Inverses

Another topic which can be represented graphically and explained using real life examples.

In terms of functions, let's consider the functions f(x) and g(x), g(x) is the inverse of f(x) if for all values of x we find that
f(g(x)) = x or g(f(x)) = x.
We can also say that f takes x to f(x) and then g takes f(x) back to x. (Think of reversing a process.)

Inverses exist in a lot of fields of mathematics. We can have invertible matrices, for example. If A is a matrix, then B is the inverse only if
AB = I
where we refer to I as the identity matrix (a matrix with 1's along it's diagonal and all other entries 0).

We can also think of it in terms of spaces, where the function f takes us from space A to space B, and the inverse of f takes us back from B to A.

An easy example is the inverse of a number. For example the multiplicative inverse of 3 is just ⅓ since the product equals 1.

We can also have the additive inverse of a number.

Lets consider a few possibilities of inverses in our daily lives.
Suppose we could take the inverse of time, then that would mean that we would be able to go back in time, and in principle also go to the future.
An inverse can be undoing an action, for instance tying your shoe is f(x) and untying your shoe is the inverse g(x).

Inverse(The beginning) =  The end.

(Images taken from: http://regentsprep.org)

Circles

Circles, the only closed shape with no edges or bends (except for an oval - the overweight circle). In the book Flatland, by Edwin A. Abbot, (see: http://www.geom.uiuc.edu/~banchoff/Flatland/), a story about a life in two dimensions, circles are the highest class or the priestly order, hence they are the perfect shapes.

The equation of a circle is the following:

Circles may appear in many areas of mathematics, but what about the world we live in? Just think of all the objects that are in the shape of a circle.

We wouldn't have the transportation systems that we have if it wasn't for circles (i.e. wheels). At the carnival most of the rides wouldn't exist. Would we have square plates and cups? Would we invent screws that fit into triangular holes? What would the people who sees UFO's tell us they look like?

So many things just wouldn't make sense without the circle shape. I agree with Edwin Abbot, that circles are definitely the highest class of two dimensional shapes. Lets take a moment to appreciate the circle.

All that aside, here follows a few useful formulas when performing calculations with circles.

Circumference: 2пr

Area: пr²

Diameter: 2r

Arc length:  rθ

r: radius

θ: angle between two lines both from the center of the circle to somewhere on the edge.

Happy circling!

(Flatland also available at Amazon (the book) or eBay (the CD).)

A 4x4 matrix

Recently in one of my tutorial classes we came across an interesting problem. At first glance it was merely a system of four equations with four variables which we had to solve by means of a matrix. So we (or rather the students) went about carrying out the row operations, reducing the matrix to row echolen form.
(Note that this was a first year class and the concept was still very new to them, hence many struggled to complete these operations.)

This is an example of a 3x3 matrix in row echelon form:

Now, solving this system we find that it has infinitely many solutions, and the solutions turns out to be the equation of a plane.

I'll give you a moment to consider this...

We have four objects in a four dimensional space which intersects in a three dimensional object, a plane.
What does these 4D objects look like? Even though don't know, we are able to show how they would intersect.

Interesting? I thought so.

Happy 4D thinking!

Math Joke

Orthogonality

We all (basic school math) are familiar with the basic concept of orthogonality, where one line are orthogonal to another in the xy-plane, as is shown below.

We also use the words perpendicular or normal. A basic definition is that the angle between two orthogonal lines are 90 degrees. So if we add the blue and orange angles above we get 180 degrees, which indicates a straight line.

Furthermore, orthogonality between lines and planes in higher dimensions are defined in a similar way.

Now lets consider the idea of orthogonal polynomials. A polynomial is a function with the general form
f(x) = a_1 x^n + a_2 x^(n-1) . . . a_(n-1) x + a_n
where the x^i 's are the variables and the a_i 's are the coefficients, where n can be any positive integer value. As long as n is greater than 1 we are no longer consider straight lines and a whole new definition of orthogonality is needed, especially since we want to consider polynomials of any degree.

Orthogonal polynomials is on its own a field of mathematics. They have various properties and a lot of applications in engineering, physics and biology. I completed my Masters dissertation in this field, considering the convexity of the zeros of these types of polynomials.

A lot of research has been done concerning orthogonal polynomials and they are divided into certain classes, depending on certain attributes. They form part of the so called special functions.

So now the question, how do we define orthogonality between polynomials? We make use of an inner product formula, where two polynomials are orthogonal with respect to a certain measure (or a weight function).

Read more on this topic at: http://sydney.edu.au/science/chemistry/~mjtj/CHEM3117/Resources/poly_etc.pdf