We have reached the point where we can properly introduce infinite processes, and show that the set of rational numbers cannot include all of the numbers we’ll need. You can read this lesson here.
Now that the field axioms have been established, we can formally define the rational numbers. Most elementary school students refer to them as “fractions.” You can read this lesson here.
The second volume of Math From Scratch is here, with the intent to reach the definition of the real numbers. We will cover some Euclidean geometry along the way. This lesson covers the axioms of an algebraic field.
One of the most common applications of advanced mathematics is in encryption. The RSA encryption algorithm is the most common version, and is the basis of the security that underlies the https protocols used to connect users to online retailers, banks, and so forth. This also concludes the first volume of the Math From Scratch, which can be obtained in a single document through Graphicly. Join us when we return in September with the goal of defining rational, irrational and real numbers.
The Euler Totient function is the focus of this month’s lesson, and is the final piece of the puzzle required before we reach the RSA encryption algorithm in June.
We are finally able to cover the Chinese Remainder Theorem, one of the oldest recorded mathematical theorems and one that is of great importance to many computer science applications.
We are now equipped to define and solve linear diophantine equations. We are on track to cover RSA encryption algorithms in June.
Our teaching tidbits and Math From Scratch are overlapping for the first time with the Euclidean algorithm. If all goes as planned, RSA encryption algorithms will be derived and explained in June.
It turns out there needed to be more groundwork than I had anticipated before hitting the Chinese Remainder Theorem. This lesson introduces modular arithmetic, and the Chinese Remainder Theorem will be covered later. I will also be taking a detour to cover all of the math inherent to RSA encryption algorithms in the coming months.
The results for testing numbers for divisibility in base 10 are laid out here. It appears we’ll be reaching the rational numbers at some point in the spring of 2012.