Significance and Applications

Applications

There are many applications of abstract algebra and number theory. The main area that abstract algebra is mentioned is in the feild of computer science. Consider the following example. Use the applet linked below explore the use of modular arithmetic in cryptography.

Ceasar Cipher 2

The applet just presented shows how messages can be coded by using modular arithmetic. These techniques have been used for coding messeges during times of war. Cryptography is still used today for computer programming and creating security for computer systems. Although computer science is one field that uses abstract algebra extensively, there are several other areas of study that have benefitted from using abstract algebra and number theory.

"Integers have repeatedly played a crucial role in the evolution of the natural sciences. Thus, in the 18th century, Lavoisier discovered that chemical compounds are composed of fixed proportions of their constituents which, when expressed in proper weights, correspond to the ratios of small integers. This was one of the strongest hints to the existence of atoms; but chemists, for a long time, ignored the evidence and continued to treat atoms as a conceptual convenience devoid of physical meaning. (Ironically, it was from the statistical laws of large numbers, in Einstein's and Smoluchowski's analysis of Brownian motion at the beginning of our own century, that the irrefutable reality of atoms and molecules finally emerged.)"

Uses for abtract algebra can be found in almost every field of science. There are many different discoveries throughout history that have resulted from the use of techniques of abstract algebra.

"One of the most dramatic consequences of congruential arithmetic is the existence of the chemical elements as we know them. In 1913, Nicls Bohr postulated that certain integrals associated with electrons in "orbit" around the atomic nucleus should have integer values, a requirement that 10 years later became comprehensible as a wave interference phenomenon of the newly discovered de Broglie matter waves: In essence, integer-valued integrals meant that path differences are divisible by the electron's wavelength without leaving a remainder."

These are just some of the different areas in which modular arithmetic or abstract algebra have helped develop new advancements.

Significance

From all of these different applications presented, hopefully it is apparent that these applications are significant enough for us to keep developing new mathematics in this field. It stands to reason that in order to develop more advancements in the field of abstract algebra, we must start developing more advancements for teaching these materials to students. I believe that if we develop students understanding of these key concepts earlier in the educational system, then we will increase the potential for greater advancements in this field. Abstract algebra and number theory are very crucial fields of mathematics and these are areas that should be developed more. The following lesson plan is one that I developed during my work over this semester. The objective of this lesson was to have students discover the relationship of prime factors.

This lesson on prime factors is designed so that the students can discover that all numbers can be broken down to their prime factors. Teaching this topic in this specific way will allow the students to gain a better understanding of prime numbers and factors of integers. In conclusion, this is the main objective of my semester project and I hope that I can convince my colleagues that developing well thought out teaching strategies for teaching abstract algebra will only benefit our study of abstract algebra in the future.

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