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## Advanced Mathematics – What Does it All Mean?

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

What does an advanced branch of mathematics like? *Abstract algebra* to worry Simply put, this field attempts to classify and categorize mathematical sets with the end result of being able to solve problems that share certain characteristics. To clear up the above 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 real numbers and is not 0. The set of all these equations forms a mathematical class, and as a result any member of this set shares a number of similar properties. The variable constants aib determine differences such as the slope of the line and the point at which the line graphically crosses the y-axis, also known as the y-intercept.

By studying this set of objects, mathematicians can categorize the properties inherent in the class and thus draw conclusions about what is and is not possible with respect to this set. For example, in the linear equation class y = ax + b, we can rewrite this as *ax + by = c*again where a, bic are real numbers and aib are not 0. (If they are 0, we no longer have a linear equation in xi and y.) Now, if we restrict a, bic to a subset of the real numbers, the integers, we have a new class named *linear Diophantine equations*. These become a curious set of objects, which are found abundantly in real life. For example, many real-world applications require the solution of these linear equations with the constraint that a, bic are integers. An example would be in agriculture, where this equation could represent the production of bovine milk.

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

This 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 these “forest”-provoking questions. When studying this class, mathematicians will find certain rigid or invariant properties that bind the class together. These rigid properties become theorems that can be used to decide whether a given problem can be solved or not. In fact, it was the study of second-order Diophantine equations that led to Fermat’s historic Last Theorem, which was only recently solved. This problem remained unsolved for hundreds of years, having been left aside in a manuscript by the French mathematician Pierre de Fermat.

So if you think more advanced math exists just to confuse, think again. It is this higher realm that allows us to move forward endlessly in our technology-driven world. For you to learn to appreciate this higher realm, I will continue to explore this topic in more detail in future articles. Now start with what you have and start appreciating this extraordinary field.

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