On-line Resources: Discrete Math
Think about our world and name some things that can be counted; the number of pixels in an image, the votes cast in an election, the number of fingers on your hand, the number of different ways in which a network can be traversed. From a mathematical perspective, in each of these examples we are talking about sets of discrete objects. The study of discrete objects and how they can be counted is fundamental to the study of discrete mathematics.
Now consider things that we measure; the speed of a cannon ball when it leaves the cannon, the distance from the earth to moon, the volume of blood in your body. In these cases the results of the measurement -the number that you get- depends on the accuracy and the precision of the measurement. The speed of a car could be 35 mph, 35.1 mph, 35.111 mph,and so on. Given a better measuring device you could always add another digit of precision to your measurement. For this reason we say that measurement usually gives us a “continuous” result. In terms of our study of mathematics, algebra and calculus are among the mathematical tools that we use to analyze and understand questions that involve continuous quantities.
Discrete math is also commonly associated with computers. That’s because, believe it or not, computers cannot handle those messy imprecise numbers that go along with continuous math. Sure you probably use your computer to work with numbers containing decimal points all the time, but at the lowest level your computer sees 31.111 as five integers with a decimal point after the first two. That’s a discrete quantity. The messy part comes in when we use our computers to do calculations with irrational numbers like Pi. As you probably know, Pi cannot be defined as a exact value; the decimal places go on forever. When a computer does math with numbers like that it has to decide how many digits of precision will be used for each calculation. And when lots of these “rounded off”numbers come together in some sort of calculation problems can occur.
Another area where discrete math is used is in the definition of the recipes that are used to write computer programs; called algorithms.Students who pursue degrees in information technology or computer science are usually required to take one or more courses in discrete mathematics.
||Image Analysis Using Maple may sound like an arcane topic, but if you own a digital camera you already have an advanced digital image analysis system. Image analysis is central to how digital imaging works. Some cameras even give you access to some of the image processing features allowing you to shift the contrast, the color balance, or the luminosity of an already captured image.
For Empire State College students studying discrete math, and using Maple math software, you can apply these sorts of image transforms using Maple. Maple provides the basic image processing functions found in your camera and more advanced ones not typically found in graphics software.
You can download a worksheet from the Maple Applications Center (free registration required) that demonstrates these features. On the Application Center home page search for “image” and select “image tools” from the list. You can download the worksheet for a demonstration of the different ways that Maple can be used to manipulate images.