Month: September 2011

Sets: A Gentle Introduction

College Mathematics, Set Theory

There are people who are fond of buying things in sets. For example, a pen collector may collect pens of different colors.   Other people are obsessed with brands.   For instance, Apple fans may buy the following an iPhone, iPod, iPad, and an iMac. If you are fond of reading Dan Brown, you probably have read or own some of the following books: The Lost Symbol, Deception Point, Digital Fortress, and The Da Vinci Code, and others.

Pens

In mathematics, a set is not very different.  A set is can be described as a collection of objects with common characteristics. These objects are called elements. Given a set and an object, it is clear if the object is an element of a set or not.  If we let D be a set of Dan Brown novels, then D = { The Lost Symbol, Deception Point, Digital Fortress, The Da Vinci Code}. Also, if we let V be the set of vowel letters in the English alphabet, then V = {a, e, i, o, u}, and if we let R be the set of polygons whose number sides is less than 9, then the set would include the polygons in the figure below. » Read more

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Guest Post: A Hundred Years Down the Road

Miscellaneous

Technology has evolved tremendously, especially more so over the past decade in a breathless, continuous sprint. Think: from landlines and pagers to cell phones to smart phones. From paperbacks and newspapers to Kindle and iPads.  From compact discs and cumbersome CRT television sets to miniature MP3 devices and sleek 3D-enabled high definition flat screens. From physically bulky tower servers to nimble cloud computing. So on and so forth. This transformation ultimately and inevitably spilled over into various genres of livelihoods. On the education front, classroom learning hauled ass with Mathematics leading the revolution. Once upon a time there were only scientific calculators, now advanced handheld graphing machines dominate the student market. With an internet connection, the entire world library is brought right to your doorstep. Digesting geometry, algebra and calculus has been made a million times more fun and easier with online interactive applications such as Wolfram Alpha and GeoGebra. A gigantic virtual maths instructional video vault at Khan Academy is always ready at your disposal should you find certain new concepts taught in school less than palatable and desire immediate clarifications. Facebook and Twitter now enable both educators and students to share/ discuss ideas instantaneously without having to meet up face to face. » Read more

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Is 0 a natural number?

High School Number Theory

When I was still teaching,  several of my students asked me this: Is 0 a natural number?

Short Answer

It depends on the convention. In some references, the set of natural numbers includes 0. In other references, the set of natural numbers does not include 0.

Long Answer

When we define a particular mathematical concept, we come to some sort-of agreement. For example, we might want to call the sequence  1, 4, 7, 10, … “cool numbers.” If all of us agree on this, we will have no problem.  However, other group of uncool guys might give it another name. Or, they might keep the name, but change the sequence.  This was what happened to the set of  natural numbers. » Read more

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