In this article, I take a look at classic mathematics. I discuss intuitionism and constructivism and the uses of classical mathematics throughout time. Then, from the Islamic Golden age to European developments, I review some of its histories. I also briefly explain some of the elementary fields of classic mathematics. In the end, mathematics concerns itself with the search for truth.
This definitive guide covers proofs, examples, algorithms, applications, and the Chinese Remainder Theorem history. It also includes links to additional resources such as online articles, courses, books, and tutors to help students learn from various sources. Professionals can also use these resources to increase their knowledge of the field or help structure courses for their students.
So you probably know a divisibility test for 2, 3, and 5. But what about 7, 11, 13, or even larger primes? In this article, I go over divisibility tests. Including how to create your own. I also discuss the Days of the Week problem, where you are to determine the day of the week from a given date very quickly.
The idea behind solving polynomial congruence equations is that we can reduce a congruence equation to an equivalent system of congruence equations using prime factorization. We then 1) solve each equation modulo a prime number (by brute force), 2) use Hensel’s Lifting theorem, and then 3) piece together the solutions using the Chinese Remainder Theorem. We provide several nontrivial examples many of which are workable by hand.
The Euclidean Algorithm is to find the greatest common divisor of two given integers. In this article, you will see this critical algorithm proven in detail. Further, I will show you how to use these computations to solve linear congruence equations and linear Diophantine equations. While this algorithm has been around a while, it is the key to much success.
We discuss several simple lemmas for greatest common divisors and linear combinations. I then prove Bezout’s identity to show that the greatest common divisor of two integers is the smallest linear combination. We also work through several elementary facts concerning relatively prime integers, and I present many examples.