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…]## 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.

## Natural Numbers and Arithmetic: Intuition

Welcome to the third post from my series on numbers and arithmetic. In the previous two posts, I gave an overview of **real numbers** (and their main subsets), as well as the main operations between them. And in the rest of the series, we’re going to build all these concepts from scratch. Well, the focus in today’s post is the most basic subset of the real numbers: the **natural numbers**.

*This post is part of my series* Numbers, Arithmetic, and the Physical World.

## Arithmetic Properties: A Comprehensive Breakdown

Welcome to the second post from my series on numbers and arithmetic. As a sort of continuation to the introductory post, here I want to talk about the most important properties of arithmetic operations.

*This post is part of my series* Numbers, Arithmetic, and the Physical World.