Search results “Foundations of algebra”
Foundations and Algebra Review 1
Georgia High School Graduation Test Content: http://www.stephenwelchtutoring.com/GHSGT.html
Algebra Introduction - Basic Overview - Online Crash Course Review Video Tutorial Lessons
This math video tutorial provides a basic overview of concepts covered in a typical high school algebra 1 & 2 course or a college algebra course. This video contains plenty of lessons, notes, examples, and practice problems for you to get a good foundation in algebra. Subscribe: https://www.youtube.com/channel/UCEWpbFLzoYGPfuWUMFPSaoA?sub_confirmation=1 Basic Math Review: https://www.youtube.com/watch?v=nTn9gVqRfKY How To Receive Tutoring and Get Paid At The Same Time: https://www.youtube.com/watch?v=J8A8JTpOWCQ Epic Music Mix: https://www.youtube.com/watch?v=qeljbZhx9bY Top 30 Excel 2016 Tips: https://www.youtube.com/watch?v=UMad9_-4rOU Top 10 Side Hustles You Can Do To Make Extra Money: https://www.youtube.com/watch?v=gu9YIchkmSc My Online Courses: Algebra Course: https://www.udemy.com/algebracourse7245/learn/v4/content Trigonometry: https://www.udemy.com/trigonometry-the-unit-circle-angles-right-triangles/learn/v4/content Algebra Playlist: https://www.youtube.com/watch?v=i6sbjtJjJ-A&list=PL0o_zxa4K1BWKL_6lYRmEaXY6OgZWGE8G&index=1&t=13129s Access to Premium Videos: https://www.patreon.com/MathScienceTutor Here is a list of topics: 1. Introduction to Algebra - Online Crash Course Review 2. Monomials, Binomials, and Trinomials 3. Adding and Subtracting Polynomials with Like Terms 4. Multiplying Two Binomials - Foiling / Foil Method 5. Monomial & Trinomial Multiplication - The Distributive Property 6. Adding, Subtracting and Multiplying Exponents - Rules of Algebra 7. Negative Exponents - Variables and Expressions 8. Simplifying Expressions and Dividing Fractions with Variables 9. Solving Linear Equations With Parenthesis, Fractions, and Decimals 10. Simplifying Complex Fractions 11. How To Solve Quadratic Equations By Factoring 12. Solving Quadratic Equations Using The Quadratic Formula 13. Factoring Trinomials With Leading Coefficient 1 14. How To Factor By Grouping 15. Factoring Polynomials 16. Factoring Binomials Using Difference of Squares Methods 17. Factoring the GCF - Greatest Common Factor 18. Solving Equations With Variables on Both Sides 19. Solving Multi Step Equations 20. Graphing Linear Equations 21. Slope Intercept Form, Point Slope Form and Standard Form 22. How To Write The Equation of the Line 23. Parallel and Perpendicular Lines
The beauty of algebra | Introduction to algebra | Algebra I | Khan Academy
Why the abstraction of mathematics is so fundamental Watch the next lesson: https://www.khanacademy.org/math/algebra/introduction-to-algebra/overview_hist_alg/v/descartes-and-cartesian-coordinates?utm_source=YT&utm_medium=Desc&utm_campaign=AlgebraI Missed the previous lesson? https://www.khanacademy.org/math/algebra/introduction-to-algebra/overview_hist_alg/v/abstract-ness?utm_source=YT&utm_medium=Desc&utm_campaign=AlgebraI Algebra I on Khan Academy: Algebra is the language through which we describe patterns. Think of it as a shorthand, of sorts. As opposed to having to do something over and over again, algebra gives you a simple way to express that repetitive process. It's also seen as a "gatekeeper" subject. Once you achieve an understanding of algebra, the higher-level math subjects become accessible to you. Without it, it's impossible to move forward. It's used by people with lots of different jobs, like carpentry, engineering, and fashion design. In these tutorials, we'll cover a lot of ground. Some of the topics include linear equations, linear inequalities, linear functions, systems of equations, factoring expressions, quadratic expressions, exponents, functions, and ratios. About Khan Academy: Khan Academy is a nonprofit with a mission to provide a free, world-class education for anyone, anywhere. We believe learners of all ages should have unlimited access to free educational content they can master at their own pace. We use intelligent software, deep data analytics and intuitive user interfaces to help students and teachers around the world. Our resources cover preschool through early college education, including math, biology, chemistry, physics, economics, finance, history, grammar and more. We offer free personalized SAT test prep in partnership with the test developer, the College Board. Khan Academy has been translated into dozens of languages, and 100 million people use our platform worldwide every year. For more information, visit www.khanacademy.org, join us on Facebook or follow us on Twitter at @khanacademy. And remember, you can learn anything. For free. For everyone. Forever. #YouCanLearnAnything Subscribe to Khan Academy’s Algebra channel: https://www.youtube.com/channel/UCYZrCV8PNENpJt36V0kd-4Q?sub_confirmation=1 Subscribe to Khan Academy: https://www.youtube.com/subscription_center?add_user=khanacademy
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Foundations of Algebra
Introductory video to algebra
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F. William Lawvere - What are Foundations of Geometry and Algebra? (2013)
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: 3992 Matt Earnshaw
Algebra Basics: What Is Algebra? - Math Antics
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!
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Foundations of Algebra webinar
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]
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Introduction to Algebra | Arithmetic and Geometry Math Foundations 46 | N J Wildberger
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
Algebra - Basic Algebra Lessons for Beginners / Dummies (P1) - Pass any Math Test Easily
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
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An introduction to abstract algebra | Abstract Algebra Math Foundations 213 | NJ Wildberger
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.
College Algebra Introduction Review - Basic Overview, Study Guide, Examples & Practice Problems
This college algebra introduction / study guide review video tutorial provides a basic overview of key concepts that are needed to do well in a typical algebra course. High school students taking Algebra 1 and 2 can benefit from this video. It contains plenty of examples and practice problems. Trigonometry: https://www.youtube.com/watch?v=g8VCHoSk5_o Epic Music Mix: https://www.youtube.com/watch?v=qeljbZhx9bY Algebra Online Course: https://www.udemy.com/algebracourse7245/learn/v4/content Algebra Video Playlist: https://www.youtube.com/watch?v=i6sbjtJjJ-A&list=PL0o_zxa4K1BWKL_6lYRmEaXY6OgZWGE8G&index=1&t=13129s Access to Premium Videos: https://www.patreon.com/MathScienceTutor Here is a list of topics: 1. Properties of Exponents - Multiplication and Division Rules 2. Negative Exponents 3. Adding and Subtracting Polynomial Expressions such as binomials and trinomials 4. Foil Method - Multiplying Two Binomials 5. Solving Linear Equations 6. Solving Absolute Value Equations and Inequalities 7. Graphing Inequalities on a Number Line Using Interval Notation 8. Graphing Linear Equations In Slope Intercept Form and In Standard Form 9. Identifying the Slope and Y-intercept in a linear equation 10. Graphing Absolute Value Equations Using Transformations 11. Graphing Quadratic Functions Using Transformation - Horizontal & Vertical Shift with Reflection over X - axis 12. Solving Quadratic Equations By Factoring 13. Factoring Quadratic Expressions - Difference of Perfect Squares Method 14. Factoring trinomials with a leading coefficient of 1 15. How to factor a trinomial when the leading coefficient is not 1 16. Factoring Polynomials By Grouping 17. Solving Quadratic Equations Using the Quadratic Formula 18. Factoring Quadratic Expressions with the Quadratic Formula 19. Complex Imaginary Numbers 20. Simplifying Radical Expressions With Complex Numbers 21. Composition of Functions 22. Inverse Functions & Graphs 23. Evaluating Functions Using Synthetic Division 24. Solving Systems of Equations Using Elimination and Substitution
Foundations & Pre-Calc 10 - Algebra & Number Sample Questions
Solutions of Sample Questions for Algebra and Number from BC Ministry of Education Foundations of Mathematics and Pre-Calculus 10 Provincial
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The Foundations of Algebra
ethan explains math for hsiehzam.com
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Linear spaces and spans I | Abstract Algebra Math Foundations 219 | NJ Wildberger
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.
Baby Algebra | Arithmetic and Geometry Math Foundations 47 | N J Wildberger
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
The fundamental dream of algebra | Abstract Algebra Math Foundations 216 | NJ Wildberger
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!
1.1 Foundations for Algebra
This project was created with Explain Everything™ Interactive Whiteboard for iPad.
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Hoglund - U1F2 Day 1- Foundations of Algebra
U1F2 Day 1- Foundations of Algebra
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The Algebra of Boole is not Boolean Algebra! (I) \ Math Foundations 255 | N J Wildberger
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!
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Boolean algebra and set theory | Math Foundations 259 | N J Wildberger
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!
Foundations of Algebra Unit 3 Test
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Foundations of Data Science - Lecture 1
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: 13094 Microsoft Research
Transistors, Logic Gates and Boolean algebra | Math Foundations 261 | N J Wildberger
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!
What is the Fundamental theorem of Algebra, really? | Abstract Algebra Math Foundations 217
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.
Algebra and number patterns | Arithmetic and Geometry Math Foundations 50 | N J Wildberger
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
GRE | Algebra Foundations - Linear Inequalities | GRE Test Prep | ArgoPrep | GRE Math
To purchase the book please visit: http://amzn.to/2G8ZCMU To have an access to online free test visit: https://argoprep.com The GRE is comprised of three core assessment areas: • Analytical Writing • Verbal Reasoning • Quantitative Reasoning These areas are designed to measure your aptitude for critical reasoning and quantitative analysis, and to assess your ability to write coherent, well-supported arguments based on provided evidence and instructions. The core assessment areas are spread out across the exam in five scored sections: one analytical reasoning section, two verbal reasoning sections, and two quantitative reasoning sections. You will have 30 minutes to complete each Analytical Writing prompt and Verbal Reasoning section, and 35 minutes for each Quantitative Reasoning section. The exam can take anywhere from 31⁄2 - 4 hours depending on the version of the exam administered. Your exam will consist of one scored Analytical Writing section that includes two prompts: Analyze an Argument prompt The Analyze an Argument prompt will present you with an argument and ask you to evaluate its merits and logical sound­ness. Unlike the Analyze an Issue prompt, you will not choose a side for this prompt. Instead, you will write a critical as­sessment of the argument presented. The Analytical Writing section will always appear first on the exam. You will have 30 minutes to complete each prompt. The prompts are separately timed, and you can only work on one prompt at a time. The section is scored on a scale of 0-6, in half-point increments. A “6” is the highest possible score. Analyze an Issue prompt The Analyze an Issue prompt will present you with specific instructions on how to analyze a given topic. The topic lends itself to multiple perspectives, and there is no correct answer. What is important is that you construct a well-reasoned, cohesive argument that both supports your stance on the issue and closely follows the instructions given in the prompt. Verbal Reasoning In the two Verbal Reasoning sections, you will be asked to read and synthesize information presented in various forms from short sentences to multi-paragraph passages. This assessment area is designed to test your ability to comprehend and evalu­ate written material. The Verbal Reasoning sections also measures your understanding of sentence structure, punctuation, and proper use of vocabulary. Your exam will consist of two scored Verbal Reasoning sections that include the following question types: Reading Comprehension Reading Comprehension questions require you to read the given passages and select the answer choice that best com­pletes the question task. Content of the passages can come from a wide-range of subject matters, and there is often more than one question that corresponds to each passage. Text Completion Text Completion questions require you to identify the appropriate term (or terms) that best completes a given sentence. Text Completions can have anywhere from one to three terms that need to be identified. A strong vocabulary and the ability to understand context clues are both essential in this section. Sentence Equivalence Sentence Equivalence questions require you to identify two terms for a single blank in a sentence that will create two similar sentences that express the same main idea. Similar to the Text Completion questions, this section requires a strong vocabulary and command of context clues. Quantitative Reasoning In this section, you will be asked to solve mathematical problems drawn from the subject areas of arithmetic, geometry, al­gebra, and data analysis. This section tests your ability to solve quantitative problems, understand real-world applications of mathematical principles, and interpret statistical data from charts and graphs. Your exam will consist of two scored Quantitative Reasoning sections that include the following question types: Quantitative Comparisons Quantitative Comparison questions require you to analyze the relationship between two given quantities and select the answer choice that best describes the relationship. These questions focus more on understanding mathematical relation­ships and less on actual mathematical calculations. Mathematical Problem-Solving Mathematical Problem-Solving questions require you to use various mathematical formulas and processes to solve for the correct answer to the given problems. These are multiple-choice questions that can have either one or multiple cor­rect answers. These questions may also require you to input your own answer without being provided any answer choices to select from. Data Interpretation Data interpretation questions require you to interpret data from charts and graphs in order to solve for the correct an­swer. These questions are a sub-set of the Mathematical Problem-Solving questions and occur as part of a set where you will use one chart or graph to answer multiple questions.
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The Algebra of Boole is not Boolean Algebra! (II) | Math Foundations 256 | N J Wildberger
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.
A broad canvas: algebra with maxels from integers | Data Structures Math Foundations 209
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