The topic of numeral systems is something I wanted to cover in the previous post on number theory. That post got too long before I could even get to it, however. So, here we are, talking about it now. This topic is about the concrete ways of representing the otherwise abstract notions of numbers. Both in written form, as well as when stored as information in other types of media, like computer memory.[Read more…]
Introduction to Number Theory: The Basic Concepts
Hi, everyone. Today I want to talk about number theory, one of the most important and fundamental fields in all of mathematics. This is a field that grew out of arithmetic (as a sort of generalization) and its main focus is the study of properties of whole numbers, also known as integers.
Concepts and results from this field are important enough to have implications for (and be used in) almost every branch of mathematics. And not just mathematics. Number theory is also extremely fundamental to fields like cryptography and computer science and has applications in many other sciences, like physics and chemistry.[Read more…]
Cryptography During World War I
Welcome to part 5 of my series on cryptography! Today, the focus is going to be on codes and ciphers used during World War I. With a special focus on the most notable ciphers, as well as a particular code and the message it was used to encode. The latter — the Zimmermann Telegram — was a secret message whose interception had a dramatic effect on the dynamic of the war.
This post is part of my series Cryptography: Historical Intro & Combinatoric Analysis.[Read more…]
Mean and Variance of Discrete Uniform Distributions
In today’s (relatively) short post, I want to show you the formal proofs for the mean and variance of discrete uniform distributions. I already talked about this distribution in my introductory post for the series on discrete probability distributions. Well, this is a pretty simple type of distribution that doesn’t really need its own post, so I decided to make a post that specifically focuses on these proofs. More than anything, this is going to be a small exercise in algebra.
This post is part of my series on discrete probability distributions.[Read more…]
Euclidean Division: Integer Division with Remainders
In today’s post I want to talk about Euclidean division. This is a more general definition of division between two integers because, unlike regular division, it’s defined for any pair of integers (except when the divisor is 0). I originally wanted to cover this topic in my post on negative numbers. But I decided to take it out in a separate post in order to keep the length below some reasonable limit.[Read more…]
Negative Numbers and Arithmetic: Intuition
Hi everyone, and welcome to the next post from our journey in the world of numbers. Where we last left off, I was telling you about natural numbers — the objects that most naturally appear when we make our first steps in the world of Mathematics. Well, today’s post is about the next most natural kind of mathematical objects: negative numbers and the new structure they form together with natural numbers called integers!
This post is part of my series Numbers, Arithmetic, and the Physical World.[Read more…]
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