Fractions (mathematics) | Elementary mathematics | Rational numbers | Field (mathematics)
In mathematics, a rational number is a number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. For example, −3/7 is a rational number, as is every integer (e.g. 5 = 5/1). The set of all rational numbers, also referred to as "the rationals", the field of rationals or the field of rational numbers is usually denoted by boldface Q, or blackboard bold A rational number is a real number. The real numbers that are rational are those whose decimal expansion either terminates after a finite number of digits (example: 3/4 = 0.75), or eventually begins to repeat the same finite sequence of digits over and over (example: 9/44 = 0.20454545...). This statement is true not only in base 10, but also in every other integer base, such as the binary and hexadecimal ones (see Repeating decimal § Extension to other bases). A real number that is not rational is called irrational. Irrational numbers include √2, π, e, and φ. Since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers are irrational. Rational numbers can be formally defined as equivalence classes of pairs of integers (p, q) with q ≠ 0, using the equivalence relation defined as follows: The fraction p/q then denotes the equivalence class of (p, q). Rational numbers together with addition and multiplication form a field which contains the integers, and is contained in any field containing the integers. In other words, the field of rational numbers is a prime field, and a field has characteristic zero if and only if it contains the rational numbers as a subfield. Finite extensions of Q are called algebraic number fields, and the algebraic closure of Q is the field of algebraic numbers. In mathematical analysis, the rational numbers form a dense subset of the real numbers. The real numbers can be constructed from the rational numbers by completion, using Cauchy sequences, Dedekind cuts, or infinite decimals (see Construction of the real numbers). (Wikipedia).
Determine Rational or Irrational Numbers (Square Roots and Decimals Only)
This video explains how to determine if a given number is rational or irrational.
From playlist Functions
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From playlist Common Core Standards - 6th Grade
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2.11117 What is a rational function Functions
http://www.freemathvideos.com presents: Learn math your way. My mission is to provide quality math education to everyone that is willing to receive it. This video is only a portion of a video course I have created as a math teacher. Please visit my website to join my mailing list, downloa
From playlist Rational Functions - Understanding
Construction of the rational numbers In this video, I construct the rational numbers starting from the integers, using equivalence relation and equivalence classes. After this video, you can finally understand what 1/2 really means! Enjoy! Check out my Real Numbers Playlist: https://www.
From playlist Real Numbers
Classify Numbers as Rational or Irrational (Common Core Math 7/8 Ex 4)
This video explains how to classify real numbers as rational or irrational. http://mathispower4u.com
From playlist Common Core Grade 7/8 Practice Standardized Test Math Problems
Rational and Irrational Numbers - N2
A review of the difference between rational and irrational numbers and decimals - including square rootes and fraction approximations of pi.
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Description of the rational numbers. Proof that the square root of two is not rational. Definition of algebraic number.
From playlist Course 6: Introduction to Analysis (Fall 2017)
http://www.freemathvideos.com In this video playlist you will learn everything you need to know with complex and imaginary numbers
From playlist Simplify Rational Expressions
Putnam Battle part 1!!! @Michael Penn vs @ProfOmarMath
This is the first in a series of videos with @ProfOmarMath where we look at a total of four solutions to Question A6 from the 2018 William Lowell Putnam Mathematic Competition. Our strategy for this solution is to place everything in the coordinate plan and use algebraic manipulation. Ch
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Manjul Bhargava: What is the Birch-Swinnerton-Dyer Conjecture, and what is known about it?
Abstract: The Birch and Swinnerton-Dyer Conjecture has become one of the central problems of number theory and represents an important next frontier. The purpose of this lecture is to explain the problem in elementary terms, and to describe the implications of Andrew Wiles' groundbreaking
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From playlist Algebra 1 Regents August 2014
Construction of the Real Numbers
Dedekind Cuts In this video, I rigorously construct the real numbers from the rational numbers using so-called Dedekind Cuts. It might seem complicated at first, but the advantage is that we can construct the real numbers without using any axioms. More importantly, in the next video, we u
From playlist Real Numbers
Set Theory (Part 14): Real Numbers as Dedekind Cuts
Please feel free to leave comments/questions on the video and practice problems below! In this video, we will construct the real number system as special subsets of rational numbers called Dedekind cuts. The trichotomy law and least upper bound property of the reals will also be proven. T
From playlist Set Theory by Mathoma
Learn Rational Numbers In 7 min
Rational Numbers are numbers that can be expressed as fractions. This video will go over the basics of real numbers and define rational numbers. TabletClass Math Academy Math Courses: Pre-Algebra: https://tabletclass-academy.teachable.com/p/tabletclass-math-pre-algebra1 Algebra 1: htt
From playlist GED Prep Videos
RA1.1. Real Analysis: Introduction
Real Analysis: We introduce some notions important to real analysis, in particular, the relationship between the rational and real numbers. Prerequisites may be found in the Math Major Basics playlist.
From playlist Real Analysis
Set Theory (Part 18): The Rational Numbers are Countably Infinite
Please feel free to leave comments/questions on the video and practice problems below! In this video, we will show that the rational numbers are equinumerous to the the natural numbers and integers. First, we will go over the standard argument listing out the rational numbers in a table a
From playlist Set Theory by Mathoma
Set Theory (Part 13): Constructing the Rational Numbers
Please feel free to leave comments/questions on the video and practice problems below! In this video, we will use the integers to construct the rational numbers as a quotient set, just as we constructed the integers. We will also introduce arithmetic on the rational numbers and show that
From playlist Set Theory by Mathoma
Rational and Irrational Numbers
This math video tutorial provides a basic introduction into rational and irrational numbers. My Website: https://www.video-tutor.net Patreon Donations: https://www.patreon.com/MathScienceTutor Amazon Store: https://www.amazon.com/shop/theorganicchemistrytutor Subscribe: https://www.yo
From playlist GED Math Playlist