Author: Marco Damele
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Six proofs that √2 is irrational
Dimostriamo che √2 è irrazionale usando 6 dimostrazioni diverse. 1° Proof: Suppose that √2 ∈ ℚ then there exist two coprime integers a and b such that √2=a/b. It follows that 2b²=a². So a² is even, that is, a is even, or a=2k with k being an integer. But then 2b²=4k², or b²=2k², from which…
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Che forma ha il sottoinsieme dei punti di ℝ³ soddisfacenti l’equazione -Z⁴+8Z³+X²+Y²-16Z²=0 ?
Consideriamo il sottoinsieme S dei punti (X,Y,Z) di R³ soddisfacenti l’equazione -Z⁴+8Z³+X²+Y²-16Z²=0 . Che forma ha S ? Per capirlo prendiamo un punto (X,Y,Z) in S, allora: -Z⁴+ 8Z³ + X² + Y² -16Z² =0. Tale condizione diventa: X²+ Y² = Z⁴ -8Z³ + 16Z² = (Z² -4Z)² quindi posto v=Z, esiste u in [0,2pi)…
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Every year admits a Friday the 13th
Theorem: Every year admits a Friday the 13th Proof: Note that a year admits a Friday the 13th if there is a month that begins with Sunday. We denote the days of the week with numbers. Specifically, 0 denotes Sunday, 1 Monday, up to 6, a number that denotes Saturday. If k is a natural…
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n-agonal numbers
Numbers are awesome. Here’s why. Yesterday I made a video about the so-called “n-gonal numbers”. I’ll explain what they are as we go. Let’s start by saying that a natural number k is said to be “square” if k=n² for some natural number n. This definition, which is now completely algebraic, was given in the…
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A Diophantine sistem.
We want to determine all integers x₁,…,x₂₀₁₁ such that If such integers exist, by Fermat’s little theorem we would have that xi⁷=xi (mod 7) for every i=1,…,2011. So 123=321 (mod 7) or 4=6 (mod 7) which is absurd.
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If K is a field (K,+) it is not isomorphic to (K{0},•).
Let (K,+,•) be a field, let us prove that its additive group (K,+) is not isomorphic to the group of invertibles of K, that is (K{0},•). Let us suppose that there is an isomorphism between them, then, since isomorphisms preserve the order of an element, we would have that: |{x ∈ K : x+x=0}|=|{x∈ K{0}…
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Ring with 4 elements
Let R be a unit ring with 4 elements. Let us prove that R is commutative. R ={0,1,a,b}. Let x,y be in R and let us prove that xy=yx. If x=0 then obviously xy=0=yx. If x=1 then xy=y=yx. We then have to prove that ab=ba. We have a+1∈ R. So a+1=0 or a+1=b. If a+1=0…
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Ring with p elements
Let us prove that every (unitary) ring R with p elements, p being prime, is isomorphic to ℤp. Let us consider the homomorphism φ:ℤ—>R , 1|—>1. We have that Ker(φ) is an ideal of ℤ so Ker(φ)=nℤ for some n∈ℕ. Now by the first isomorphism theorem ℤ/Ker(φ) is isomorphic to Im(φ). By Lagrange’s theorem we…
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Can a polynomial of degree n have more than n roots?
Let’s start with some definitions. Let R be a ring and f∈R [X] a polynomial of degree n≥0, then f=a0+a1X+…+anXⁿ where a0,a1,…,an are elements of R. An element α∈R is called a root of f if f(α)=0, that is, if a0+a1α+…+anαⁿ=0. For example, the polynomial X²-2 ∈ ℝ[X] has as its only roots √2 and…
Marco Damele 