Mathematics

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The cross product is ubiquitous throughout linear algebra and vector calculus. It plays a major role in transformations of coordinate systems and is intimately related to the determinant. In fact, its definition in linear algebra courses is often given in terms of the determinant, which can seem mysterious and arbitrary. In this post, I want to ...

A postdoctoral fellow once laughingly told me, “Every mathematician needs to construct the real numbers at least once.” For most people in science and engineering, the existence of the real numbers is obvious. And that is a good thing; mathematics should describe things as they are or must be, not as we would like to see them. For the ...

This is a blog post about understanding linear maps and a special number associated to them called the determinant. A linear map f from Rn (n-dimensional Euclidian space) to Rm (m-dimensional Euclidian space) is any map which satisfies the following properties:

The unit circle is a handy tool that can help students learn the trigonometric values, sine, cosine, and tangent, of certain angles (30°, 45°, and their multiples) that the math “Powers-That-Be” have determined to be important. Unfortunately, for a lot of students, the unit circle can feel like tedious rote memorization with unfamiliar numbers ...

The mention of mathematics often evokes mixed emotions among biologists. While some embrace it as a powerful tool, most merely view it as a black box for their collaborators to navigate for them or as intimidating and unrelated to their study. However, as a Biology PhD student, I firmly believe that math is not scary and an essential and ...

Mathematics is a topic that is notoriously difficult to learn on one’s own. Sadly, this often causes people to think they’re not “math people” or that learning mathematics perhaps isn’t for them. In the contrary, I think that learning math is something that is accessible to everyone, but requires a different approach from reading a novel, or ...

Set theory is the branch of mathematics that studies the infinite. The discipline was founded by Georg Cantor in the late 1800s. Cantor is responsible for many of the notions we discuss here. A set, according to Cantor, is a collection of definite, distinguishable objects conceived as a whole. A set consists of its elements. There is exactly one ...

If you’re reading this, you probably already know what a matrix is. But just to be clear, a matrix is a rectangular array of numbers. Here is an example:

When learning about lines in Algebra 1, you likely learned how to solve a system of (linear) equations, such as the following, by graphing:

Behind every mathematician is a beautiful mind: one that has been forged through years of critical thinking. Their minds are molded by countless failed attempts at solving problems and refined by the exposure to remarkable ideas explored along a lifetime of learning. How can we train our minds to see the world more like mathematicians do? The map ...

Learning math often feels like it’s all about right or wrong, like success or failure are the only two possible options and that all of your math expertise is visible as soon as you take a test. What I experienced as a student studying math and now as a math teacher is that this couldn’t be further from the truth. Some of my greatest learning ...

I'm going to introduce you to my favorite math puzzle. It's a doozy, and I hope you'll find it as intriguing as I do. And maybe a bit more intuitive than I did when I first encountered it.

What is the sum of the first n positive integers? Phrased mathematically: 1 + 2 + 3 … + n -1 + n = ?. The answer, it turns out, is n * (n + 1) / 2. How do we show this is true though? How do we prove this?

I always tell my students not to be afraid to ask why. In so many parts of our lives, we are asked to defend our opinions and ideas—to offer evidence and to explain our thinking or reasoning. But sometimes, it feels this is missing from math education, especially in middle school and high school. Math becomes about memorizing formulas rather than ...

The Pythagorean Theorem plays an essential role in many facets of math from Euclidean Geometry to complex numbers to trigonometry. Today we’ll explore one of its many proofs.

We’ve all been there: on a homework set or in an exam, you turn to the final page and, to your dismay, it’s a wall of text. The dreaded Word Problem. Some of the words are useful, but some of them are meant to distract. Let’s look at a strategy for answering initial value word problems.

In the year 1202, Italian mathematician Leonardo Fibonacci published the extremely influential Liber Abaci (Book of Calculations). The book's most significant contribution was to bring to Europe the Hindu-Arabic number system that we all use today. But it also contained a curious thought experiment about the reproductive patterns of rabbits, which ...

Probability is one of those topics that haunt children from grade school days, asked to determine the likelihood of picking out red marbles from a box. Even my most advanced math tutoring students sometimes feel bamboozled by it. Why? Because probability and statistics can quickly become overwhelming with the many different distributions and ...

I thought that it would be fun to tackle a geometry problem on today's post. I know what you're thinking, fun and geometry – can these two words even go in the same sentence? But yes, geometry can be fun; let's see as we work together through the details of a particular problem.