Top Tweets for #proofInATweet
L'aire de AA'C est d'une part :
c²/2 (avec base=AC=c)
d'autre part :
Aire(triangle vert BCC') + Aire(AA'BC' hachuré en bleu)
=a²/2 + b²/2
Le quadrilatère AA'BC' en bleu peut être vu comme deux triangles de même base AB=b et dont la somme des hauteurs vaut b.
#proofinatweet
@jamestanton Exactly the multiples of 4.
1️⃣Necessity :
Suppose n=a² - b² is even.
n= (a+b)(a-b) with an even factor. Then factors are both even since their difference (2b) is even. So n is a multiple of 4.
2️⃣Sufficient:
Say 4|n, write n=2p×2q.
n=(p+q)²-(p-q)²
#proofinatweet
#ProofInATweet that 100!>10¹⁰⁰
Proof: 10¹⁰⁰=100⁵⁰ and 100!=(100x1)(99x2)(98x3)...(51x50). Since each factor is ≥100 the result follows.
I just got to asked to prove that the least prime factor of an composite number is at most √n. I think this quite a neat problem as it unravels so nicely when you think about it and I thought my proof of it was quite pretty. (Proof on next tweet)
#ProofInATweet
Very nice proof !
Here's a #proofinatweet using complex numbers and the same notations.
z:=a+ib, z':=c+id
|zz'|=|z|.|z'| so |zz'|²=|z|².|z'|²
So N.M=|zz'|² and is therefore the sum of two squares.
Here's a geometric proof that if N=a^2+b^2 and M=c^2+d^2 are each a sum of two squares, then so is their product N*M. Give the puzzles of this week, what is the matching result from considering equilateral triangles on a triangular lattice? [Any other versions? 3D lattices?]
![jamestanton's tweet photo. Here's a geometric proof that if N=a^2+b^2 and M=c^2+d^2 are each a sum of two squares, then so is their product N*M. Give the puzzles of this week, what is the matching result from considering equilateral triangles on a triangular lattice? [Any other versions? 3D lattices?] https://t.co/Pxozz7J0Ut](https://pbs.twimg.com/media/EjaQen7VcAEqSMf.png)
d(eˣ)=eˣ: the tangent to y=eˣ at (a,eᵃ) has gradient eᵃ/1. ln x is the inverse of eˣ: the tangent to y=ln x at (eᵃ,a) has gradient 1/eᵃ ⇒ d(ln x)/dx=1/x ⇒ ∫(1/x)dx=ln x+c #proofinatweet
The set of integers which are the sum of two squares is closed under multiplication #proofinatweet :
∀ (a, b, c, d) ∈ ℤ⁴, |a+ib|² |c+id|²=|(a+ib)(c+id)|²
@RufusWilliam @ThainMike A Level proof by contradiction:
If n is non-square positive integer, assume (for contradiction) that root n is rational, a/b
Squaring, we get n = a^2/b^2
Hence nb^2=a^2. Comparing prime factors, n must contain only repeated factors
Hence n is square
Contradiction
#proofinatweet
The Pythagorean Theorem has many proofs, usually either graphical or going through the properties of a right angle. Dijkstra proved this identity about all triangles and involving all the angles, for which the Pythagorean Theorem is just a particular case #ProofInATweet #MathType
Abel prize winners announced. Furstenburg's 1955 proof of the infinitude of primes is one of my favorite proofs ever, using #topology! He puts an interesting topology on Z, the integers. Here's how it goes: (next tweet #proofinatweet) https://t.co/hktloAvnXy
@NookiedV @geogebra If f(n) = 1/n, the sum from 1 to N of f(n) equals the integral of 1/floor(n) from 1 to N, which is greater than the integral of 1/n from 1 to N, which is ln(N). This increases without limit as N gets large #proofInATweet
The Pythagorean Theorem has many proofs, usually either graphical or going through the properties of a right angle. Dijkstra proved this identity about all triangles and involving all the angles, for which the Pythagorean Theorem is just a particular case #ProofInATweet #MathType
Pascal's Combinatorial Rule can be proved in a sentence:
"If from a set of n+1 we choose k objects, we either exclude the last one, in which case we choose k from the first n, or include it, in which case we choose k-1 from the remaining n"
#ProofInATweet #Combinatorics #MathType
Pascal's Combinatorial Rule can be proved in a sentence:
"If from a set of n+1 we choose k objects, we either exclude the last one, in which case we choose k from the first n, or include it, in which case we choose k-1 from the remaining n"
#ProofInATweet #Combinatorics #MathType
@michelvanbaal @mkeulemans @LaurensVHG Kijk nog eens naar #proofinatweet (uit de goede oude tijd).
Tegenwoordig speel ik vals: #bewijsopeenusem
Take the roots of a polynomial and plot them in the complex plane. The Gauss-Lucas Theorem ensures that the roots of the derivative of the original polynomial lie on the convex hull of those points, the area enclosed by them. Here's a short proof #ProofInATweet #Algebra #MathType
Have you ever wondered: how does one come up with the formula to solve a quadratic equation? Well, this is surprisingly easy to show and doesn't require any complex reasoning, just the very common move of "completing the square" #ProofInATweet #Algebra #Quadratic #MathType
The Substitution Rule is a powerful trick that can transform hard integrals into easy ones. It encodes the Chain Rule for derivatives into integrals via the almighty idea that integration and differentiation are inverse operations #Calculus #ProofInATweet #MathType
The Pythagorean Theorem has many proofs, usually either graphical or going through the properties of a right angle. Dijkstra proved this identity about all triangles and involving all the angles, for which the Pythagorean Theorem is just a particular case #ProofInATweet #MathType
#Combinatorics is the art of counting. Looking at numbers like n choose k as the number of ways to choose k objects from a bag of n objects gives birth to some awesome proofs. Not a single equation, just a simple reasoning about counting #ProofInATweet #MathType
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