The Importance of Pure Mathematics

            In today’s world of mathematics, there are two main fields of mathematics. Those fields are pure, and applied mathematics.  Pure mathematics in its simplest definition is math purely for the sake of math. Pure mathematics would include developments is number theory, Algebra, Control theory, and Calculus analysis. Applied mathematics in its simplest definition is math that has a purpose outside math itself. Applied mathematics would include math done in, physics, chemistry, biology, and any other math intensive science. For the most part pure mathematics is not seen as an important part of society. Sometimes pure mathematics is given the label of useless, or pointless. Although it might seem like a waste of time to some people, pure mathematics has an immensely important part, in the everyday workings of society as a whole.

            Pure mathematics is important because it develops the tools for applied mathematics. The groundwork of all Calculus, Algebra, Topology, and mathematics as a whole, rely on pure mathematics. Farida Kachapova wrote in the Journal of Mathematics and Statistics that, “ . . . without fundamental mathematics there would be nothing to apply.”(Kachapova) This means that all of the applied mathematics need pure mathematics to facilitate them. For example, a huge chunk of Newtonian physics relies on the understanding of basic algebra.   G. H. Hardy was once quoted saying, “Pure mathematics is on the whole distinctly more useful than applied. For what is useful above all is technique, and mathematical technique is taught mainly through pure mathematics.” (Hardy). What Hardy is saying is that the tools used for mathematics are more important then the products of mathematics. The tools of mathematics can be used to make all the results, while the results only help with that specific issue. That is why pure mathematics is important to applied mathematics.

            Pure mathematics is, and was used to develop many of the technologies used in the daily life of individuals.  The most commonly mentioned invention, which came about because of pure mathematics, is the computer. This is because the computer is revolutionary to society as a whole. Phones, laptops, Gameboys, and any other electronic available all rely on pure mathematics. Ben Orlin wrote about the invention of the computer saying that,” . . . one of the purest mathematical enterprises ever undertaken . . . It gave us the computer, which in turn gave us… well… the world we know (Orlin).  The invention of the computer changed the world, and it came from pure mathematics. Not only does the processing of the computer use pure mathematics, but also the components powering it use pure mathematics. Electrical engineering requires the use of imaginary numbers. For the longest time imaginary numbers were thought to be useless, it was thought that imaginary numbers meant nothing. Jim Lesurf wrote regarding complex numbers in electronics, saying that,” Complex numbers are used a great deal in electronics. The main reason for this is they make the whole topic of analyzing and understanding alternating signals much easier. “(Lesurf). This shows that even a pure mathematical idea can be used in today’s world. In understanding complex numbers, it allows for a better understanding of electronics, which are an integral part of technology today.

            Pure mathematics is an important part of society today, because pure mathematics is still relevant, and will always be relevant. Farida Kachapova also wrote that,” . . . even the oldest known mathematical formulae . . . known 2400 years ago by Babylonians, Chinese and later the Greeks …are the bread and butter of present-day elementary mathematics.” (Kachapova). This means that pure mathematics is still used today. The math the Greeks were using is still being used in the classroom today. Now not only is pure mathematics relevant today, but it will be relevant tomorrow, and for the future to come. The universe will never be fully explained, and there will always be new mysteries, new problems, and a need for new mathematics to try to explain it. Going back to G. H Hardy, “For what is useful above all is technique, and mathematical technique is taught mainly through pure mathematics.” (Hardy). Pure mathematics will always be the tool of applied mathematics. This meaning that no matter whatever problems arise, pure mathematics will be used. The fundamentals of math will always be used to understand new forms of applied mathematics.

            Pure mathematics is often times seen as non-useful form of mathematics. This is simply because pure mathematics is math for the sake of math. The common misconception is that there are not any benefits from working in pure mathematics, and only applied mathematics gets real world results. Pure mathematics is the root of all applied mathematics. Any applied mathematics goes back to number theory, algebra, or calculus roots. Although pure mathematics might not be making the break through, or discoveries, it is facilitating them. Pure mathematics is incredibly important in society today. This is because pure mathematics is the fundamentals of all mathematics, pure mathematics helped developed many of the technological advances of today, and pure mathematics with always be relevant in mans quest of understanding the universe.

           

      Works Cited

Kachapova, Farida. "On the Importance of Pure Mathematics." Science Publications. N.p., 2014. Web. 15 Mar. 2017.

Lesurf, Jim. "Complex Numbers." Complex Numbers. University of St. Andrew, n.d. Web. 18 Mar. 2017.

Orlin, Ben. "Why Do We Pay Pure Mathematicians?" Math with Bad Drawings. Word Press, 25 Feb. 2015. Web. 18 Mar. 2017.

Unknown. "Quotations of G H Hardy." History.mcs.st-andrews. N.p., Dec. 2013. Web. 12 Mar. 2017.

Squaring The Circle

            This week’s post is going to be about a problem called squaring the circle. The problem dates back to the time of the Greeks. The problem itself is not the difficult part, the difficult part of the problem is that one of the restrictions of the problem is to solve it only using the technology available to the Greeks. First, this post will cover the problem itself. Secondly, the post will cover how pi relates to the problem. Finally, the post will cover how pi was proven to be transcendental.

            Squaring the circle is a problem about finding a circle with the same area as a square. At first this problem may not seem hard, but not impossible. After a few minutes of thinking, it might become apparent that a circle of radius one, and a square of side length root pi would have the same are. That would be correct, but the Greeks did not have the square root of pi. The difficult part of the problem is that, the problem can only be solved with the abilities of the Greeks. That means there is no algebra, there are straight edges, and compasses. Straight edges, and compasses limit the ability of available math. With this Greek Technology the four basic functions of mathematics, addition, subtraction, multiplication, and division, can be performed. Addition and subtraction are straight forward. Adding is putting two line segments together, and subtraction is removing a line segment. Multiplication, and division require some scaling with similar triangles, but are also possible. There is one more operation that can be performed with a straight edge and a compass, and that is square rooting. Imagine a line of length x, and then a distance of one is added to it, then a semi-circle is drawn to the end of the line. If a line is drawn vertically from where the line x and line of length one meet, up to the semi-circle, the length of the line is the square root of x. Looking back at one of the solution to the problem, one might ask then why could the Greeks not square root pi? The answer is that pi is not a constructible number, a constructible number refers to a number that can made with a compass, and a straight edge.

            Pi is a rather important number in mathematics, but at the same time it is equally complex, and mysterious. Pi is a transcendental number. The name for transcendental comes from the idea that pi is beyond other numbers, that it has transcended to a high plane of understanding. Or in other words, it cannot be expressed as an algebraic number. This is why the Greeks could not solve the problem. Since pi cannot be an algebraic number, then automatically it cannot be a construable number. However, pi was not known to be transcendental until the late 1800’s. Even though pi was known, and could be calculated, it was still unclear whether or not it was an algebraic number. It was proven to be a transcendental number by a combined understanding from Euler, and a proof of e’s transcendentalism.

            Two important facts about the number e were proven, e is transcendental, and e to any algebraic term is transcendental. This was proven via contradictions. What that means is that if e was algebraic, certain properties would apply. Since those properties do not apply, e is transcendental. Now the same thing is done with pi, but there needs to be a relationship between e and pi. This is where Euler’s Identity comes to the rescue. Euler’s Identity is e raised to the (pi) x (i) is equal to negative one. Going back to the proof of e raised to an algebraic power, if pi is algebraic then e raised to it, should result in a transcendental number. Negative one is not transcendental, so by contradiction it must not be algebraic, so if pi is not algebraic, then it is transcendental.

            Going back to the problem itself, this means that there was not a way for the Greeks to find pi. Even if the Greeks tried to use a square with a side length not having pi in it, the radius circle would have to multiple by a ratio of pi to get a whole number answer. Pi is crucial to this problem, and so it was proven that under the given rules of the problem, Squaring the circle is an impossible problem to solve.

Sources https://www.youtube.com/watch?v=CMP9a2J4Bqw
              https://www.youtube.com/watch?v=seUU2bZtfgM
              http://mathworld.wolfram.com/CircleSquaring.html