Real closed field | Mathematics of infinitesimals | Field (mathematics) | Nonstandard analysis | Infinity | Mathematical analysis
In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the form (for any finite number of terms). Such numbers are infinite, and their reciprocals are infinitesimals. The term "hyper-real" was introduced by Edwin Hewitt in 1948. The hyperreal numbers satisfy the transfer principle, a rigorous version of Leibniz's heuristic law of continuity. The transfer principle states that true first-order statements about R are also valid in *R. For example, the commutative law of addition, x + y = y + x, holds for the hyperreals just as it does for the reals; since R is a real closed field, so is *R. Since for all integers n, one also has for all hyperintegers . The transfer principle for ultrapowers is a consequence of Łoś' theorem of 1955. Concerns about the soundness of arguments involving infinitesimals date back to ancient Greek mathematics, with Archimedes replacing such proofs with ones using other techniques such as the method of exhaustion. In the 1960s, Abraham Robinson proved that the hyperreals were logically consistent if and only if the reals were. This put to rest the fear that any proof involving infinitesimals might be unsound, provided that they were manipulated according to the logical rules that Robinson delineated. The application of hyperreal numbers and in particular the transfer principle to problems of analysis is called nonstandard analysis. One immediate application is the definition of the basic concepts of analysis such as the derivative and integral in a direct fashion, without passing via logical complications of multiple quantifiers. Thus, the derivative of f(x) becomes for an infinitesimal , where st(·) denotes the standard part function, which "rounds off" each finite hyperreal to the nearest real. Similarly, the integral is defined as the standard part of a suitable infinite sum. (Wikipedia).
What are Hyperbolas? | Ch 1, Hyperbolic Trigonometry
This is the first chapter in a series about hyperbolas from first principles, reimagining trigonometry using hyperbolas instead of circles. This first chapter defines hyperbolas and hyperbolic relationships and sets some foreshadowings for later chapters This is my completed submission t
From playlist Summer of Math Exposition 2 videos
Algebra Ch 40: Hyperbolas (1 of 10) What is a Hyperbola?
Visit http://ilectureonline.com for more math and science lectures! To donate: http://www.ilectureonline.com/donate https://www.patreon.com/user?u=3236071 We will learn a hyperbola is a graph that result from meeting the following conditions: 1) |d1-d2|=constant (same number) 2) the grap
From playlist THE "HOW TO" PLAYLIST
Computations with homogeneous coordinates | Universal Hyperbolic Geometry 8 | NJ Wildberger
We discuss the two main objects in hyperbolic geometry: points and lines. In this video we give the official definitions of these two concepts: both defined purely algebraically using proportions of three numbers. This brings out the duality between points and lines, and connects with our
From playlist Universal Hyperbolic Geometry
Calculus 2: Hyperbolic Functions (1 of 57) What is a Hyperbolic Function? Part 1
Visit http://ilectureonline.com for more math and science lectures! In this video I will explain what are hyperbolic functions and how it compares to trig functions. Next video in the series can be seen at: https://youtu.be/c8OR8iJ-aUo
From playlist CALCULUS 2 CH 16 HYPERBOLIC FUNCTIONS
The Opposite of Infinity - Numberphile
Continuing to talk Infinitesimals, this time with Dr James Grime. See last week's video: https://youtu.be/BBp0bEczCNg More links & stuff in full description below ↓↓↓ Dividing by Zero: https://youtu.be/BRRolKTlF6Q James Grime: http://singingbanana.com Support us on Patreon: http://www.p
From playlist Infinity on Numberphile
Hyperbola 3D Animation | Objective conic hyperbola | Digital Learning
Hyperbola 3D Animation In mathematics, a hyperbola is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other an
From playlist Maths Topics
9.999... reasons that .999... = 1
Point Nine Repeating Equals One! 9.999... reasons in 9.999... minutes. Bonus points if you can name all 9.999... lords a-leaping. Dear YouTube, wouldn't it be nice if I could include the full script with this video? A larger character limit would not be unreasonable. My personal website
From playlist Doodling in Math and more | Math for fun and glory | Khan Academy
Pascal's wager and real numbers
My entry for 3blue1brown's contest, talking about Pascal's wager and how it leads to interesting questions about (hyper)real numbers. A big shoutout to Grant for coming up with this wonderful idea. Link to Thierry Platinis channel for more on hyperreal numbers: https://www.youtube.com/cha
From playlist Summer of Math Exposition Youtube Videos
How real are the real numbers, really?
We usually say that infinity isn't real, but here we'll see how crucial it is to have one very big infinity for the real world; there is an infinite number of numbers. But why do we need real numbers at all? Aren't rational numbers enough? And what about hyperreal numbers? What we'll see
From playlist Some fun math videos about approximation
Hyperreality: How Will Your Brain Handle the Future Internet? (Part 3 of 3)
We are warned about the long hours we spend in front of screens, but what if the internet is actually a better way to experience reality? Part 1 of 3 - https://youtu.be/73fhpKkdHN4 Part 2 of 3 - https://youtu.be/WveOaob3PbE Did you know Seeker Plus is also a podcast? Check it out on A
From playlist Seeker Plus
Hyperreal Marvel Sculptures at Comic-Con 2022!
We had to do a double take at these incredibly lifelike sculptures of Spider-Man, Dr. Strange, Loki, and Captain America on display at this year's Comic-Con. They're the work of Queen Studios, and we learn how their artists create these hyperreal silicone busts with the likeness of the act
From playlist Toys, Models and Collectibles
11_3_8 Example problem calculating a tangent hyperplane
Let's look at an example where we calculate the function of a tangent hyperplane to a point on a higher dimensional curve.
From playlist Advanced Calculus / Multivariable Calculus
Infinitely large numbers that act like they’re finite - Alok Singh
In this talk Alok gives a hands-on introduction to the idea of infinitely small and large numbers, in the framework of the hyperreals. He shows how this point of view allows you to do mathematics in a "natural" way with infinitely large numbers, including Taylor series expansions, differen
From playlist Anything At All seminar
Hyperbola with Foci (-3, 0), (1, 0) and Vertices (-2, 0), (0, 0)
We find the equation of the hyperbola with foci (-3, 0), (1, 0), and vertices (-2 0), (0,0). Please Subscribe here, thank you!!! https://goo.gl/JQ8Nys
From playlist Conics
The circle and projective homogeneous coordinates (cont.) | Universal Hyperbolic Geometry 7b
Universal hyperbolic geometry is based on projective geometry. This video introduces this important subject, which these days is sadly absent from most undergrad/college curriculums. We adopt the 19th century view of a projective space as the space of one-dimensional subspaces of an affine
From playlist Universal Hyperbolic Geometry
What is the definition of a hyperbola
Learn all about hyperbolas. A hyperbola is a conic section with two fixed points called the foci such that the difference between the distances of any point on the hyperbola from the two foci is equal to the distance between the two foci. Some of the characteristics of a hyperbola includ
From playlist The Hyperbola in Conic Sections
Leighton, an Athlete Wrestling with a Python
Sir Frederic Leighton, An Athlete Wrestling with a Python, 1877, bronze, 1746 x 984 x 1099 mm (Tate Britain, London). Created by Beth Harris and Steven Zucker.
From playlist Art in 19th century Europe | Art History | Khan Academy
Catching a Cunning Corporate Criminal: Little Joe's Greatest Challenge | Real Stories
In episode 4, Joe walks us through the difficulties of a complex repossession from a corporate criminal who’s use of muscle and cunning consistently keeps Joe and the banks one step behind. ‘Little Joe’ is a short-form documentary series set in the hyperreal world of Debt Collector Joseph
From playlist Little Joe (A Real Stories Original Series)
A-Level Further Maths H4-04 Hyperbolic Inverse: Logarithmic Form of y=artanh(x)
https://www.buymeacoffee.com/TLMaths Navigate all of my videos at https://sites.google.com/site/tlmaths314/ Like my Facebook Page: https://www.facebook.com/TLMaths-1943955188961592/ to keep updated Follow me on Instagram here: https://www.instagram.com/tlmaths/ Many, MANY thanks to Dea
From playlist A-Level Further Maths H4: Hyperbolic Inverse
Australia’s Most Unorthodox Debt Collector | Little Joe | Real Stories
‘Little Joe’ is a short-form documentary series set in the hyperreal world of Debt Collector Joseph. With his unique voice and compelling stories, Joe gives the viewer a voyeuristic insight into the grinding gears of a debt dependent society, through the eyes of a guy who grew up and is no
From playlist Little Joe (A Real Stories Original Series)