Search results “Foundations of algebra”

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.
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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

Views: 742038
The Organic Chemistry Tutor

Georgia High School Graduation Test Content:
http://www.stephenwelchtutoring.com/GHSGT.html

Views: 356
Stephen Welch Tutoring

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:
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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

Views: 253605
The Organic Chemistry Tutor

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: 3142
Matt Earnshaw

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
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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
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Views: 3528147
Khan Academy

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: 8613
njwildberger

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: 1963
njwildberger

U1F2 Day 1- Foundations of Algebra

Views: 190
Beast Algebra

Views: 23
kosherfan

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: 3469
njwildberger

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: 917
hand2mind

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: 8056
Microsoft Research

Views: 197
DrDoveMath

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: 11817
njwildberger

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: 6680
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: 3137
njwildberger

Solutions of Sample Questions for Algebra and Number from BC Ministry of Education Foundations of Mathematics and Pre-Calculus 10 Provincial

Views: 5525
mathjohnson

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.

Views: 3035
Argo Brothers

This project was created with Explain Everything™ Interactive Whiteboard for iPad.

Views: 21
Justin Walls

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: 5333
njwildberger

Recorded with http://screencast-o-matic.com

Views: 70
Candace Loudermilk

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: 8425
njwildberger

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: 10076
njwildberger

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: 6870
njwildberger

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: 3616
njwildberger

Views: 23
TSmithECCS

* * * 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: 2126
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: 6195
njwildberger

Having introduced the important idea of an integral linear space, or ilinear spaces, we see the continued importance of having a good multiset orientation towards basic structure in mathematics. Here we continue that exploration, by introducing the idea of a spanning mset of an ilinear space, and then define a basis as a spanning mset of minimal size. Note carefully that the usual condition of linear independence is not part of the of definition; rather it will eventually be seen as a consequence. The dimension of an integral linear space is then also defined along the way.
In order to smoothly work with vexels as examples, we also introduce some convenient notations for connecting our general vexel notation with more standard conventions with vectors.

Views: 2027
njwildberger

Views: 21
A. Owens

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