222. Integers - JulTob/Mathematics GitHub Wiki
Natural numbers are closed under addition, meaning that adding two natural numbers always results in another natural number. This property is essential to our algebraic structure.
However, we run into a limitation when we try to subtract one natural number from another:
- Example: There’s no solution for
$3+x=1$ within$ℕ$ , as$x$ would need to be negative. - This limitation leads to the discovery of integers
$ℤ$ , the set that includes all natural numbers and their negatives.
Reflection: Why is (-1) x (-1) = +1?
The reason behind the rule that the product of two negative numbers is positive may seem puzzling at first. However, this rule maintains the consistency of the distributive property across integers. Reflecting on this question deepens our understanding of the algebraic structures we rely on.
-3 -2 -1 0 1 2 3
───╂───╂───╂───╂───╂───╂───╂───
The Integer numbers are built from the extension of the naturals under the subtraction operation. This is the inverse of addition.
Sea 𝑎,𝑏,𝑐,𝑑 ∈ ℕ,
(𝑎,𝑏)𝓡(𝑐,𝑑) ⟺ 𝑎+𝑑=𝑏+𝑐 -- Reflexiva, Simétrica, transitiva
⇒ 𝑧∊ℤ=ℕ×ℕ
𝑧=(𝑎,𝑏)
-𝑧=(𝑏,𝑎)
Un grupo es un conjunto G en el que está definida una ley de composición *
que tiene las siguientes propiedades
(a * b) * c = a * (b * c)
a*e=e*a=a : ∃e ∀a e∈G
a * á = á * a = e : ∀a ∈ G ⟹ ä⊂G
a*b=b*a : ∀a ∀b
Aditivo
- Neutro:
0
- Simétrico: a̚
Conmutativo
- Neutro:
1
- Simétrico: ä
Un anillo es un conjunto A en el que están definidas dos leyes de composición, una suma respecto de la cual A es un grupo abeliano, y un producto que es asociativo y distributivo respecto de la suma.
Como todo anillo es un grupo aditivo, en un anillo A se verifican todas las propiedades válidas para grupos. La siguiente proposición establece otras propiedades válidas en cualquier anillo. Proposición: En todo anillo A se verifican las siguientes propiedades:
- a • 0 — 0 • a — 0 para todo a ∈ A. (0 es el elemento neutro para la suma en A).
- (—a) • b = a • (—b) = — ab cualesquiera que sean los elementos a y b de A.
- (—a) • (—b) = ab cualesquiera que sean los elementos a y b de A El conjunto 1L de los números enteros es un anillo conmutativo con uni dad. Ya sabemos que 7L es un grupo aditivo abeliano. Habrá que probar que el producto de números enteros tiene las siguientes propiedades:
Asociativa: (ab)c = a(bc)
Conmutativa: ab=ba
Distributiva respecto de la suma: a(b+c)=ab+ac
Existencia de neutro: a·1=1·a=a
#Ordenación de enteros
a<b ∧ b<c ⟹ a<c
a<b ⟹ a+c<b+c
a<b ∧ c<d ⟹ a+c < b+d
a<b ∧ c>0 ⟹ ac<bc
a<b ∧ c<0 ⟹ ac>bc
a²≥0
1>0
A set is well-ordered if every non-empty subset has a least element. This property holds for
-
Absence of a Least Element in Unbounded Subsets:
- Consider the entire set of integers
$( \mathbb{Z})$ . It contains negative values that extend infinitely in both the positive and negative directions:$( { \dots, -3, -2, -1, 0, 1, 2, 3, \dots } )$ . - For any non-empty subset of
$( \mathbb{Z} )$ that includes all negative numbers (e.g., $( { \dots, -3, -2, -1 } )$), there is no smallest element in this subset. - Therefore,
$( \mathbb{Z} )$ cannot be well-ordered, as certain subsets (such as those containing all negative integers) do not have a least element.
- Consider the entire set of integers
-
Contrast with
$( \mathbb{N} )$ :- In
$( \mathbb{N} )$ , any non-empty subset has a least element because all elements are positive and ordered from smallest to largest without bounds in only one direction (toward infinity).
- In
Here are key notes extracted from Chapter 2: The Integers, covering topics such as division, the Division Theorem, the concept of greatest common divisors (GCD), Euclid's Algorithm, and the Fundamental Theorem of Arithmetic.
-
Basic Idea: For any integers
$( a )$ and$( b )$ , we can express$( b )$ as$( b = aq + r )$ , where$( q )$ is the quotient and$( r )$ is the remainder, with$( 0 \leq r < |a| )$ . - Example:
$( 326 = 21 \cdot 15 + 11 )$ .
-
Theorem: For any integer
$( a \neq 0 )$ and any integer$( b )$ , there exist unique integers$( q )$ and$( r )$ such that$( b = aq + r )$ and$( 0 \leq r < |a| )$ . -
Proof Outline:
-
Existence: Proved by induction on
$( b )$ . -
Uniqueness: If there are two pairs
$( (q, r) )$ and$( (q', r') )$ such that$( b = aq + r = aq' + r' )$ , it follows that$( q = q' )$ and$( r = r' )$ .
-
Existence: Proved by induction on
-
Definition: The GCD of two integers
$( a )$ and$( b )$ (not both zero) is the largest positive integer$( d )$ that divides both$( a )$ and$( b )$ , denoted$( \gcd(a, b) = d )$ . -
Properties:
- If
$( a \neq 0 )$ , then$( \gcd(a, 0) = |a| )$ . - Every pair of integers has a GCD, which is unique.
- If
- Purpose: Efficiently computes the GCD of two integers using repeated applications of the Division Theorem.
-
Steps:
- For integers
$( b )$ and$( a )$ with$( |b| \geq |a| )$ , set$( b_0 = b )$ and$( a_0 = a )$ . - Repeat
$( b_i = a_i q_i + r_i )$ where$( r_i )$ is the remainder until$( r_i = 0 )$ . - The GCD is the last non-zero remainder
$( r_{i-1} )$ .
- For integers
-
Theorem: For integers
$( a )$ and$( b )$ (not both zero), there exist integers$( x )$ and$( y )$ such that:
-
Constructive Proof: Using Euclid’s Algorithm, the GCD can be expressed as a linear combination of
$( a )$ and$( b )$ .
- Statement: Every non-zero integer (other than $( \pm 1 )$) can be factored uniquely as a product of prime (irreducible) integers, up to order and sign.
-
Definitions:
-
Irreducible: An integer
$( p )$ (not $( \pm 1 )$) is irreducible if whenever$( p = ab )$ , either$( a )$ or$( b )$ is$( \pm 1 )$ . -
Prime: An integer
$( p )$ (not 0 or $( \pm 1 )$) is prime if$( p | ab )$ implies$( p | a )$ or$( p | b )$ . -
Equivalence: In
$( \mathbb{Z} )$ , irreducibility and primeness are equivalent.
-
Irreducible: An integer
-
Theorem: If an integer
$( x = a_1 a_2 \dots a_n = b_1 b_2 \dots b_m )$ , where the$( a_i )$ and$( b_j )$ are all irreducibles, then$( n = m )$ and the irreducibles$( b_j )$ can be rearranged such that$( a_i = \pm b_i )$ . - Proof Outline: Uses induction on the number of irreducible factors and the property that irreducibles are prime.
- Euclid’s Algorithm: Originally seen as a geometric procedure for finding a common measure (line segment) that divides two given line segments.
- Prime Factorization: Euclid stated that if a number is the least common multiple of primes, then it cannot be divided by any primes other than those in its factorization, aligning with the Fundamental Theorem of Arithmetic.
-
Division and Remainders: Division in
$( \mathbb{Z} )$ always produces a unique quotient and remainder. - GCD and Euclid’s Algorithm: The GCD can be computed efficiently using Euclid’s Algorithm, and the GCD can be written as a linear combination.
- Fundamental Theorem of Arithmetic: Every integer can be uniquely factored into primes, a foundational result in number theory.
To find the quotient and remainder for each pair, we’ll apply The Division Theorem, which states:
For any integers
$( a )$ and$( b )$ with$( a \neq 0 )$ , there exist unique integers$( q )$ (quotient) and$( r )$ (remainder) such that:$$b = aq + r \quad \text{where} \quad 0 \leq r < |a|.$$
Let’s go through each case.
Here, we want to find integers
-
Calculate
$( q )$ :- Dividing
$( -120 )$ by$( 13 )$ gives approximately$( -9.23 )$ , so we choose$( q = -10 )$ to keep$( r )$ within the bounds.
- Dividing
-
Calculate
$( r )$ :- Substitute
$( q = -10 )$ into the equation:
- Substitute
Here, we want to find integers
-
Calculate
$( q )$ :- Dividing
$( 120 )$ by$( -13 )$ gives approximately$( -9.23 )$ , so we choose$( q = -9 )$ .
- Dividing
-
Calculate
$( r )$ :
- Substitute
$( q = -9 )$ into the equation:
Here, we want to find integers
-
Calculate
$( q )$ :- Dividing
$( -120 )$ by$( -13 )$ gives approximately$( 9.23 )$ , so we choose$( q = 9 )$ .
- Dividing
-
Calculate
$( r )$ :
- Substitute
$( q = 9 )$ into the equation:
Simplify: