FoldUnfold Table of Contents Modular Arithmetic Modular Arithmetic Definition: If $a, b, m \in \mathbb{Z}$ then we say that $a$ is Congruent to $b$ modulo $m$ denoted $a \equiv b \pmod m$ if $m \mid (a – b)$. The congruence of two integers is an equivalence relation which we can describe in other words. We […]

## The Unique Prime Factorization Theorem

FoldUnfold Table of Contents The Unique Prime Factorization Theorem The Unique Prime Factorization Theorem We will now look at a very important theorem which says that any integer $n > 1$ can be written uniquely as a product of primes $p_1, p_2, …, p_k$: (1) \begin{align} \quad n = p_1p_2…p_k \end{align} Theorem 1 (The Unique […]

## Euclid’s Theorem of the Existence of Infinitely Many Primes

FoldUnfold Table of Contents Euclid’s Theorem of the Existence of Infinitely Many Primes Euclid’s Theorem of the Existence of Infinitely Many Primes Recall from the Prime and Composite Numbers page that an integer $p > 1$ is a prime number if the only divisors of $p$ are $\pm 1$ and $\pm p$, and an integer […]

## Prime and Composite Numbers

FoldUnfold Table of Contents Prime and Composite Numbers Prime Numbers Composite Numbers Prime and Composite Numbers Prime Numbers Definition: An integer $p > 1$ is said to be Prime if the only divisors of $p$ are $\pm 1$ and $\pm p$. For a long time it was debated whether the number $1$ should be classified […]

## Integer Divisibility Review

FoldUnfold Table of Contents Integer Divisibility Review Integer Divisibility Review We will now review some of the recent material regarding integer divisibility. On the The Well-Ordering Principle of the Natural Numbers that the Well-Order Principle for the set of natural numbers $\mathbb{N}$ states the following important fact: Every NONEMPTY subset of the natural numbers has […]

## The Greatest Common Division Between Integers

FoldUnfold Table of Contents The Greatest Common Division Between Integers The Greatest Common Division Between Integers Definition: If $a, b \in \mathbb{Z}$ then the Greatest Common Divisor of $a$ and $b$ denoted $\gcd (a, b)$ or $(a, b)$ is an integer $d \in \mathbb{Z}$ such that: 1. $d \mid a$ and $d \mid b$. 2. […]

## The Division Algorithm

FoldUnfold Table of Contents The Division Algorithm The Division Algorithm One rather important aspect of the divisibility of integers is that if $a, b \in \mathbb{Z}$ then $a$ can be written as the product of some quotient $q$ with $b$ plus a remainder $r$. For example, if $a = 11$ and $b = 3$, then […]

## Integer Divisibility

FoldUnfold Table of Contents Integer Divisibility Integer Divisibility Definition: Let $a, b \in \mathbb{Z}$. Then $b$ is said to be Divisible by $a$, or, $a$ is said to Divide $b$ written $a \mid b$ if there exists a $q \in \mathbb{Z}$ such that $aq = b$. The number $a$ is said to be a Divisor […]

## The Well-Ordering Principle of the Natural Numbers

FoldUnfold Table of Contents The Well-Ordering Principle of the Natural Numbers The Well-Ordering Principle of the Natural Numbers We begin our look through abstract algebra with a rather simple theorem regarding the set of natural numbers known as The Well-Ordering Principle of the Natural Numbers. Consider the following set which we define to be the […]

## Normal Linear Operators

FoldUnfold Table of Contents Normal Linear Operators Normal Linear Operators We are now going to look at another important type of linear operator known as Normal linear operators which we define below. Definition: Let $V$ be a finite-dimensional inner product space. Then a linear operator $T \in \mathcal L(V)$ is said to be Normal if […]

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