Proof that log 2 is an irrational number

Before doing the proof, let us recall two things: (1) rational numbers are numbers that can be expressed as \frac{a}{b} where a and b are integers, and b not equal to 0; and (2) for any positive real number y, its logarithm to base 10 is defined to be a number x such that 10^x = y. In proving the statement, we use proof by contradiction.

Theorem: log 2 is irrational

Proof:

Assuming that log 2 is a rational number. Then it can be expressed as \frac{a}{b} with a and b are positive integers (Why?).  Then, the equation is equivalent to 2 = 10^{\frac{a}{b}}. Raising both sides of the equation to b, we have 2^b = 10^a. This implies that 2^b = 2^a5^a.  Notice that this equation cannot hold (by the Fundamental Theorem of Arithmetic) because 2^b is an integer that is not divisible by 5 for any b, while 2^a5^a is divisible by 5. This means that log 2 cannot be expressed as \frac{a}{b} and is therefore irrational which is what we want to show.

Diagonals of a Parallelogram

Coordinate geometry was one of the greatest inventions in mathematics.  Aside from connecting geometry and algebra, it has made many geometric proofs short and easy.  In the example below, we use coordinate geometry to prove that the diagonals of a parallelogram bisect each other.

The proof can be simplified by placing a vertex of the parallelogram at the origin and one side coinciding with the x-axis.  If we let a and b be the side lengths of the parallelogram and c as its altitude, then, the coordinates of the vertices can be easily determined as shown below.

In addition, if we label the vertices P, Q, R, and S starting from the origin and going clockwise, then the coordinates of the vertices are P(0,0), Q(b,c), R(b,c) and S(a,0). » Read more

2011 GeoGebra Institute Activities

In 2011, we at the GeoGebra Institute of Metro Manila (GIMM) have squeezed our schedule and have given GeoGebra trainings to teachers and students from all over the country. Here are the list of trainings that we have conducted.

GeoGebra Basics
Date: February 16, 2011
Venue: EARIST College, Manila
No. of Participants: 30

Introductory Course on the Use of GeoGebra in Teaching and Learning Mathematics
Date: May 5-6, 2011
Venue: UP NISMED, University of the Philippines, Diliman, Quezon City
No. of Participants: 21

Lesson Study for Teaching through Problem Solving with GeoGebra
Date: May 9-13,2011
Venue:  NISMED, University of the Philippines, Diliman, Quezon City
No. of Participants: 20

Introduction to GeoGebra
Date: October 1, 2011
Venue: St. Therese College, Pasay City
No. of Participants: 41

GeoGebra Fundamentals
Date: November 16, 29, 2011
Venue: Santa Lucia High School, Pasig City
No. of Participants: 18

GeoGebra Fundamentals In-House Seminar
Date: November 25, 2011
Venue: UP NISMED, University of the Philippines, Diliman, Quezon City
No. of Participants: 15

In addition, we are also working on a book on College Geometry with GeoGebra (the title has not been finalized yet).  Hopefully, it will be tried out this school year  and will be published late this year or early next year.

We are also planning to have Level 1 and 2 GeoGebra trainings this summer (April-May, no definite date yet). If you are interested, please email me at gip@gmail.com or mathandmultimedia@gmail.com.

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