and finding the best fit line on a graph and interpreting rates of change.
For our group's experiment, we used a Hulk doll instead of a Barbie to drop. The Hulk doll was heavier so we knew we needed a stronger bungee. We measured how high the bridge was so that we had a good estimate of how far the Hulk would fall. We used a meter stick in the classroom to set up simulation examples to help. We made a prediction of 23 rubber bands needed, which was a little bit of a guess. In the experiment, the first time the Hulk did just barely hit the cement. To improve, we should have had just one or two less rubber bands on our bungee. We did end up taking a couple off and launching again and it was perfect, just centimeters from hitting the ground.
Math and science are both integrated into this activity. Without mathematics, nothing can be calculated in this experiment and it wouldn't be able to exist.
Mathematics is the study of anything that obeys the rules of logic, using the rules of logic. Barbie agrees, math is an important element of the scientific method in order to carry out science.
Two numbers, k and j, 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.
Amicable numbers are two different numbers so related that the sum of the proper divisors of each is equal to the other number. (A proper divisor of a number is a positive factor of that number other than the number itself.For example, the smallest pair of amicable numbers is (220, 284); for the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, of which the sum is 284; and the proper divisors of 284 are 1, 2, 4, 71 and 142, of which the sum is 220.
The first few amicable pairs are: (220, 284), (1184, 1210), (2620, 2924), (5020, 5564)...
Ancient philosophers/mathematicians used friendly numbers to describe friendships between people (Aristotle in his work Ethics, for example). A pair of friendly numbers became a symbol of friendship.
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. Qurra was also one of the first mathematicians to describe the phenomenon that we now know as the Pythagorean Theorem.
Thabit ibn Qurra Ibrahim, who lived in Baghdad in the 9th century, came across an algorithm by which he was able to find more amicable pairs. He did operations on rows starting with the powers of two. He then found "friendly pairs" or amicable numbers by neighboring numbers and corresponding primes.
Pierre de Fermat and Marin Mersenne discovered, in 1636, the amicable pair
17296 = 16*23*47 and 18416 = 16*1151
and Rene Descartes found the third pair
9363584 = 128*191*383 and 9437056 = 128*73727.
In 1747 Euler, as usual, went into overdrive and produced more amicable pairs than anyone had done before him. He published a paper "On the amicable numbers" adding 30 more pairs; and then three years he had extended the list to 60 amicable pairs. 16-year old Nicolo Paganini found, in 1866, another amicable pair which was missed by the great mathematicians before him: 1184 and 1210.
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.
I have struggled with this question since I first started studying math as my major concentration. I feel it's because it is a loaded question with many interpretations. To me, math is the study of everything around us. It is how we quantify structures. It's a science that deals with logic. It is a measurement of the physical space around us. It is so much more then just a simple discipline or school subject.
I am not all that knowledgable when it comes to the early discoveries of mathematics.
5 milestones I can recollect:
Quantifying time and number systems in Egyptian times
Pythagoras and his theorem
Newton and his Laws
Math of the Scientific Revolution- Galileo etc.