#Eulers Totient Function

Let’s take a moment to appreciate how beautifully this Euler’s Totient Function behaves. #eulertotient #eulerfunction #numbertheory #coprimes #mathxmatrix

Leonhard Euler was a Swiss polymath, famous for his numerous contributions to mathematics. Some of them, which are included in this video are 1)e^i𝛑 +1 = 0 : Euler's Identity. It relates the mathematical constant e, the complex unit i (√-1), the circle constant 𝛑, the unit quantity 1 and the null quantity 0. It is more correct to write it as e^i𝛑 = -1. 2)lim n→∞ (1 + 1/n)ⁿ : the formula for the aforementioned mathematical constant e. e is a unique number. Aproximately 2.71828, it is the value such that the derivative of eˣ is eˣ. 3)The 7 bridges of königsberg : a classic 18th century mathematical puzzle regarding whether a walker could cross each of the city's seven bridges over the Pregel River exactly once and return to the start. It is not possible, as Euler would go on to proove. To do this, he used graph theory to do so; reducing the four landmasses and 7 bridges involved into a graph. 4)eˣ = 1 + x/1! + x²/2! + x³/3! ... : This is the Taylor Series expansion for eˣ. It can be derived mathematically from the properties of e and eˣ. Differentiating it term by term reveals that the derivative of eˣ is eˣ. 5)The next image is of a line called the Euler line of a triangle. It is the line that passes through the Centroid, Orthocenter and Circumcenter of a non-equilateral triangle. 6)The Euler-Lagrange equation : A fundamental differential equation in calculus that determines the function that makes a functional, such as action, stationary. It is used in optimization and feild theory, among other things. 7)E-V+F=2 : The Euler Polyhedron formula. It describes a fundemental relation that all polyhedra must have.E,V and F are number of Edges, Verticies and Faces respectively. 8)Euler's alternative polar form is a way to rewrite trig functions with the exponential function, eˣ. 9)This image shows the above as formulae. 10)Euler's theorem in number theory states that if a and n are coprime positive integers ( gcd(a,n) = 1 ), then a^ϕ(n)≡1(mod n) , where ϕ(n) is Euler's totient function (the count of integers up to n coprime to n). It is a generalization of Fermat's little theorem. Have a nice day :) #memes #meme #stemeducation #mathematics #sciencemem

📐🔄 Ecuación de Euler: exp(iωt) = cos(ωt) + i·sen(ωt) 🔍 La fórmula que conecta el movimiento circular con las funciones seno y coseno. Este contenido visual presenta la forma más utilizada de la ecuación de Euler, fundamental en física e ingeniería para describir oscilaciones, ondas y señales periódicas. --- ▶️ La ecuación: e^(iωt) = cos(ωt) + i·sen(ωt) donde: · ω es la frecuencia angular (radianes por segundo) · t es el tiempo · i es la unidad imaginaria --- 📌 Interpretación geométrica: En el plano complejo, e^(iωt) representa un punto que gira sobre un círculo unitario: · Parte real: cos(ωt) → proyección horizontal · Parte imaginaria: sen(ωt) → proyección vertical A medida que t avanza, el punto describe un movimiento circular uniforme. --- 🔍 Visualización en el vídeo: · Se muestra un círculo unitario en el plano complejo. · Un vector rojo gira con velocidad angular ω. · Su proyección sobre el eje real (horizontal) traza una curva coseno. · Su proyección sobre el eje imaginario (vertical) traza una curva seno. · Las dos curvas se desplazan en el tiempo desfasadas 90°. --- ⚙️ Aplicaciones prácticas: ➡️ Circuitos eléctricos (AC): Voltajes y corrientes como fasores. ➡️ Ondas mecánicas: Cuerdas vibrantes, sonido. ➡️ Procesamiento de señales: Series de Fourier, filtrado. ➡️ Mecánica cuántica: Función de onda de partícula libre. ➡️ Control automático: Respuesta en frecuencia de sistemas. ➡️ Óptica: Polarización circular de la luz. --- 🎯 Dirigido a: Estudiantes y profesionales de física, ingeniería eléctrica, mecánica, telecomunicaciones y matemáticas aplicadas. --- 💬 ¿En qué contexto usas esta ecuación: circuitos, ondas, control o procesamiento de señales? Comparte tu experiencia en los comentarios. . . . . . #maths #mathematics #euler #fypシ❤️💞❤️ #mathidea

Euler’s identity is often described as one of the most beautiful results in mathematics because it connects five fundamental constants in a single, simple statement: 0, 1, π, e, and i. At its core, it comes from Euler’s formula, which links exponential growth with circular motion by showing that complex exponentials trace out rotations in the plane. When you plug in π into this relationship, something remarkable happens: the expression collapses perfectly to -1, and rearranging gives the famous identity. What makes this so striking is that it ties together ideas from completely different areas of math, including algebra, geometry, and calculus, in a way that feels almost unexpected yet deeply natural. Intuitively, Euler’s identity works because of how exponential functions behave when extended into the complex world. Instead of just growing or shrinking, they begin to rotate, encoding both magnitude and direction at once. The number e controls continuous growth, π represents a full rotation in a circle, and i introduces a perpendicular direction, allowing movement in a new dimension. When these pieces interact, they don’t just coexist—they fit together perfectly, revealing a hidden structure beneath many mathematical concepts. This is why Euler’s identity shows up in fields like signal processing, physics, and engineering, wherever waves, oscillations, or rotations are involved. Like this video and follow @mathswithmuza for more! #math #science #euler #scientist #maths

Euler so tuff gng 💔🥀 @foofee_da_gamer #meme #funny #maths #euler #eulerstheorem #integration #foryoupage #fyp #tuff #graphs

Euler's identity is a mathematical equation that links five fundamental constants: the number 𝑒, the imaginary unit 𝑖, the number 𝜋, 1, and 0, in a simple yet profound way. It is expressed as 𝑒^𝑖𝜋+1=0. This equation is important because it bridges several key areas of mathematics, such as complex numbers, exponentiation, and trigonometry. Its beauty lies in how it connects seemingly unrelated concepts in an unexpectedly simple and elegant form, making it a symbol of mathematical harmony. . . . . . #edit #edits #math #meme #tutor #calculus #school #mathematics #learning #brainrot #integration #funny #relatable #euler

La Identidad de Euler en 3D En un vídeo anterior explicamos qué significa realmente la Identidad de Euler. Ahora estamos visualizando la curva real que traza "e elevado a i por t", pero en 3D. Esta espiral es la Identidad de Euler, llevada al espacio. El ángulo t avanza a lo largo del eje x. Los ejes y y z muestran las partes real e imaginaria: coseno de t y seno de t. Así que, en lugar de ver solo el seno y el coseno como ondas separadas, ahora puedes ver cómo surgen de una única exponencial compleja: de una sola trayectoria en forma de hélice. Luego la proyectamos: sobre el plano del coseno, la parte real sobre el seno, la parte imaginaria y sobre el plano complejo, donde dibuja un círculo Un solo movimiento, tres perspectivas: Así se ve la Identidad de Euler como curva en 3D. #mathematics #physics #maths #math

Leonard Euler, born on April 15, 1707, in Basel, Switzerland, was a brilliant Swiss mathematician and physicist who transformed science despite losing his eyesight in his thirties. He produced over 800 papers on calculus, graph theory, mechanics, optics, and astronomy, introducing notations like π, f(x), Σ, and e. Famous for Euler’s formula (e^(ίπ) + 1 = 0) and solving the Basel problem, he founded graph theory with the Seven Bridges of Königsberg. Euler worked in Saint Petersburg, Russia, until his death on September 18, 1783, leaving a legacy that shapes modern math and engineering.

Mathematics is the language of nature, and Euler’s Identity is its most elegant poem. 🌿🔢 Have you ever wondered how the most disparate branches of mathematics—trigonometry, calculus, and complex numbers—all connect? Enter Euler's Identity. This video breaks down the ingredients of this mathematical masterpiece: \pi (Pi): The "limitless ratio" of a circle’s circumference to its diameter. i: The "imaginary number," defined as the square root of -1. e: The base of "unbounded growth," or Euler’s number, emerging from the limit of (1 + \frac{1}{n})^n. When you combine these with 1 and 0, you get e^{i\pi} + 1 = 0. It’s a moment where everything in the universe seems to click into place. From the growth of populations to the waves in the ocean, these constants govern the world around us. 🌊🏙️ Whether you're a student struggling through integration or a professional engineer, there's no denying the sheer elegance of this proof. It's not just math; it's art. Which constant is your favorite? Let me know in the comments! 👇 #️⃣ Viral Hashtags #EulersIdentity #MathIsBeautiful #STEM #Calculus Physics ScienceExplained Mathematician Pi ImaginaryNumbers Euler Education StudyMotivation ScienceRules MathematicalProof DeepThoughts

Ululation is a long, wavering, high-pitched vocal sound produced by trilling the tongue and uvula, commonly used to express strong emotion. #dandy #wordoftheday #wordoftheweek #posh #femboy

Fórmula de Euler - La ecuación más hermosa de las matemáticas existe (y conecta todo). La fórmula de Euler une en una sola ecuación cosas que parecen totalmente distintas: exponenciales, números imaginarios y trigonometría. Dice que: 👉 e elevado a i·x es igual a cos(x) + i·sin(x). Esta relación explica cómo funcionan las ondas, las rotaciones y los movimientos periódicos, y aparece en ingeniería eléctrica, física cuántica y señales. Pero lo más increíble ocurre cuando x = π: e^(i·π) + 1 = 0 Una sola ecuación conecta cinco números fundamentales: e, i, π, 1 y 0. Por eso muchos la llaman la ecuación más bella de las matemáticas. 🎥 Video de @mathswithmuza Si no es el creador original del video, este espacio está reservado para atribuir el contenido al creador original. Si eres el autor y deseas que se añadan créditos o se retire el video, puedes contactarme por mensaje privado. Se comparte con fines educativos y divulgativos. #FórmulaDeEuler #Matemáticas #Ciencia #Curiosidades #Trigonometría #NúmerosComplejos #Aprender #CienciaReal

Euler’s number (e ≈ 2.718) isn’t just math trivia, it powers exponential growth, decay, and nearly every continuous system in science and engineering. Stan explains why e is everywhere. #MechanicalStan #StanExplains #EulersNumber #ExponentialGrowth #NaturalLogarithm #EngineeringMath #RCcircuits #AskStan #STEMContent #CalculusInRealLife #EulerWasRight