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Georgia High School Graduation Test Content: http://www.stephenwelchtutoring.com/GHSGT.html
Views: 3563739 Khan Academy
Introductory video to algebra
Views: 295 Blanco Tutes
This video gives an overview of Algebra and introduces the concepts of unknown values and variables. It also explains that multiplication is implicit in Algebra. The first video in the Algebra Basics Series: https://www.youtube.com/watch?v=NybHckSEQBI&list=PLUPEBWbAHUszT_GebJK23JHdd_Bss1N-G Learn More at mathantics.com Visit http://www.mathantics.com for more Free math videos and additional subscription based content!
Views: 2055551 mathantics
Here we give restatements of the Fundamental theorems of Algebra (I) and (II) that we critiqued in our last video, so that they are now at least meaningful and correct statements, at least to the best of our knowledge. The key is to abstain from any prior assumptions about our understanding of continuity and "real " or "complex" numbers, and state everything in terms of rational numbers. For this we briefly first review some rational complex arithmetic, crucially the concept of quadrance of a complex number which ought to be a core definition in undergraduate courses. These restatements were first proposed some years ago in my AlgTop series of videos. It should be emphasized that we do NOT currently have proofs for these "theorems", so there is a huge opportunity here for people to make a significant contribution to mathematics. But new and deeper understanding is required, at least I believe so, and hopefully we can aspire to computationally oriented proofs, that actually tell us how to go about finding approximate zeroes to a prescribed level of accuracy. Working this out satisfactorily will be as significant an accomplishment as any 20th century mathematical achievement.
Views: 9473 njwildberger
ethan explains math for hsiehzam.com
Views: 76 sargeant45two
This video reveals the unfortunate truth about the "Fundamental Theorem of Algebra": namely that it is not actually correct. This is meant to be a core result in undergrad mathematics, but curiously undergrads don't see much in the way of proof. Why? Because none of the many current arguments are actually convincing once one stops and looks carefully at them. Modern mathematics students: prepare for some disruption to your thinking! Modern mathematicians: is it not time to admit the harsh reality? This entire topic is intimately connected with what I consider the fundamental problem in mathematics, which I discuss in the Famous Math Problems 19 lectures. And so we need some seriously new thinking. We need to peel back the layers of conformity, imprecise thinking and wishful dreaming that characterize so much of modern pure mathematics. Help support this channel by becoming a patron, at https://www.patreon.com/njwildberger. Even just \$1 per video will let you share the excitement of building up a new and better mathematics for the coming millennium. And happy new year!
Views: 7332 njwildberger
Views: 25 kosherfan
Index: Module 1 (17:00) Module 2 (54:45) Module 3 (1:13:05) Module 4 (1:26:00) Module 5 (1:38:18) Secondary Index (by manipulative): Intro to Manipulatives/What’s In the Kit 12:26 Base Ten Blocks/Factor Track 17:06 Fraction Towers/Fraction Number Lines 25:45 Place Value Chips 50:50 Area Model of Multiplication (Base Ten and Algeblocks) 55:00 Cuisenaire Rods 1:13:22 XY Coordinate Board with Slope 1:20:21 AngLegs 1:22:38 Algeblocks (Expressions/Equations) 1:26:10 XY Coordinate Board (Coordinate Plane/4 Quadrants ) 1:33:30 XY Coordinate Board (Functions) 1:38:18 Teaching Foundations of Algebra? Find FREE resources on www.hand2mind.com! Review our recorded webinar that breaks down each Module and shows how to use manipulatives to help build concrete understanding of mathematical concepts. Currently only the preliminary version is available for viewing, but check back soon for the edited searchable webinar! For more Foundations of Algebra resources, including Small Group Manipulative Kits, contact Carolyn Cutts at [email protected]
Views: 995 hand2mind
There are three main branches of mathematics: arithmetic, geometry and algebra. This is the correct order, both in terms of importance and of historical development. Here we introduce our program for setting out foundations of algebra. This video belongs to Wildberger's MathFoundations series, which sets out a coherent and logical framework for modern mathematics. A screenshot PDF which includes MathFoundations46 to 79 can be found at my WildEgg website here: http://www.wildegg.com/store/p101/product-Math-Foundations-screenshot-pdf
Views: 7037 njwildberger
We begin to introduce the Algebra of Boole, starting with the bifield of two elements, namely {0,1}, and using that to build the algebra of n-tuples, which is a linear space over the bifield with an additional multiplicative structure. This important abstract development played a key role in the application of logic to circuit and logic gate analysis. Surprisingly it is not quite the same as Boolean algebra, which is closer to the arithmetic of sets. We will move towards understanding the critical difference between these two mathematical approaches to logic. However in both cases, the situation is that mathematics was introduced to make logic more precise and rigorous---- not the other way around! This understanding has major ramifications for an appreciation of why 20th century mathematics got things so fundamentally wrong!
Views: 5005 njwildberger
How do we set up abstract algebra? In other words, how do we define basic algebraic objects such as groups, rings, fields, vector spaces, algebras, lattices, modules, Lie algebras, hypergroups etc etc?? This is a hugely important question, and not an easy one to answer. In this video we start by giving a bird's eye view of some basic examples, namely the first four kinds of objects on this list. We will not attempt complete definitions, but just rather provide intuitive example based descriptions, using the standard thinking current in mathematics these days. In some videos, we will be looking to reorganize our understanding of all of these topics by being much more precise and careful, and utilizing our knowledge of data structures.
Views: 15226 njwildberger
This project was created with Explain Everything™ Interactive Whiteboard for iPad.
Views: 22 Justin Walls
One important use of letters in algebra is to describe patterns in a quantitative and general way. We look at the `sequences' of square numbers and triangular numbers, and derive formulas for the nth terms. A table of differences shed light on these and other number patterns. This video belongs to Wildberger's MathFoundations series, which sets out a coherent and logical framework for modern mathematics. Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary. A screenshot PDF which includes MathFoundations46 to 79 can be found at my WildEgg website here: http://www.wildegg.com/store/p101/product-Math-Foundations-screenshot-pdf
Views: 10465 njwildberger
Algebra starts with the natural and simple problem of trying to solve an equation containing an unknown number, or `variable'. Here we start with simple examples familiar to public school students. This video belongs to Wildberger's MathFoundations series, which sets out a coherent and logical framework for modern mathematics. The idea is to transform an equation with a variable into a simpler but equivalent equation, which can be more easily solved. We review examples of such manipulations--that go back to Hindu and Arab mathematicians. My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/Norman_Wildberger. I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?ty=h . A screenshot PDF which includes MathFoundations46 to 79 can be found at my WildEgg website here: http://www.wildegg.com/store/p101/product-Math-Foundations-screenshot-pdf
Views: 6406 njwildberger
Keynote lecture at the Fifty Years of Functorial Semantics conference, Union College, October 2013. http://www.math.union.edu/~niefiels/13conference/Web/ Transcript: http://www.math.union.edu/~niefiels/13conference/Web/Slides/Fifty_Years_of_Functorial_Semantics.pdf Abstract From observation of, and participation in, the ongoing actual practice of Mathematics, Decisive Abstract General Relations (DAGRs) can be extracted; when they are made explicit, these DAGRs become a guide to further rational practice of mathematics. The worry that these DAGRs may turn out to be as numerous as the specific mathematical facts themselves is overcome by viewing the ensemble of DAGRs as a ’Foundation’, expressed as a single algebraic system whose current description can be finitely-presented. The category of categories (as a cartesian closed category with an object of small discrete categories) aims to serve as such a Foundation. One basic DAGR is the contrast between space and quantity, and especially the relation between the two that is expressed by the role of spaces as domains of variation for intensively and extensively variable quantity; in that way, the foundational aspects of cohesive space and variable quantity inherently includes also the conceptual basis for analysis, both for functional analysis and for the transformation from continuous cohesion to combinatorial semi-discreteness via abstract homotopy theory. Function spaces embody a pervasive DAGR. The year 1960 was a turning point. Kan, Isbell, Grothendieck and Yoneda had further developed the Eilenberg-Mac Lane Theory of Naturality. Their work implicitly pointed towards such a Foundation as a foreseeable goal. Although the work of those four great mathematicians was still unknown to me, I had independently traversed a sufficient fragment of a similar path to encourage me to become a student of Professor Eilenberg. As I slowly became aware of the importance of those earlier developments, I attempted to participate in the realization of a Foundation in the sense described above, first through concentration on the particular docrine known as Universal Algebra, making explicit the fibered category whose base consists of abstract generals (called theories) and whose fibers are concrete generals (known as algebraic categories). The term ’Functorial Semantics’ simply refers to the fact that in such a fibered category, any interpretation T 0 → T of theories induces a map in the opposite direction between the two categories of concrete meanings; this is a direct generalization of the previously observed cases of linear algebra, where the abstract generals are rings and the fibers consist of modules, and of group theory where the dialectic between abstract groups and their actions had long been fundamental in practice. This kind of fibration is special, because the objects T in the base are themselves categories, as I had noticed after first rediscovering the notion of clone, but then rejecting the latter on the basis of the principle that, to compare two things, one must first make sure that they are in the same category; when the two are (a) a theory and (b) a background category in which it is to be interpreted, comparisons being models., the category of categories with products serves. Left adjoints to the re-interpretation functors between fibers exist in this particular doctrine of general concepts, unifying a large number of classical and new constructions of algebra. Isbell conjugacy can provide a first approximation to the general space vs quantity pseudo-duality, because recent developments (KIGY) had shown that also spaces themselves are determined by categories (of figures and incidence relations inside them). My 1963 thesis clearly explains that presentations (having a signature consisting of names for generators and another signature consisting of names for equational axioms) constitute one important source of theories. This syntactical left adjoint directly generalizes the presentations known from elimination theory in linear algebra and from word problems in group theory. No one would confuse rings and groups themselves with their various syntactical presentations, but previous foundations of algebra had underemphasized the existence of another important method for constructing examples, namely the Algebraic Structure functor. Being a left adjoint , it can be calculated as a colimit over finite graphs. Fundamental examples, like cohomology operations as studied by the heroes of the 50’s, show that typically an abstract general (such as an isometry group) arises by naturality; to find a syntactical presentation for it may then be an important question. This extraction, by naturality from a particular family of cases, provides much finer invariants, and as a process bears a profound resemblance to the basic extraction of abstract generals from experience.
Views: 3617 Matt Earnshaw
Views: 4 Rachel Jackson
Views: 11 Sarah Cook
We investigate further the Algebra of Boole, consisting of vectors of 0 and 1 of a given size, with operations pointwise mod 2. The idempotent law x^2=x of Boole is distinguished. To illustrate the geometry, we look at a 5 dimensional example and the span of three vectors, along with the algebra generated by them, giving both a 3 dimensional cube and a 4 dimensional hypercube. Then we introduce the square of opposition going back to medieval philosophers, now in algebraic form by re-interpreting Aristotle's propositions using Boole's algebraic reformulation. Then armed with this mathematical framework, we begin the fun task of proving Aristotle's syllogistic rules using just mathematics! Including Barbara, Celarent, Cesare and others.
Views: 2670 njwildberger
The development of circuit analysis in the 20th century had strong connections to the theory of logic. In this video we discuss Huntington's general formulation of Boolean algebra as an abstract theory, and then investigate Claude Shannon's important 1938 paper which first lays out the connections between Boolean algebra and circuit analysis. There are some surprises here, as Shannon's initial orientation was in some sense opposite to the modern point of view --- he sees the logical laws as applying to what he calls "hindrances" to flow in circuits. Nevertheless he shows that Huntington's Boolean algebra with its many interesting reduction formulas was well suited to simplify, and more generally understand, complicated circuits. Unfortunately he appears not to have read the original paper of Boole; and this ends up affecting the development of electrical engineering in the 20th century.
Views: 1702 njwildberger
Description
Views: 1046 The Hunt Institute
The simplest and most common examples of abstract algebraic objects are probably linear spaces. They occur in many areas of mathematics, and are pillars of linear algebra, where they are often called vector spaces. Our approach will be to generalize simple aspects of Nat, Int and Rat from a data structure orientation, prominently using multisets, or msets. In this lecture we start with the idea of generalizing Nat with addition, to form spans of msets.
Views: 3837 njwildberger
After George Boole's introduction of an algebraic approach to logic, the subject morphed towards a more set theoretic formulation, with so called Boolean algebra initiated by John Venn and Charles Peirce. Venn diagrams (originally going back to Euler), give us a visual way of representing relations between subsets of a universal set. The operations of meet and join, or intersection and union, together with taking complements become replacements for the product and sum of the Algebra of Boole. In this set theoretic context, de Morgan's laws clarify how to compute complements of unions and intersections, and the two distributive laws involve 3 sets, and either a union of an intersection, or the intersection of a union. We illustrate how to verify these laws from, first of all a truth table perspective, and then with computations using the Algebra of Boole. There is a heretical message here: professors teaching circuit analysis to engineers might want to start thinking about revamping their subject, and replacing Boolean algebra with the original, more powerful and simpler Algebra of Boole!
Views: 2248 njwildberger
While abstract algebra is not as problematic logically as modern analysis, it still suffers from very serious difficulties. In this video we begin laying out some of these logical reefs that we will have to steer clear from. And we look at the first one: which is distinguishing between descriptions, definitions and specifications of abstract algebraic objects. For example, how do we officially define a group? It turns out that there are distinctly different approaches to answering this, and notably the theory of groups that we build up is highly dependent on which definition we choose.
Views: 5819 njwildberger
We introduce transistors and how they combine to create logic gates. These include prominently the gates called NOT, AND, OR, XOR, NAND, NOR and XNOR which input a pair of signals A and B and output a single signal. So these gates act as connectives, or even operations, on pairs of numbers {0,1}. Transistors were introduced by Bardeen, Brattain and Shockley in 1947, although a patent in a similar direction goes back to J. E Lilienfeld in 1925. They rely on remarkable properties of the semiconductor silicon. By putting together transistors in clever ways, we can create logic gates, which are then the building blocks for more elaborate circuits, these days involving possibly millions of transistors. The question of how to mathematically model what is happening with logic gates--in other words how to analyse them, is a separate question from how to construct them physically. We will be pursuing the idea that the original Algebra of Boole is in fact superior for the majority of applications involving logic gates over the classical Boolean algebra. This is quite a novel position, but there are strong arguments for it. It has major ramifications for the education of electrical engineers around the world!
Views: 1548 njwildberger
There is a very big jump in going from finite algebraic objects to "infinite algebraic objects". For example, there is a huge difference, if one is interested in very precise definitions, between the concept of a finite group and the concept of an "infinite group". We illustrate this important distinction in this video by looking at a rich and interesting example: SL(2) with just 0,1 entries mod 2, which is a lovely and special finite group, and SL(2) with arbitrary integer entries, which is also an interesting and important mathematical object, but which ought to be defined and treated differently. Of course one of the advantages in the modern sloppy approach to the definition of a "set" is that it does not distinguish at all between a collection that can be explicitly listed and one that cannot be. So this meaningful distinction can be left under the carpet.
Views: 3295 njwildberger
We want to tackle the explicit computational question of how to find a convenient basis of an integral linear space. Given some msets, they span an integral linear space, but we would like to find a simpler basis for such a space. The algorithm which we introduce is a variant of the row reduction algorithm from linear algebra, also sometimes called Gaussian elimination. However one big difference is that we are only working with integral combinations, not rational ones. It is somewhat pleasant that such an important algorithm figures prominently in this first foray into abstract algebra!
Views: 1941 njwildberger
Roots of unity as regularly spaced points on the unit circle in the complex plane are common-place objects in algebra, but they implicitly rely on the Fundamental theorem of algebra for their existence. But their existence, in fact their definition, is fraught with logical difficulty, at least when we view them in the complex plane. In this video we summarize some basic algebraic and geometrical properties of complex numbers, centered on the rational parametrization of the unit circle. There is a huge difference between specifying a unit quadrance complex number by an expression like 3/5+4i/5 and cos 4+i sin 4. Once we understand that, we can start to appreciate the logical difficulty in supposing that something like the seventh roots of unity really do exist, perfectly spaced around a regular septagon.
Views: 9333 njwildberger
Engineers and computer scientists create complicated circuits from the basic logic gates, typically NOT, AND, OR, XOR, NAND, NOR, XNOR and sometimes others. Such a circuit typically has a number of inputs, say A,B,C,etc and yields a single output. [More complicated circuits will have several outputs.] A fundamental problem in circuit analysis is to encode the effect on the inputs by associating to the entire circuit an algebraic expression. In classical engineering, taught in schools and universities around the world, that is a Boolean expression, based on the gates NOT, AND and OR. But we are advocating an exciting new way of thinking about this entire subject, by expressing the workings of a circuit rather based on the Algebra of Boole, which treats AND and XOR as primary, and for which we reserve the multiplication and addition symbols. For us the OR operation will be treated as a "circle sum": please note this departure from conventional courses. The reason for this is that we will show that the Algebra of Boole approach is simpler and more powerful for the majority of questions and problems that an engineer wants to tackle. We start this discussion by an in depth look at two simple but important examples, both from the Algebra of Boole and the Boolean algebra point of view.
Views: 1516 njwildberger
Views: 21 A. Owens
Modern data often consists of feature vectors with a large number of features. High-dimensional geometry and Linear Algebra (Singular Value Decomposition) are two of the crucial areas which form the mathematical foundations of Data Science. This mini-course covers these areas, providing intuition and rigorous proofs. Connections between Geometry and Probability will be brought out. Text Book: Foundations of Data Science. See more on this video at https://www.microsoft.com/en-us/research/video/foundations-of-data-science-lecture-1/
Views: 10846 Microsoft Research
* * * PREVIOUS VIDEO IS UP! * * * MathFoundations224: Lattice relations and Hermite normal form https://www.youtube.com/watch?v=nypI4o7bxoo&t=1s Given some msets, we can consider their integral linear span. But complementary to this is the question of the relations satisfied by the relations. This balancing between relations among an mset of msets and the ispan of that mset of msets is directly analogous to the rank -nullity theorem in linear algebra. Here we explore how to obtain relations from the Hermite normal form algorithm.
Views: 2230 njwildberger
U1F2 Day 1- Foundations of Algebra
Views: 192 Beast Algebra
A key problem in circuit analysis is to associate to a logical circuit, typically made of logic gates such as AND, OR, NOT, XOR, NAND and NOR, an algebraic expression that captures the effect of that circuit on all possible inputs. Such an effect is called a Boolean function, and it acts on the space of possible 0/1 values that can be assigned to the input values. For example if we have three inputs A,B and C then there are 2^3=8 different possible values of 0's and 1's that can be assigned to these three, and so then there are 2^8 Boolean functions representing all possible circuits with these inputs. Such a Boolean function has a geometric interpretation, and in this video we show how to get an algebraic version, both using the technology of Boolean algebra and the alternate, more powerful technology of the original Algebra of Boole. The advantages of the latter start becoming pretty obvious, once you start doing the computations.
Views: 1525 njwildberger
Algebra the easiest way for Dummies/Beginners. For GED, AccuPlacer, COMPASS, SAT, ASVAB and more. Master Algebra without even Learning anything math.(DUMMY PROOF. Follow the steps and get the answer). Algebra lessons here are well taught so that you can familiar with Algebra basics. From our Algebra Introduction through our Basic Algebra Lessons, our Algebra 1 Lessons and more, this Algebra video is made well to make sure that you get a good Algebra review and pass any test with ease. Go to http://ultimate-algebra.com and get the complete course and more. Follow the steps and get the answer. (DUMMY PROOF). Can be used for Pre-Algebra Lessons Covered in Part 1 1. Addition and Subtraction in Algebra 0:00 2. Addition and Subtraction of Multiple terms 2:41 3. The Invisible One 4:23 4. Multiplication and Division 5:21 5. Multiplication and Division of Negative Numbers 6:31 6. Multiplication and Division in Algebra 9:15 7. Multiple Multiplication 11:47 8. Division in Algebra 12:46 Check out on Facebook - http://facebook.com/ultimatealgebra #UltimateAlgebra
Views: 1652337 UltimateAlgebra
Given two binary inputs p and q, there are four possible assignments of 0's and1's to them, and correspondingly 16 different possible connectives or operations on these four assignments. The systematic study of these 16 connectives forms the foundation for both Boole's algebra, Boolean algebra, and modern circuit analysis. We want to position these subjects so that Boole's algebra is primary. We introduce AND, OR, XOR gate terminology, as well as the negations NOT, NAND, NOR, XNOR and also the implication operations IMP and NIMP. We illustrate the situation with a study of Modus Ponens both through truth tables and a more algebraic analysis based on Boole's algebra. This will give us a powerful new insight that allows us to apply very elementary polynomial algebra to solving complex problems in electrical engineering.
Views: 2234 njwildberger
Matrix theory is just a shadow of the more fundamental and far-reaching maxel theory. In our last video we introduced maxels from integers, which gives us a broad canvas to restructure matrix theory, extending to integer indices. In this video we begin to discuss the ramifications of this larger two-dimensional view of linear algebra. We review some constructs from earlier videos, and extend them to this more general set up, such as the partial identity maxels e_J corresponding to a set J of integers that allow us to identify matrix subalgebras inside our maxels. A lovely feature of this view is that a fundamental shift invariance comes into focus, which is not available in the more classical matrix view. The entire integer screen supporting maxel theory has a symmetry which allows us to shift up or down along the main diagonal. Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary. A screenshot PDF which includes MathFoundations184 to 212 can be found at my WildEgg website here: http://www.wildegg.com/store/p105/product-Math-Foundations-C2-screenshots-pdf
Views: 1658 njwildberger
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Views: 5 BreAnna Martinez
Views: 10 Beth Nichols
Views: 6 Matthew Wilson