February 2012 - Page 4 of 7 - Math and Multimedia

Week in Review – February 2012 Week 3

February 19, 2012 GB Post Summary

Good morning from Baguio City (6 hours from Manila), the summer capital of the Philippines. I am here for a one-day lecture-workshop on “Mathematics as a Way of Thinking” at the AEIRS National Conference.

After sleeping for 7 hours, I feel rejuvenated, so I am going to list the most recent posts for this week.

Math and Multimedia

My other blogs

That’s all for this week. Have a blessed Sunday!

Leave a comment summary of posts, weekly summary

Using Area to Prove the Arithmetic-Geometric Mean Inequality

February 16, 2012 GB High School Algebra, High School Mathematics, High School Number Theory

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. » Read more

2 comments AM-GM inequality, arithmetic mean, arithmetic mean geometric mean inequality, geometric mean

Heart Graph for the Mathematically In Love

February 13, 2012 GB Math Lite, Web 2.0 Technology

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

5 comments google search, plotting functions, valentines day

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