Author: Thought Thrill
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The Game With Infinite Expected Value – The St Petersburg Paradox Explained
Imagine a game beginning with a stake of $2. The stake is the amount the player will be paid at the end. The player flips a coin, and if it lands on tails, the stake doubles. Otherwise, the game ends and the player collects the stake. Calculating the Expected Value The read more
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Every Proof That 0^0 is 1 but they get increasingly complex
Why Mathematicians Often Define 0⁰ as 1 The Empty Product Combinatorics and Tuples Functions and the Empty Set Further reading Why Mathematicians Often Define 0⁰ as 1 Throughout mathematics, 0⁰ is a notoriously problematic expression. Is it defined to have a value of 1, or does it have no mathematical meaning read more
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Complex & Imaginary Numbers Explained
From Squaring to Square Roots The Imaginary Unit The Complex Plane Operations on Complex Numbers Polynomial Equations and Complex Roots A Brief History of Complex Numbers Modeling Rotations From Squaring to Square Roots To start, consider multiplying a number by itself. For example, 4 × 4 = 16. This can be read more
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The Basic Math Iceberg Explained
The Math Iceberg Explained, Part 1 Counting Addition Subtraction Multiplication Division Equations Basic Geometry Order of Operations Decimals Factors Fractions Powers and Radicals Data Plots Variables Cartesian Coordinates Functions Additional Context on Number Systems Counting Counting is a method of finding what number of something you have. The number zero indicates read more
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The Discovery of the Atom Explained (From Ancient Greece to Bohr)
The Ancient Idea of Atoms John Dalton and Chemical Ratios J. J. Thomson and the Electron Rutherford and the Gold Foil Experiment The Problem with Classical Physics Niels Bohr and Quantized Energy Levels Toward Quantum Mechanics Further Reading The Ancient Idea of Atoms Our understanding of matter, everything we can touch, read more
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Where the Quadratic Formula Actually Comes From (Derived Step by Step)
What Are Complex Numbers? A Note on the Names Solving Quadratic Equations with Complex Numbers Working with Imaginary Solutions A Quick Aside on the Definition of i Deriving the Quadratic Formula What Are Complex Numbers? Complex numbers often arise when we are solving quadratic equations. We’ll begin with a simple read more
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Every Geometric Shape Used in 3D Printing Explained
The Geometry Behind 3D Printing 2D Profiles and Extrusion Platonic Solids Prisms and Pyramids Solids of Revolution The Torus Boolean Operations Parametric Surfaces The Geometry Behind 3D Printing 3D printing would not exist without geometry. Each generated object begins as a three-dimensional representation constructed from vertices, edges, and faces. This structure read more
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Every Unsolved Circle Problem that Sounds Easy
Five Circle Problems That Are Way Harder Than They Look The Erdős-Oler Conjecture The Gauss Circle Problem The Kissing Number Problem Unequal Circle Packing Sendov’s Conjecture The Erdős-Oler Conjecture A unit circle is a circle whose radius (the distance from the center to the edge) is 1. An equilateral triangle read more
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Unsolved Math Problems in Calculus That Sound Easy
The Casas-Alvero Conjecture The Riemann Hypothesis Navier-Stokes Existence and Smoothness The Jacobian Conjecture The Casas-Alvero Conjecture If an integer k can be expressed as the product of two integers m and n, then m and n are called factors of k. A common factor of two integers is a factor shared read more
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Unsolved Math Problems That Took Centuries to Solve
Famous Math Problems That Took Centuries to Solve Squaring the Circle Euler’s Sum of Powers Conjecture The Four Color Map Theorem Legendre’s Constant Fermat’s Last Theorem Squaring the Circle We begin with a sheet of paper, a pencil, and only two other tools. The first is a straight edge, a ruler read more
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