2000 years unsolved: Why is doubling cubes and squaring circles impossible?
Mathologer・28 minutes read
The video discusses ancient mathematical problems involving ruler and compass constructions that were proven impossible in the 19th century, aiming to make the complex proofs accessible to a wider audience. It demonstrates the impossibility of tasks such as doubling a cube, trisecting angles, and constructing regular heptagons using only ruler and compass due to the nature of square root and irrational numbers.
Insights
- The video delves into ancient mathematical problems that persisted for centuries, showcasing the impossibility of tasks like doubling a cube, trisecting angles, and squaring a circle using only a ruler and compass.
- Through a detailed exploration of constructible numbers and the limitations of ruler-and-compass constructions, it becomes evident that certain geometric challenges, such as trisecting angles and constructing regular heptagons, are fundamentally unachievable within these constraints, shedding light on the intricacies of mathematical proofs and impossibility theorems.
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Recent questions
What are some ancient unsolved mathematical problems?
Doubling a cube, trisecting angles, squaring a circle.
How are geometric constructions done with a ruler and compass?
By drawing lines and circles to create shapes.
What are constructible numbers in mathematics?
Integers, midpoints, sums, differences, products, quotients.
How can square roots of numbers be constructed with ruler and compass?
By drawing lines and circles to create geometric shapes.
Why is it impossible to double a cube with only a ruler and compass?
Due to the properties of irrational numbers and geometric constructions.
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Summary
00:00
"Unsolvable Ancient Math Problems Explained Simply"
- The video discusses ancient mathematical problems that remained unsolved for over 2,000 years, focusing on geometric constructions using a ruler and compass.
- The problems include doubling a cube, trisecting angles, constructing regular heptagons, and squaring a circle, all challenging tasks.
- In the 19th century, it was proven that these tasks are impossible using only a ruler and compass.
- The video aims to simplify these complex proofs into an understandable format for a wider audience.
- The video is structured into levels of enlightenment, with each level tackling different mathematical challenges.
- The rules of constructing with a ruler and compass involve drawing lines and circles to create geometric shapes.
- The aim of doubling a cube involves constructing points that are the cube root of 2 apart.
- Squaring the circle involves constructing points that are root pi apart.
- Constructible numbers include all integers, midpoints, and their sums, differences, products, and quotients.
- The video demonstrates how to construct square roots of numbers using ruler and compass, expanding the range of constructible numbers.
15:42
"Square Rooty Coordinates and Cube Roots"
- Points with square rooty coordinates lead to equations with square rooty coefficients.
- Circles with square rooty centers and points have square rooty radii and equations.
- New points are obtained by intersecting lines and circles, solving linear and quadratic equations.
- Starting with 0 and 1 on the x-axis, ruler and compass can create any square rooty coordinates.
- The cube root of 2 cannot be expressed as a mess of square roots, proven through rationality.
- The cube root of 2 cannot be written as a fraction, being an irrational number.
- The proof by contradiction shows that the cube root of 2 is not of the form a plus b root 7.
- The cube root of 2 is not equal to any square rooty number, proven through a series of rational and irrational number properties.
- The pink numbers, constructed from rational numbers and rooty expressions, do not include the cube root of 2.
- Any square rooty number cannot be the cube root of 2, as proven through iterative extension processes of subfields.
31:03
Impossibility of Doubling a Cube and Trisecting Angles
- The proof that no square root number is equal to the cube root of 2 is established, demonstrating the impossibility of doubling a cube using only ruler and compass.
- Trisecting angles and constructing a regular heptagon with ruler and compass are shown to be similarly impossible, with the roots of specific cubic equations needing to be non-square root expressions.
- The impossibility of trisecting angles is illustrated by proving that constructing 20-degree angles is unattainable, leading to the conclusion that not all angles can be trisected.
- Constructing a regular heptagon is also deemed impossible, as it would involve constructing the cosine of 360/7 degrees, which is a solution to a cubic equation.
- The proof that all square root numbers are algebraic is explained by starting with a square root expression, setting it equal to x, and eliminating square roots to reveal integers and powers of x.
