History

In the world of abstract algebra, arithmetic is what algebra originally grew from. It is very crucial for students to learn the key characteristics of arithmetic before moving on to abstract algebra. The Division Algorithm is a very crucial part of abstract algebra that students should learn. This was presented in Euclids Elements, which was a compilation of books. The main portion of the books were in relation to geometry, but some of these books were devoted to algebra.

Book VII begins with a variety of definitions that serve all three arithmetical books, including those of prime and composite numbers. Where Euclid phrased these in terms of line segments, we shall use modern notation and wording.

Thus, 39 is divisible by 13, since 39 = 13 x 3. However, 10 is not divisible by 3; for there is no integer c that makes the statement 10 = 3c true.

Euclids work on the fundamental aspect of arithmetic have helped in every realm of mathematics and abstract algebra is one area of mathematics that is no exception. Although the concept of division is something that is covered in earlier grade levels, it is presented differently. However, this does not mean that students would not be able to understand the algorithm. Euclid also presented a process of finding the greatest common divisor of two integers (GCD (a,b)).

To obtain the greatest common divisor of two integers, we could always proceed as in the last example by listing all their positive divisors and picking out the largest one common to each; but this is cumbersome for large numbers. A more efficient process is given early in the seventh book of the Elements. Although there is historical evidence that this method predates Euclid by at least a century, it today goes under the name Euclidean algorithm. Euclids procedure relies on a result so basic that it is often taken for granted: the division theorem. Roughly, the theorem asserts that an integer a can be divided by a positive integer b in such a way that the remainder is smaller than b. An exact statement of this fact follows.



The integers q and r are called the quotient and the remainder in the division of a by b. We accept the division theorem without proof, noting that b is a divisor of a if and only if the remainder r in the division of a by b is zero.

In todays world there are many applications of modular arithmetic and this area of study of mathematics has many contributors, but there is one mathematician that is considered crucial. Carl Friedrich Gauss developed the modern approach to modular arithmetic. He first presented these ideas in his book Disquisitiones Arithmeticae, published in 1801.

Gauss was one of those remarkable infant prodigies whose natural aptitude for mathematics soon becomes apparent. He used to say, laughingly, that without the help or knowledge of others he had learned to reckon before he could talk. As a child of three, according to a well-authenticated story, he corrected an error in his fathers weekly payroll calculations. His arithmetical powers so overwhelmed his schoolmasters that by the time Gauss was nine years old, they admitted that there was nothing more they could teach the boy. It is said that in his first arithmetic class, Gauss astonished his teacher by instantly solving what was intended as a long busy work problem, Find the sum of all the numbers from 1 to 100. The young Gauss later confessed to having recognized the pattern.

Anyone that has studied in the fields of mathematics, astronomy, or physics more than likely has heard of Carl Friedrich Gauss. Although he is more publicly known for his work in other fields, from this story from his childhood it is obvious that he was a brilliant mathematician as well. His work in mathematics has become the basis of modular arithmetic and is a strong presence in the field of abstract algebra.

In the first chapter of Disquisitiones Arithmeticae, Gauss introduced the concept of congruence and the notation that made it such a powerful technique. He explained that he was induced to adopt the symbol = because of the close analogy with algebraic equality. According to Gauss, If a number n measures the difference between two numbers a and b, then a and b are said to be congruent with respect to n; if not, incongruent.

This concept that Gauss introduced is the basis for congruence in Z and modular arithmetic. In terms of abstract algebra, this is a very fundamental topic. These two great mathematicians, along with many others, have contributed much to this area of study within mathematics and it is from this that our world has been able to advance in a wide variety of fields.

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