Mathematics and Multimedia

Using Area to Prove the Arithmetic-Geometric Mean Inequality

Posted on February 16, 2012 by Guillermo Bautista | Leave a comment

If we have real numbers $a$ and $b$ , we call $\frac{a + b}{2}$ the arithmetic mean (AM) and $\sqrt{ab}$ the geometric mean (GM) of $a$ and $b$ . In this post, we are going to examine the relationship of these two means.

To start off, let’s have a few examples. If $a = 3$ and $b = 12$ , then $GM = 6$ and $AM = 7.5$ ; if $a = 4$ and $b=16$ , then $GM = 8$ and $AM = 10$ ; if $a = 3$ and $b = 27$ , then $GM = 9$ and $AM = 15$ . What do you observe? Try a few example and see if your observations hold.

From a few examples above, and from your trials, you have probably observed that $GM \leq AM$ which means that $\sqrt{ab} \leq \frac{a + b}{2}$ for some positive real numbers $a$ and $b$ . This is actually true for all positive real numbers $a$ and $b$ . In the following discussion, we are going to use the concept of area to prove that the statement is true.

To begin the proof, we construct a square with side length $a + b$ made up of four rectangles and a square at the center (technically, a square is also a rectangle). Clearly, the area of each of the four rectangles is $ab$ , and the square at the center has are $(a - b)^2$ (Can you see why?). If we remove the square at the center, the remaining area is represented by the equation $(a + b)^2 - (a - b)^2=4ab$ . Note that $4ab$ is the total area of the four rectangles. Continue reading →

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Posted in High School Algebra, High School Mathematics, High School Number Theory

Tagged AM-GM inequality, arithmetic mean, arithmetic mean geometric mean inequality, geometric mean

Heart Graph for the Mathematically In Love

Posted on February 13, 2012 by Guillermo Bautista | 3 Comments

It’s the first day of our GeoGebra training tomorrow, so let us celebrate Valentine’s day in advance. To all who are mathematically in love, you can celebrate Valentine’s day by doing the following:

Copy this: sqrt(cos(x))*cos(300x)+sqrt(abs(x))-0.7)*(4-x*x)^0.01, sqrt(6-x^2), -sqrt(6-x^2) from -4.5 to 4.5
Go to Google.com
Paste it in the Google search box and click the Search button. You will see the graph below.
Send the graph to your loved one.

Surprised? Yes, Google Search has now the ability to plot mathematical functions.

Happy hearts day in advance!

H/T: Math Concepts Explained

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Posted in Math Lite, Web 2.0 Technology

Tagged google search, plotting functions, valentines day

Introduction to the Complex Plane

Posted on February 12, 2012 by Guillermo Bautista | Leave a comment

Imaginary numbers had no use when it was invented except for intellectual exploration until it was linked to coordinate geometry. The complex number $a + bi$ can be plotted as the ordered pair $(a,b)$ where $a$ is the real part and $b$ is the imaginary part. Therefore, the complex numbers $3 + 2i$ , $3- 2i$ , $-2 + 0i$ , and $0 - i$ can be plotted as $(3,2)$ , $(3,-2)$ , $(-2,0)$ , and $(0,-1)$ in a “coordinate plane” respectively as shown below. In the figure, we can see that the origin is $0 + 0i$ and that $3 - 2i$ , the complex conjugate of $3 + 2i$ , is its reflection along the x-axis (Can this be generalized?).

The plane where the complex numbers are plotted above is called the complex plane.

In the complex plane, we can observe that all numbers of the form $(a,0)$ are real numbers and these numbers are represented by the horizontal axis. The numbers of the form $(0,bi)$ are all imaginary numbers and it is represented by the vertical axis. Thus, we can call them the real axis and the imaginary axis respectively.

In addition, we can conclude that all real numbers are complex numbers since for all real numbers $a$ , $a = a + 0i$ . So, the set of real numbers is a subset of the set of complex numbers.

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Posted in High School Mathematics

Tagged complex numbers, complex plane, imaginary numbers, real numbers

Popularizing Lesson Study

Posted on February 10, 2012 by Guillermo Bautista | 1 Comment

The reason that I was away for two days was that I observed the lesson implementation in Nueva Ecija High School (NEHS). It was part of lesson study, one of the components of the 2-year project of our institute and NEHS.

Lesson study is a professional development program for teachers that originated in Japan. In lesson study teachers collaborate with one another in developing and implementing a lesson. If you have not heard about this type of professional development program, it is now gaining popularity worldwide and is already practiced by elementary school and high school teachers in many countries.

The process in lesson comprises of the following steps: