Sample Essay
Question: What is your favorite theorem in mathematics? Justify your answer.
Some terminology that I use in my response:
Definition: A polynomial is an expression of the form y=anxn+an-1xn-1+an-2xn-2+…+a1x+a0,where the aiare some numbers and n is a positive integer. Some examples include: y=2x3+4x4+7, y=x2+1, or y=3x-1.
Definition: A complex number is a number of the form a+bi, where a and b are real numbers and i is the square root of -1.
Response:
Mathematics is an area in which the subject matter builds upon itself – we start with a basic principle and build to more complicated situations. For instance, the most basic math involves counting – if Jenny has 4 apples and 3 oranges, how many pieces of fruit does Jenny have? From there we build to problem solving – If Jenny has 4 apples and I ruthlessly steal two of her apples, how many does she have left?
As we take the step from counting to problem solving, the ability to solve an equation becomes important. My favorite theorem in mathematics is called the Fundamental Theorem of Algebra and it is a theorem that tells you about solving equations. The Fundamental Theorem of Algebra states that every polynomial has a solution. This solution may be an integer, it may be a rational number (fraction), it might be irrational, or it might be uglier yet – a complex number. (For instance, the polynomial y=x2+1 has two solutions, i and –i.) Regardless of how nasty this solution may be, you are at least guaranteed that a solution exists, which is a powerful statement. Perhaps the statement seems obvious. However, in the research that I do, we look at certain equations that may not even have a solution. You can imagine how fun that is!
Although the statement of the theorem seems simple enough, its applications are enormous. When you are trying to solve most problems that come up in nature, you can generally model your problem using polynomials. Thus, this theorem guarantees that you can find an answer to your problem. This theorem is also useful in many general areas of mathematical theory including linear algebra, algebraic geometry, and analysis.
The history behind the Fundamental Theorem of Algebra is also intriguing. It was originally stated as a conjecture in the 16th century. In 1799, Gauss, one of the most prominent mathematicians of all time, first proved the theorem. Throughout his life, he revisited the proof three different times in an effort to make it less clumsy and complicated. Today, there are several different ways to prove this theorem. Even though its title might lead you to believe that this is merely a theorem about algebra, every major area of mathematics has a different proof for this theorem. There is a proof involving set theory and topology. There is a proof involving analysis, which is the theory of approximations that builds to calculus. There is even a proof using algebra techniques, but this proof is ironically the most complicated! Thus we see that the effort to prove this theorem has led to the development of interesting theory in all three major branches of mathematics.
The Fundamental Theorem of Algebra is a wonderful theorem. Its statement is simple enough that you could use it in a high school algebra class, yet its applications and proofs have led to a wealth of knowledge in every branch of mathematics