Algebrais on JustToThePoint

Writing Proofs

Sat, 14 May 2022 00:00:00 +0000

Mathematical induction proves that we can climb as high as we like on a ladder, by proving that we can climb onto the bottom rung (the basis) and that from each rung we can climb up to the next one “(the step).” — Concrete Mathematics, by Ronald Graham, Donald Knuth, and Oren Patashnik.

The Architecture of Truth: A Guide to Mathematical Proofs

In mathematics, a proof is more than just a verification of facts—it is a logical narrative. It explains not just that something is true, but why it is inescapably true. To build these narratives, mathematicians rely on a specific toolkit of logical frameworks. Below, we explore the four foundational methods of mathematical proof.

The Natural Numbers. The Binomial Theorem.

Sat, 14 May 2022 00:00:00 +0000

There is nothing more practical than a good theory, Leonid Brezhnev.

The Architecture of the Natural Numbers

When we first learn mathematics as children, we learn to count: $1, 2, 3, 4\dots$ and so on. We accept these numbers as self-evident facts of the universe. However, in formal mathematics, the natural numbers are not assumed; they are rigorously constructed from foundational logic.

A natural number is a mathematical object. It is a number that occurs commonly in nature. As such, it is a whole, non-negative number, that is, a member of the sequence $0, 1, 2, 3, \cdots,$ obtained by starting from zero and adding 1 successively.

Euler's Totient Function and Modular Exponentiation

Fri, 14 Jan 2022 00:00:00 +0000

Introduction

In number theory and modern cryptography, dealing with astronomically large numbers is a daily necessity. To efficiently compute large powers in modular arithmetic, mathematicians rely on a powerful two-step combination: Euler’s Totient Function (to shrink the exponent) and the Method of Repeated Squares (to compute the final value).

Euler’s Totient Function

Euler's totient function, denoted by $\varphi(n)$, counts the number of positive integers up to $n$ that are relatively prime (coprime) to $n$. Two numbers are relatively prime if their greatest common divisor (GCD) is $1$.

Integers. Well Ordering Principle. Division & Euclidean Algorithm.

Fri, 14 Jan 2022 00:00:00 +0000

When you have eliminated the impossible, whatever remains, however improbable, must be the truth — Sherlock Holmes

The Integers, Divisibility, and the Euclidean Algorithm

While the natural numbers ($\mathbb{N}$) allow us to count, they possess a fatal algebraic flaw: we cannot always subtract. To solve an equation like $x + 5 = 2$, we must expand our mathematical universe. We must invent the integers.

Constructing the Integers ($\mathbb{Z}$)

An integer is a whole number that can be positive, negative, or zero. But mathematically, how do we build negative numbers using only the natural numbers?

The Foundation of Mathematics: An Introduction to Set Theory

Sat, 14 May 2022 00:00:00 +0000

There are two ways to do great mathematics. The first is to be smarter than everybody else. The second way is to be stupider than everybody else — but persistent, Raoul Bott.

An Introduction to Set Theory

Set Theory is the fundamental language of modern mathematics. Almost every mathematical concept—from numbers and functions to geometry and probability—is built on the foundation of sets.

At its most basic level in Naive Set Theory, a set is a well-defined collection of distinct objects. The objects within the collection are called the elements (or members) of the set.

Cartesian Products, Relations, and Partitions

Tue, 12 Jul 2022 00:00:00 +0000

The Cartesian Product

The Cartesian product of two sets $A$ and $B$, denoted $A \times B$, is the set of all possible ordered pairs $(a, b)$, where the first element $a$ belongs to $A$, and the second element $b$ belongs to $B$.

Formally, we write: $A \times B = \{(a, b) \mid a \in A \text{ and } b \in B\}$. Example: Let $A = \{1, 2\}$ and $B = \{\text{Red, Blue}\}$. $A \times B = \{(1, \text{Red}), (1, \text{Blue}), (2, \text{Red}), (2, \text{Blue})\}$