cs241

Continuing Where We Left Off On this page we will discuss some theorems and topics that are slightly more advanced and would likely scare 👻 anyone away. Remember that equivalence $\mod n$ is literally an equivalence relation, that means we can actually create equivalence classes of items that are all equivalent to each other.

Smallest Numbers Consider the natural numbers $\mathbb{N}$, what is the smallest number? Clearly the answer is $0$, as it is smaller than every positive number, and there are no negative numbers in $\mathbb{N}$ so it must have $0$ as the smallest.

Euler’s Totient Function In this page we will discuss some interesting properties, and get us to the point that we need in order to understand RSA Encryption from scratch. Now that we have a distinction between residue classes and prime residue classes, we can see that some numbers seem to be “less prime” than other numbers.

We’re Still on Grade School Stuff Remember back in grade school when the teachers taught you about functions? They were always represented as $f(x)$ (which apparently is according to Euler? Idk some YouTube video said its his notation) and were discussed as some sort of “input/output machine”.

What is RSA Encryption 📬 We’ve gone through a solid amount of number theory so far, and its crazy to think that this actually has a direct application in computer science, but it does.

Some of Math’s Coolest Terms We have the general idea of functions, but as with anything in math, we can define some useful properties to have. Injectivity We have as a propery of functions that every input must have exactly one output, but what if that output is always unique to the given input?

Data Structures We have already talked about how functions in computer science, and how they are also just as functions are in mathematics, but there are many more applications of functions than just this.

The Hardest Math Theorem ☠️ The pigeonhole principle is probably the theorem in math that scares me more than anything else in the world, because it is so powerful and fundemental, but actually applying it can be some of the most challenging problems a math student can face.

What does it mean to count? When I ask you “How many apples are there” depending on however many apples you are counting, you’ll mentally provide some number. $1,2,3,4,\ldots$ etc. Counting is counting, you’ve done it since you were a child.

The Rationals and the Reals On the previous page we showed (kinda) how the rationals $\mathbb{Q}$ was a countable set the same size as the naturals $\mathbb{N}$. Right now we will take some time to how $\mathbb{Q}$ is quite intertwined with the set of real numbers $\mathbb{R}$.