# Triangle Trigonometry

I haven’t forgotten about my pledge to focus more content here towards some of the areas I’ve been asked to tutor on recently. This latest one is designed to help users understand the properties of triangles. It is based on two laws that we learn in trigonometry: the law of sines and the law of cosines. Assume that we have a triangle with sides of lengths a, b, and c and respective angles A, B and C (where the angle A does not touch the side a, the angle B does not touch the side b, and the angle C does not touch the side c). These laws are as stated as follows:

Law of Sines

 a sin(A)
=
 b sin(B)
=
 c sin(C)

Law of Cosines
c2 = a2 + b2 – 2*a*b*cos(C)

We can use these laws to determine the sides of a triangle given almost any combination of sides and angles of that triangle (the only one we cannot determine properties from is if we are given all three angles, as this leads to many solutions).

The script generates random triangles, with different combinations of sides and angles revealed and the user’s job is to try to determine the missing sides. There is a button to reveal the solution, or if you’d like to see how we arrive at these values, you can check the “Show work” box.

Hope you enjoy.

Other Blogs covering this topic:
Mathematical!
Algebra 2 Trig

# The PageRank Algorithm

I think one of the best recent examples of the importance of mathematics is the rise of the search engine Google. I remember the world of search engines before Google and it was dominated by names like AltaVista, Yahoo, WebCrawler, Excite, and the likes. The standard way these search engines ranked the order that pages would be listed on a search query was basically to count the number of times that query appeared on pages in their database. The pages with the most listings were considered the most important, the second most listings were second most important and so on and so forth.

This sounds like a feasible way of doing things but let me show you an example of how this can be tricked. Suppose I wrote my first web page and it looked like the following:

That’s a basic web page that may not garner much attention, and it wouldn’t rank highly in most search engines as no work appears more than once. Suppose that, this being a math web page, I wanted it to rank higher on the query “math”. Then I could just edit the source code of the page to be as follows:

This second page says not much more than the first, but the fact that the word math appears 9 additional times would increase the ranking of this page among math pages. This is a very simple example, but it shows how these search engine rankings did not have a useful metric for determining the important sites on the web.

The way Google solved this problem of determining the importance of a web page is basically by counting the number of links into a web page – the theory being that the more important a web page is, the more people will be talking about it and thus linking to it. Also, the more important the people talking about (linking to) a web site, the more important that site is. This can be expressed mathematically by the following formula:

In the above formula, the variable d is called the damping factor, which helps to capture some of the random nature of the internet by saying that every site should have at least some minimal worth because of the idea that a random surfer could still get to these sites.

I have written a script to implement the algorithm here.

Other Blogs that have covered this topic
Blue Onion

# The Risk of Competition

I’m not a competitive person. Let me correct that. I try not to be a competitive person. I’ve recently been playing some of my favorite games from childhood like Monopoly, Chess, Spades, and Madden and I’ve been reminded of the competitive streak in me that hates losing. This streak has been relaxed in much of my adult life, and I tend to think that’s been to my benefit. Two pieces come to mind as I write this. One is a piece written by Slim Jackson of Single Black Male on the importance of asking questions. The second is what I heard on WAMU’s “Tell Me More” while driving home from work regarding the competitive nature generally assumed by (or dictated to) men.

These two concepts go hand in hand in my opinion because I’ve found that at times where I see a person as “my competition” I’m less likely to seek advice or help from that person. Doing this limits my resources and the set of people who can help me. I know that one of the hardest things to do while playing a game like Madden is asking my competition “how’d you just make that play” or listen to this competition explain how he knew to make such a play. However, its just this ability to swallow my pride and ask these questions that I’ve gotten better in Madden. I will admit that I’m not the most humble person on the face of the earth and so there are some things that I haven’t asked how to do. Generally in those things, I’ve found myself repeatedly playing the game trying to figure things out for myself.

There are applications of both sides of this to the real world. Some who love competition will say that it brings out the best in us, and they’d be sure to point out the feeling of satisfaction we get by working independently on a problem. As I look through my lists of inspirational quotes, I’m reminded of this with often repeated statements like “failure is the key to success”.

As rewarding as a competition can be, there’s also an important saying that we don’t need to re-invent the wheel. And what often gets lost in the do-it-yourself nature of competition is the ability to utilize all possible resources available to us. In particular, the skills of how to acknowledge the things we do not understand and how to formulate questions aimed at gaining understanding.

This brings us to is the other side of the competitive spirit. As rewarding as it is to be able to say, “wow, I can’t believe I was able to figure that out on my own”, it is also a stressful situation and there are many who are never able to say those words. Should these people be satisfied with failure?

I do not ask this in some devil’s advocate type of way. I ask coming from the point of view of a mathematician, an educator, and as a former student. I had a pretty dark moment in graduate school where I realized that my mere “love” of mathematics would not get me through qualifying exams. It wouldn’t suddenly make text books and academic papers instantly understandable. Suddenly I was placed in an uncomfortable position. Instead of always being the one who was the first to get the concept and who was leading the study groups on it, I’d be the one asking the questions. Looking at this in a “competitive” frame of mind (as I did then), I felt like I was losing the game.

This same moment though, is where my thinking was really changed. There was one thing, and one thing only that I would consider a failure and that was not finishing. I view everything else as a matter of swallowing my pride and readjusting my thought process to help get to that point.

Unfortunately though, many others do not get to this point. Many get lost in the scramble of the competition and do the equivalent of folding your hand in poker. They realize that at their current pace there is little to no chance that they’d win and so they just leave the game. And this is a real risk that we’re running with competition, particularly as STEM fields are becoming more and more important and we’re trying to encourage students to focus on these areas. What may be necessary to bring this about is a more cooperative approach to these things.

# Learn Duality in Linear Programming

I have just completed a script to help learn about duality in linear programming.

Many real world problems can be formulated as Linear Programming problems. There are often many different ways to formulate a single problem. Some of these alternative formulations can easily be proven to be equal via simple algebra and arithmetic. For example, one person may see a problem as maximizing profit while another may see the same problem as minimizing losses. The relationship between the two alternative formulations can then be shown to be that one simply has the negative objective function value of the other.

Sometimes, though, this relationship between alternative formulations is not as easy to detect. Two alternative formulations that arise regularly in linear programming problems are a primal problem and a dual problem. The dual of a linear programming problem is the problem of finding the best bound on the objective function in terms of the constraints. This dual is formulated so that every feasible solution to the dual is a bound on the primal objective function. The Weak Duality Theorem for Linear Programming says that the optimal solution for a minimization problem is always an upper bound for its dual (which is a corresponding maximization problem). Correspondingly, the optimal solution for a maximization problem is always a lower bound for its dual. This leads to the Strong Duality Theorem for Linear Programming which says that if we can find feasible solutions to the primal and dual problems with matching objective function values, then both of these solutions must be optimal for their respective problems.

This shows an importance of duality. What my script provides is a means of formulating the dual of a given problem to help understand the concept.

The rules for constructing a dual linear program are as follows:

 Primal: Dual: The ith constraint is [<=] Variable yi is [>=] 0 The ith constraint is [>=] Variable yi is is [<=] 0 The ith constraint is = Variable yi is unbounded Variable xj is [>=] 0 The jth constraint is [>=] Variable xj is [<=] 0 The jth constraint is [<=] Variable xj is unbounded The jth constraint is =

Other Blogs that have covered this topic:
A Narrow Margin
Optimization and data mining

# Learn Math Through Set Relations

Much of our world deals with relationships – both in the sense of romantic ones or ones that show some interesting property between two sets. When mathematicians think of set theory, a relation between the set A and the set B is a set of ordered pairs, where the first element of the ordered pair is from the set A and the second element of the ordered pair id from the set B. So if we say that R is a relation on the sets A and B, that would mean that R consists of elements that look like (a, b) where a is in A and b is in B. Another way of writing this is that R is a subset of A x B. For more on subsets and cross product, I refer you to my earlier script work on set operations.

Relations can provide a useful means of relating an abstract concept to a real world one. I think of things like the QB rating system in the NFL as an example. We have a set of all quarterbacks in the NFL (or really all people who have thrown a pass) and we would like some means of saying that one QB is performing better than another. The set of statistics kept on a QB is a large set, so attempting to show that one QB is better by showing that every year that they played one is better in every statistical category can be (a) exhaustive, and (b) will lead to very few interesting comparisons. Most of the really good QBs have some areas that they are really good and others that they are not. The QB rating system provides a relation between the set of all QBs in the NFL and the set of real numbers. Once this relation was defined, we can say that one QB is performing better than another if his QB rating is higher. Similarly we can compare a QB to his own statistics at different points in his career to see the changes and trends.

This is just one example, and there are countless others that I could have used instead.

Once we understand what a relation is, we have several properties that we are interested in. Below I list four, although there are many more.

Properties of Relations:
A relation R is symmetric if whenever an element (a, b) belongs to R, then so does (b, a).

A relation R is reflexive if for every element a in the universe of the relation, the element (a, a) belongs to R.

A relation R is transitive if for every pair of elements (a, b) and (c, d) and b = c, then the element (a, d) belongs to R.

A relation R is anti-symmetric if the elements (a, b) and (b, a) do not belong to the relation whenever a is not equal to b.

Once we understand what a relation is, there are a few common ones that we are interested in. Below I list four, but again, I want to stress that these are some of the more common ones, but there are several others.

Types of Relations:
A relation R is a function (on its set of defined elements) if there do not exist elements (a, b) and (a, c) which both belong to R.

A relation R is an equivalence relation if R is symmetric, reflexive and transitive.

A relation R is a partial order set if R is anti-symmetric, reflexive and transitive.

A relation R is a total order set if it is a partial order set and for every pair of elements a and b, either (a, b) is in R or (b, a) is in R.

A partial order is just an ordering, but not everything can be compared to everything else. Think about the Olympics, and a sport like gymnastics. Consider the floor and the balance beam. One person can win gold on the floor and another person wins gold on the balance beam. That puts each of them in the “top” of the order for their particular section, but there’s no way of comparing the person who won the floor exercise to the person who won the balance beam. So we say the set is “partially ordered”. More formally, lets say that two people (person X and person Y) relate if they competed in the same event and the the first person (in this case person X) received an equal or higher medal in that event than the second person (in this case person Y). Obviously any person receives the same medal as themselves, so this relation is reflexive. And if Jamie received an equal or higher medal than Bobby and Bobby received an equal or higher medal than Chris, then Jamie must have received an equal or higher medal than Chris so this relation is transitive. To test this relation for anti-symmetry, suppose that Chuck received an equal or higher medal than Charlie and Charlie received an equal or higher medal than Chuck. This means that they must have received the same medal, but since only one medal is awarded at each color for each event (meaning one gold, one silver and one bronze…if this is not true, assume it is), this must mean that Chuck and Charlie are the same person, and this relation is thus anti-symmetric.

If we have a partial ordering where we can compare everything, then we say that the set is “totally ordered”.

An equivalence relation tries to mimic equality on our relation. So, staying with that example of the Olympics, an example of an equivalence relation could be to say that two athletes relate to one another if they both received the same color medal in their event (for the sake of argument lets assume that no athlete competes in more than one event). Then obviously an athlete receives the same medal as themselves, so this relation is reflexive. If two people received the same medal, then it doesn’t matter if we say Chris and Charlie or Charlie and Chris, so the relation is symmetric. And Finally if Chris received the same medal as Charlie an if Charlie received the same medal as Jesse, then all three people received the same medals, so Chris and Jesse received the same medals and this relation is transitive. Because this relation has these three properties, it is called an equivalence relation.