Advanced Mathematics — What Does it All Mean?


If you think mathematics is a difficult subject, you should try studying some of the more advanced branches like abstract algebra before you come to such a conclusion. It is in these higher realms of this most distinguished subject that one learns about mathematical structures like groups, fields, and rings, and the properties inherent in these objects. After a jaunt through such mysterious realms, one comes away with a new appreciation of this most fascinating subject.

What does an advanced branch of mathematics like Abstract Algebra concern itself with? In a nutshell, this field attempts to classify and categorize mathematical sets with the end result of being able to solve problems that share certain characteristics. To make clear the previous declared mumbo jumbo, let’s look at some specific examples. Take the set of linear equations, which take the form y = ax + b, where a and b are any real numbers and a is not 0. The set of all such equations forms a mathematical class and as a result any member of this set shares a number of similar properties. The variable constants a and b, determine such differences as the slope of the line and the point at which graphically, the line crosses the y-axis, also known as the y-intercept.

By studying such a set of objects, mathematicians can categorize properties inherent to the class and thus draw conclusions about what is and is not possible regarding this set. For example, in the linear equation class y = ax + b, we can rewrite this as ax + by = c, again where a, b, and c are real numbers and a and b are not 0. (If they are 0, then we no longer have a linear equation in x and y.) Now if we restrict a, b, and c into a subset of the real numbers, the integers, we have a new class called linear Diophantine equations. These become a curious set of objects, and one which finds itself abounding in real life. For example, many applications in the real world require the solution of such linear equations with the restriction that a, b, and c are whole numbers. An example would be in agriculture where such an equation might represent cattle production of milk.

Suppose on a farm, there are two types of cattle, which we shall call Cattle A and Cattle B. Cattle A outputs 30 gallons of milk per week, and Cattle B outputs 40 gallons of milk per week. In order for the farm to meet its delivery quotas, 1000 gallons of milk per week are needed. How many of each type of cattle will satisfy this quota?

Such a problem requires mathematicians to study the class of linear Diophantine equations. By analytically dissecting this class and finding common properties and characteristics, mathematicians can ultimately solve such «cattle» provoking questions. When studying this class, mathematicians will come up with certain unchanging or rigid properties which bind the class together. These rigid properties become theorems that can be used to decide whether a certain problem can be solved or not. In fact, it was the study of second-order Diophantine equations that led to the historical Fermat’s last theorem, which was only solved recently. This problem lay unsolved for hundreds of years, after having been left in the margins of a manuscript by the French mathematician Pierre de Fermat.

Thus if you think more advanced mathematics exists just to confuse, think again. It is this higher realm which enables us to move relentlessly forward in our technologically oriented world. So that you come to appreciate this higher realm, in some future articles I will continue to explore this topic in more detail. For now chomp on what you have and start to appreciate this most extraordinary field.

Joe Pagano