Introduction to Number Theory

When you enter your phone's PIN or make an online purchase, you're unknowingly using one of the most fascinating and powerful branches of mathematics: number theory. This discipline, which studies the deep properties of integers, connects the elegance of pure mathematics with crucial applications in our modern digital world.

Number theory explores seemingly simple yet surprisingly profound questions: Why do some numbers divide others perfectly? What makes prime numbers so special? How can we use the "remainders" of divisions to create secret codes? In this journey, you'll discover how these fundamental concepts interweave to form one of mathematics' most beautiful theoretical edifices.

What is Number Theory?

Number theory is the systematic study of the properties of integers $\mathbb{Z}=\{\dots ,-2,-1,0,1,2,3,\dots \}$ and the relationships that exist between them. At the heart of this discipline we find concepts like:

• Divisibility: when one number divides another exactly
• Prime numbers: the "fundamental building blocks" of arithmetic
• Factorization: how to decompose every number into its prime components
• Modular arithmetic: the mathematics of "remainders" and congruences

What makes this subject so fascinating is how concepts developed out of pure intellectual curiosity later prove fundamental for technologies we use daily, from the cryptography protecting our messages to the algorithms optimizing our online searches.

Your Learning Journey

Our journey through number theory is structured in five thematic blocks that will guide you progressively from the most solid foundations to the most sophisticated applications. Each block is designed to build upon knowledge acquired in the previous one, creating a deep and interconnected understanding.

Block 1: Numerical Foundations

We begin from the deepest roots, exploring numbers in their purest form and the fundamental relationships between them.

Natural Numbers will be our starting point: these numbers we use for counting hide surprising properties and are the foundation upon which we'll build everything else. You'll discover their algebraic properties and the crucial role they play in mathematics' structure.

Divisibility will introduce you to number theory's fundamental language. You'll learn when one number divides another, how to recognize divisibility patterns, and how these concepts connect to practical applications.

Prime Numbers will take you into the world of "arithmetic's building blocks." You'll explore why these numbers are so special, how to find them, and why they're at the center of modern computer security.

Greatest Common Divisor and Least Common Multiple will complete your foundations, showing you how these concepts link pure theory with practical applications like fraction simplification.

Block 2: Expanding Numerical Universes

Once solid with natural numbers, we'll expand our horizon to embrace increasingly rich and complex numerical systems.

Integers will introduce the concept of negative numbers and zero, expanding our operational possibilities. You'll see how this extension solves problems that were impossible with natural numbers alone.

Powers and Roots will show you how "multiplying a number by itself" and the inverse operation open worlds of mathematical possibilities, from exponential growth to geometric dimensions.

The journey continues through Rational Numbers, Irrational Numbers, and Real Numbers, where you'll discover how each extension of the number system solves previously unsolvable problems, from fractions to square roots, all the way to π and e.

Block 3: Advanced Powers and Notations

This block deepens exponential operations and the notations that make very large or very small numbers manageable.

Negative Exponents will introduce you to the concept that $2^{-3}=\frac{1}{2^3}$, paving the way for understanding fractional powers and their applications.

Powers with Rational Bases will further extend your skills, while Scientific Notation will give you tools to handle numbers from astronomy and subatomic physics.

The block will culminate with an Introduction to Logarithms, giving you a first taste of how to "invert" exponential operations.

Block 4: Digital World Arithmetic

Here we enter the heart of the digital age, exploring concepts that make modern technology possible.

Modular Arithmetic will introduce you to the world of "remainders" and congruences, the mathematics behind secret codes and hash algorithms. You'll discover how $17\equiv 2(\mathrm{mod}5)$ isn't just a mathematical curiosity, but the foundation of modern cryptography.

Number Bases will show you how numbers can be represented in bases other than 10, from the essential binary of computers to the hexadecimal of programmers.

Block 5: The Language of Mathematics

The journey concludes by providing you with the linguistic and logical tools to communicate and reason mathematically with precision.

Mathematical Sets will introduce you to the fundamental language of modern mathematics, while Mathematical Language will give you mastery of the symbols and conventions that make precise expression of mathematical ideas possible.

Mathematical Reasoning will conclude the path by giving you tools to construct rigorous proofs and reason logically about mathematical problems.

Why This Order?

The sequence we'll follow reflects a precise pedagogical logic: each concept is introduced at the moment when you have all the necessary tools to understand it deeply, and when its understanding enriches and illuminates what you've already learned.

The beauty of this approach is that you'll see how apparently disconnected concepts reveal themselves to be deeply interconnected: the prime numbers you'll encounter at the beginning will reappear in modular arithmetic, divisibility properties will illuminate number systems, and the mathematical language you'll develop will allow you to express precisely all the insights you'll have acquired.

From Pure Thought to Concrete Applications

One of number theory's most fascinating characteristics is how concepts developed out of pure intellectual curiosity later prove fundamental for the modern world. During our journey, you'll see how:

• Prime number properties protect your online transactions
• Modular arithmetic makes digital calendars and clocks possible
• Number systems allow computers to "think" in binary
• Prime factorization is at the heart of compression algorithms

Beautiful Connection: Number theory demonstrates that there's no sharp separation between "pure mathematics" and "applied mathematics." The boundaries are fluid, and often the most abstract discoveries become the most practical applications.

Let's Begin the Journey

Number theory awaits you with its intellectual challenges and aesthetic rewards. In the upcoming articles, we'll start from the most solid foundations: the natural numbers we use for counting, discovering the surprising properties that hide behind the apparent simplicity of "$1,2,3,\dots$".

Prepare to look at numbers with new eyes, to discover hidden patterns, and to develop that special form of mathematical intuition that makes number theory one of mathematics' most elegant and powerful disciplines.

The first step of our journey takes us to explore Natural Numbers: the foundation upon which we'll build our entire theoretical edifice.

Next: Natural Numbers (ℕ) - Discover the fundamental properties of the numbers we use for counting and their central role in mathematics.