Communicating Mathematics – Book Summary and Notes

Summary and Review

The Joy of X: A Guided Tour of Math, from One to Infinity by Steven Strogatz looks at a variety of mathematical concepts.  It starts with the basic idea of counting and ends with the abstract idea behind what infinity really is.  Along the way Strogatz presents math in a way that is can be understood by the non-mathematician world.  He talks about math in a way that would allow someone that to view it as the vast and intriguing topic that it is.

Strogatz discusses what it means for a number to be negative and what addition, subtraction, multiplication, and division really represent.  One of the best visuals I think that is included in the book is when the concept of imaginary numbers is brought up in the section on relationships.  He shows that multiplying by i is related to just rotating 90 degrees.  Another important point of discussion during this section is about how people think when presented with a math problem.  They just want to use the numbers they are given in a familiar way, instead of logically thinking about what those numbers actually represent and what they are trying to ultimately find.  Strogatz makes math seem less mystical by working through the proof of the Pythagorean theorem.  There are so many things about math that most people have just had to accept as true and don’t ever get to really see why they are true.  By working through this proof, it makes math less mysterious.

The book begins to wrap up with more abstract ideas.  Strogatz discusses calculus and the idea of infinity.  He makes some simplifications to these rather complex ideas which make the task of investigating them seem less daunting.  The use of visuals and real world examples allows people to see the beauty and the application of mathematics.

Overall, I would recommend this book to anyone that is looking into reading more about how mathematics is useful and how can be applied to the world.  It is an interesting read for a person that has a lot of background knowledge or just the math basics.

Notes

Part One: Numbers

In the first chapter Strogatz makes a statement about math that I feel is very powerful, “the right abstraction leads to new insight, and new power.”  He also states that math allows for creativity, but within certain sets of restrictions based on what you are working with.  He says that we “invent the concepts but discover their consequences.”  We can’t determine what the answers to our questions will be, but we can decide what questions we want to ask.

In the following chapters Strogatz discusses numbers and arithmetic, including why a negative times a negative is a positive.  He uses the idea of the opposite of an opposite like we did in class, but he looks at it like an enemy of my enemy is my friend.  In the same chapter he presents diagrams of the alliances leading up to World War I.  It is interesting to see graphically how the changes in alliances lead to the start of the war. A few chapters later, while discussing division, he presents a conversation with a Verizon Wireless representative and a customer.  The representative does not understand the difference between 0.99 dollars and 0.99 cents.  I thoroughly enjoyed this because one of my pet peeves is when an advertised price for an item is 0.99 cents.

Part Two: Relationships

In this section we are getting away from just looking at numbers and performing basic operations on them, to looking at equations and functions.  Strogatz talks about the beauty that is the variable x and how algebra works.  He goes from real world examples that are easy to understand in chapter 7 to looking at imaginary roots in chapter 8.  He presents them in a way that I have never considered before.  Multiplying a number by i is like rotating 90 degrees.  He says that the imaginary numbers live on a different number line perpendicular to the real numbers.

He examines how we tend to jump to conclusions because of the way we are use to looking at problems, which is another thing we discussed in class.  Some of our brains have been wired to try to add, subtract, multiply, and/or divide the numbers that are given in a story problem in order to obtain the solution.  However, sometimes it takes more thinking than just using the given numbers and manipulating them in one of the usual ways.

Another thing that Strogatz brings up that we talked about in class is al-Khawarizmi’s solution for quadratic equations.  Strogatz illustrates completing the square, matching up edges of rectangles and squares and finding the area of the missing section.  He also mentions that al-Khawarizmi did not find that the negative of the root is also a root.  The quadratic equation and other functions are examined.  Strogatz shows how functions transform things and how the different types of functions work.  One example he talks about are power functions and while discussing how rapidly they increase, he explains that that is why it is hard to fold a piece of paper more than seven or eight times.

Part Three: Shapes

Similarly to the way we illustrated it in class, Strogatz starts this part of the book by illustrating the Pythagorean theorem by creating squares along each side of a right triangle.  He also places four triangles around the square of c-squared which creates a larger square.  He then rearranges the four triangles inside the bigger square and ends up creating two smaller squares, a-squared and b-squared.  Finally, he begins a formal proof of the Pythagorean theorem.  Strogatz then goes on to discuses writing geometric proofs and the importance of them.  During the discussion he mentions Euclid and “The Elements.”  Strogatz brings up a variety of areas that Euclid has had influence over.

Moving on to a different topic, Strogatz talks about conics.  I had never really understand all of the geometric meaning that the foci of an ellipse has.  I really enjoyed reading about how the foci of a parabola can direct sound and light in a very specific way.  During this discussion he brings up another point that I like about mathematics, “There are no accidents. Things happen for a reason.” (pg 104)  Also, he shows that circles, ellipses, and parabolas are cross-sections of a cone.  I have thought about circles and ellipses in this way, but not parabolas.  The next chapter is all about sine waves.  It is fascinating that sine waves can be used to describe the movement of the smallest things that we know to exist all the way to the largest.

In the final chapter in this section, Strogatz discusses infinity.  He talks about how infinity was banned from mathematics, until Archimedes started thinking about it.  He also talks about pi.  He uses the idea of dividing a circle into infinitely many slices to determine the area.  He lays that slices next to each other making a rectangle.  I have obviously looked at dividing the area of a curve in this way, but I have never consider the same concept could be applied to the area of a circle.

Part Four: Change

This section starts with a discussion of calculus.  One of the first points he makes is about how students don’t realize just how important calculus is and what it is really about.  I feel like that could be said about most basic/intro math classes.  A lot of my friends say that math is just memorizing, but I like math because it isn’t just memorizing.  It is crazy how long it took people to connect integrals and derivatives.  Of course, it is also amazing that the two idea were ever even found.  Strogatz presents a problem about finding the best path to walk.  At the end he says that in order to solve the problem, geometry, algebra, and calculus are all needed.  Sometimes I forget how much math can go into solving one problem and how much of it I can do now without even thinking about it.  He also talks about how nature follows calculus.  Another example of how the world and math are naturally related.   Another example would be all of the Fibonacci spirals that are found in nature, like we discussed in class.

I didn’t know that the integral sign was just an elongated “S” for summation.  After talking about integration, Strogatz moves on to talk about e.  I don’t think I ever knew that e is the limiting number approached by 1+1/2+1/(1*2*3)+1/(1*2*3*4)+… I like how when he discusses examples, he doesn’t always explain the precise answer.  Instead, he shows why that answer is reasonable by doing the extreme cases or cases that are easier to figure out which make it possible to see why the real answer is plausible. Strogatz also shows how important calculus is by looking at differential equations and the problems that they have solved, mainly physics related.

Part Five: Data

This section starts off talking about statistics and the how important it is for people to understand data.  The media presents means when they should really be talking about the median.  Depending on the distribution of the data, the mean can really misrepresent the data. He also talks about people needing to understand cumulative probability and the importance of understanding what the probability of something given something else really means.  The final chapter in this short section discusses a topic more related to graph theory.  I remember talking about in my discrete math class.  Strogatz presents a rather simplified version of how Google returns results for searches.

Part Six: Frontiers

The final section begins with discussing prime numbers and number theory.  Strogatz says that number theory is viewed is the most pure area of study.  He presents a graph of prime numbers that I have never seen before.  It is a graph of the number of prime numbers less than or equal to x.  I think it is a good visual of the primes.  He also talks about how primes are used for encryption, which we discussed in my independent study with Prof. Hodge on ring theory.

Strogatz moves on to graph theory.  He applies pretty abstract ideas to something that people can relate to, flipping a mattress.  The diagrams he has really helps show what combining two actions will result in and how the order (in this case) that you do those actions doesn’t matter.  Another abstract idea he talks about is topology.  The only expose I have had with topology was with my Calc II and III professor.  He touched on it a little bit by introducing us to Klein Bottle, but we didn’t do anything with it.  Strogatz talks about Mobius strips.  He also mentions Vi Hart in this chapter because she made music using Mobius strips.

Strogatz goes on to talk about curves on 3-dimensional geometric shapes.  He makes visualizing how the curves actually appear on the shape relatively easy.  He uses earth and plane routes in order to better illustrate the ideas.  I also really like that he used a bike and the way that the handlebars would be facing while traveling along the arc in order to show what the difference is between traveling along the equator and traveling near the north or south pole.  Fourier series is the next topic discussed.  I enjoy how he applied the idea of infinite sums to real world things in order to make it seem less abstract and untouchable.  The finale chapter is devoted to the idea of infinity.  He uses the idea of a hotel with infinite numbers of rooms, and infinite number of people on an infinite number of buses to illustrate the difference between an uncountable and countable set.  Having a representation of the guests in a matrix made the whole process more understandable.

Strogatz, S. (2012). The joy of x: A guided tour of math from one to infinity. New York, NY: Mariner Books.