Algebraic numbers | Euclidean plane geometry
In geometry and algebra, a real number is constructible if and only if, given a line segment of unit length, a line segment of length can be constructed with compass and straightedge in a finite number of steps. Equivalently, is constructible if and only if there is a closed-form expression for using only integers and the operations for addition, subtraction, multiplication, division, and square roots. The geometric definition of constructible numbers motivates a corresponding definition of constructible points, which can again be described either geometrically or algebraically. A point is constructible if it can be produced as one of the points of a compass and straight edge construction (an endpoint of a line segment or crossing point of two lines or circles), starting from a given unit length segment. Alternatively and equivalently, taking the two endpoints of the given segment to be the points (0, 0) and (1, 0) of a Cartesian coordinate system, a point is constructible if and only if its Cartesian coordinates are both constructible numbers. Constructible numbers and points have also been called ruler and compass numbers and ruler and compass points, to distinguish them from numbers and points that may be constructed using other processes. The set of constructible numbers forms a field: applying any of the four basic arithmetic operations to members of this set produces another constructible number. This field is a field extension of the rational numbers and in turn is contained in the field of algebraic numbers. It is the Euclidean closure of the rational numbers, the smallest field extension of the rationals that includes the square roots of all of its positive numbers. The proof of the equivalence between the algebraic and geometric definitions of constructible numbers has the effect of transforming geometric questions about compass and straightedge constructions into algebra, including several famous problems from ancient Greek mathematics. The algebraic formulation of these questions led to proofs that their solutions are not constructible, after the geometric formulation of the same problems previously defied centuries of attack. (Wikipedia).
Visualizing decimal numbers and their arithmetic 67 | Arithmetic and Geometry Math Foundations
This video gives a precise definition of a decimal number as a special kind of rational number; one for which there is an expression a/b where a and b are integers, with b a power of ten. For such a number we can extend the Hindu-Arabic notation for integers by introducing the decimal form
From playlist Math Foundations
Tutorial - What is an imaginary number
http://www.freemathvideos.com In this video playlist you will learn everything you need to know with complex and imaginary numbers
From playlist Complex Numbers
A Maths Puzzle: Find the nine digit number
Find a nine digit numbers, using the numbers 1 to 9, and using each number once without repeats, such that; the first digit is a number divisible by 1. The first two digits is a number divisible by 2; The first three digits is a number divisible by 3 and so on until we get a nine digit num
From playlist My Maths Videos
Ex: Determine a Real, Imaginary, and Complex Number
This video explains how decide if a number is best described by the set of real, imaginary, or complex numbers. Library: http://mathispower4u.com Search: http://mathispower4u.wordpress.com
From playlist Performing Operations with Complex Numbers
We discuss what imaginary numbers are and how they are part of the larger set of complex numbers in this free math video tutorial by Mario's Math Tutoring. This is a nice introduction to working with i. We also go through some examples. 0:26 A Hierarchy of Different Types of Numbers 1:03
From playlist Imaginary & Complex Numbers
Geometric Impossibilities, Part 2: The Field of Constructible Numbers
The second video of the series, where we complete the demonstration that constructible numbers are closed under all the field operations. We also show that, unlike arbitrary fields, the constructible numbers contain the square root of any of its elements. Constructing Products & Quotient
From playlist Geometric Impossibilities
What is the complex conjugate?
What is the complex conjugate of a complex number? Free ebook http://bookboon.com/en/introduction-to-complex-numbers-ebook
From playlist Intro to Complex Numbers
Construction of Natural Numbers In this, I rigorously define the concept of a natural number, using Peano's axioms. I also explain why those axioms are the basis for the principle of mathematical induction. Enjoy! Check out my Real Numbers Playlist: https://www.youtube.com/playlist?list=
From playlist Real Numbers
Visual Group Theory, Lecture 6.7: Ruler and compass constructions
Visual Group Theory, Lecture 6.7: Ruler and compass constructions Inspired by philosophers such as Plato and Aristotle, one of the chief purposes of ancient Greek mathematics was to find exact constructions for various lengths, using only the basic tools of a ruler and compass. However, t
From playlist Visual Group Theory
Geometric Impossibilities, Part 1: Basic Constructions
The first in a video series where we explore basic straight-edge and compass construction, and prove various impossibility theorems using modern field theory. Introduction: (0:00) Distance copying, and perpendicular bisection (5:09) Angle copying and division by naturals (11:00)
From playlist Geometric Impossibilities
2000 years unsolved: Why is doubling cubes and squaring circles impossible?
Today's video is about the resolution of four problems that remained open for over 2000 years from when they were first puzzled over in ancient Greece: Is it possible, just using an ideal mathematical ruler and an ideal mathematical compass, to double cubes, trisect angles, construct regul
From playlist Recent videos
Visual Group Theory, Lecture 6.8: Impossibility proofs
Visual Group Theory, Lecture 6.8: Impossibility proofs The ancient Greeks sought basic ruler and compass constructions such as (1) squaring the circle, (2) doubling the cube, and (3) trisecting an angle. In the previous lecture, we learned how a length or angle 'z' is constructable iff th
From playlist Visual Group Theory
A topic I've wanted to cover for some time now. #SoME1 #3b1b The terrible graphics are sort of my sort of standard for me, but hopefully the content is approachable and reasonably accurate.
From playlist Summer of Math Exposition Youtube Videos
Geometric Impossibilities , Part 4: The Finale
In the final installment of this mini-series, we prove and apply the algebraic characterization of the constructible numbers, and resolve the famous ancient conjectures.
From playlist Geometric Impossibilities
New Developments in Hypergraph Ramsey Theory - D. Mubayi - Workshop 1 - CEB T1 2018
Dhruv Mubayi (UI Chicago) / 30.01.2018 I will describe lower bounds (i.e. constructions) for several hypergraph Ramsey problems. These constructions settle old conjectures of Erd˝os–Hajnal on classical Ramsey numbers as well as more recent questions due to Conlon–Fox–Lee–Sudakov and othe
From playlist 2018 - T1 - Model Theory, Combinatorics and Valued fields
Number theory Full Course [A to Z]
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure #mathematics devoted primarily to the study of the integers and integer-valued functions. Number theorists study prime numbers as well as the properties of objects made out of integers (for example, ratio
From playlist Number Theory
3. Forbidding a subgraph II: complete bipartite subgraph
MIT 18.217 Graph Theory and Additive Combinatorics, Fall 2019 Instructor: Yufei Zhao View the complete course: https://ocw.mit.edu/18-217F19 YouTube Playlist: https://www.youtube.com/playlist?list=PLUl4u3cNGP62qauV_CpT1zKaGG_Vj5igX What is the maximum number of edges in a graph forbidding
From playlist MIT 18.217 Graph Theory and Additive Combinatorics, Fall 2019
Complex Numbers as Points (1 of 4: Geometric Meaning of Addition)
More resources available at www.misterwootube.com
From playlist Complex Numbers
This lecture is part of an online graduate course on Galois theory. As an application of Galois theory, we prove Gauss's theorem that it is possible to construct a regular heptadecagon with ruler and compass.
From playlist Galois theory