# Mathematics Courses Online

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## Building Numbers

{{}} One of the most familiar entities in maths are numbers. From pre-school to advanced levels, numbers are intimately linked with maths. But what are these numbers mathematically speaking? This little book builds numbers the way mathematicians build them up (at least the way most of them do now). We start from the bare axioms of set theory and develop numbers in the mathematical sense. The reader is assumed to be sufficiently mathematically mature, having been exposed at least to high school maths to understand the arguments that follow. Nothing can be built out of nothing. Our arguments can only proceed from some unproven assumptions. We start off with a list of the axioms of set theory, using these as our "unproven truths". We do not bother to focus much on logic here, rather give an intuitive feel of the axioms, so that our end goal of building numbers is not sidetracked. For those interested in a very logically sound discussion of these axioms (specifically in first order logic) the bibliography mentions books which carry further details. 1. Axiom of extensionality

7 hours

## Ratios, Proportions, and Their Uses

Ratio, as a basic construct in mathematical language, is used to explore the reason why of the sense of beauty, including shapes and music. Thus, ratio is a human notion of geometry and time, as a part of human perception. The reasoning on ratio is proportion, that two ratios are equivalent but different in size, from which a mathematical theory of ratio was born, and variant mathematical subjects were derived. The concept of ratio did become an important tool in the history of mathematics, especially in differential calculus, accompanied with other mathematical concepts such as divisibility. But the modern definition of ratio is mainly arithmetic, that treats a ratio as a value rather than the ratio itself, thus the meaning and possible uses of ratios are lost. This course tries to study ratios from a historical and methodological respect, to recover the possible uses of ratios in the history, thus we may be able to use similar methods and concepts on new things. That is, ratio will be treated as a conceptual tool that can help, rather than merely an arithmetic entity with a value. Before the formal discussion of ratio, we explore the occurrence of the notion itself, thus we can know where it can be used and the reason why of related talk. Let's regard ratio as an innate functionality of human mind. When we say someone is thin, it means the height is relatively larger than the extent of the person, this is where the notion of ratio functions -- the sense of thin is derived from the function of ratio. Here, the word function means an activity of mind that accepts input (height and extent), performs some process (comparison of height and extent and match of the concept of thin) and produces an output (the utterance of thin).

7 hours

## Visual physics and mathematics

This course will present images and animations which illustrate points of physics or mathematics. Anyone is invited to contribute. Gravitation: Sound waves : Flow around a cylinder: Flow around a wing: Venturi Effect: Pi and the relationship between a circle's circumference: Heisenberg indeterminacy: Wave-particle duality: Solitary wavelet or particle? Interference of a particle with itself:

7 hours

## Operations Research

Operations research or operational research (OR) is an interdisciplinary branch of mathematics which uses methods like mathematical modeling, statistics, and algorithms to arrive at optimal or good decisions in complex problems which are concerned with optimizing the maxima (profit, faster assembly line, greater crop yield, higher bandwidth, etc) or minima (cost loss, lowering of risk, etc) of some objective function. The eventual intention behind using operations research is to elicit a best possible solution to a problem mathematically, which improves or optimizes the performance of the system. This course is intended for both mathematics students and also for those interested in the subject from a management point of view.

7 hours

## Applied Mathematics

Applied Mathematics is the branch of mathematics which deals with applications of mathematics to the real world problems, often from problems stemming from the fields of engineering or theoretical physics. It is differentiated from Pure Mathematics, which deals with more abstract problems. There is also something called Applicable Mathematics, which deals with real world problems which need the techniques and mindset usually used in Pure Mathematics. These distinctions do not really become apparent during school level mathematics. Examples of topics in Applied Mathematics:

7 hours

## Calculus of Variations

This course is a transcribed version of Lectures on the Calculus of Variations (the Weierstrassian theory) by Harris Hancock in 1904. The scanned original is available here from Cornell University. PREFACE CHAPTER I: PRESENTATION OF THE PRINCIPAL PROBLEMS OF THE CALCULUS OF VARIATIONS. CHAPTER II: EXAMPLES OF SPECIAL VARIATIONS OF CURVES. APPLICATIONS TO THE CATENARY. CHAPTER III: PROPERTIES OF THE CATENARY. CHAPTER IV: PROPERTIES OF THE FUNCTION F ( x , y , x ′ , y ′ ) {\displaystyle F(x,y,x',y')} . CHAPTER V: THE VARIATION OF CURVES EXPRESSED ANALYTICALLY. THE FIRST VARIATION. CHAPTER VI: THE FORM OF THE SOLUTIONS OF THE DIFFERENTIAL EQUATION G = 0 {\displaystyle G=0} . CHAPTER VII: REMOVAL OF CERTAIN LIMITATIONS THAT HAVE BEEN MADE. INTEGRATION OF THE DIFFERENTIAL EQUATION G = 0 {\displaystyle G=0} FOR THE PROBLEMS OF CHAPTER I.

7 hours

## Calculus

This course aims to be a high quality calculus course through which users can master the discipline. Standard topics such as limits, differentiation and integration are covered, as well as several others. Please contribute wherever you feel the need. You can simply help by rating individual sections of the course that you feel were inappropriately rated! 1.1 Algebra 1.2 Trigonometric functions 1.3 Functions 1.4 Graphing linear functions 1.5 Exercises 1.6 Hyperbolic logarithm and angles 2.1 An Introduction to Limits 2.2 Finite Limits 2.3 Infinite Limits 2.4 Continuity 2.5 Formal Definition of the Limit 2.6 Proofs of Some Basic Limit Rules 2.7 Exercises 3.1 Differentiation Defined 3.2 Product and Quotient Rules 3.3 Derivatives of Trigonometric Functions 3.4 Chain Rule 3.5 Higher Order Derivatives: an introduction to second order derivatives 3.6 Implicit Differentiation 3.7 Derivatives of Exponential and Logarithm Functions 3.8 Some Important Theorems 3.9 Exercises 3.10 L'Hôpital's Rule 3.11 Extrema and Points of Inflection 3.12 Newton's Method 3.13 Related Rates 3.14 Optimization 3.15 Euler's Method 3.16 Exercises 4.1 Definite integral 4.2 Fundamental Theorem of Calculus 4.3 Indefinite integral 4.4 Improper Integrals

7 hours

## Vectors

This course introduces three-dimensional vectors as mathematical entities, though their application will be found, very likely, in physical science. For examples, velocity and acceleration of a particle in a reference frame are usually defined as vectors. As this is an elementary mathematical textbook, it is useful to state the prerequisites for readers expecting to benefit from what is written. When faced with listing prerequisites for a similar textbook in 1965, James A. Hummel of University of Maryland gave this list: Basic concepts of trigonometry, of Cartesian coordinates in the plane, and of set theory and notation. For his text he also required knowledge of the definition of a function, of the definition and properties of determinants of orders two and three, the absolute value, the field axioms, and the order axioms for real numbers. Study of the vector algebra in This course is good preparation for Linear Algebra. For students that have studied calculus of one variable, the chapter on vector analysis provides an introduction to the tools physicists use to study vector fields dependent on position in space. Consider the set of directed line segments in the plane. If two such segments are parallel, equal in length, and similarly directed, they are said to be equipollent. Such segments can be considered equivalent, and the collection of equivalence classes of directed segments in the plane provides an illustration of a planar space of vectors.

7 hours

## Geometry

The word geometry originates from the Greek words (geo meaning world, metri meaning measure) and means, literally, to measure the earth. It is an ancient branch of mathematics, but its modern meaning depends largely on context. Geometry largely encompasses forms of non-numeric mathematics, such as those involving measurement, area and perimeter calculation, and work involving angles and position. It was one of the two fields of pre-modern mathematics, the other being the study of numbers. In modern times, geometric concepts have been generalized to a high level of abstraction and complexity, and have been subjected to the methods of calculus and abstract algebra, so that many modern branches of the field are barely recognizable as the descendants of early geometry. This course is dedicated to high school geometry and geometry in general. The outline of topics reflects the California curriculum content standards. Template:Download version [1]

7 hours

## Associative Composition Algebra

This course is on associative composition algebras, structures that have been used in kinematics and mathematical physics. 1. Introduction 2. Transcendental paradigm 3. Binarions 4. Quaternions 5. Homographies This text expands the repertoire of algebra beyond real numbers R and complex numbers C to just five more algebras. The prospective reader will be well-acquainted with the utility of R and C in science, and might like to know (more) about quaternions H and related algebras, and what have been the historical invocations of these algebras. Some group theory and matrix multiplication are prerequisites from linear and abstract algebra. Attention to this text will show some concrete instances of mathematical objects, thus nailing down the abstruse nature of abstract algebra. Whereas linear algebra characteristically is concerned with n-dimensional space and n × n matrices, for this text n = 2 is the limit. Some of the content of this text was summarized in 1914 by Leonard Dickson when he noted that the complex quaternion and complex matrix algebras are equivalent, but their real subalgebras are not ! For more history of these algebras see Abstract Algebra/Hypercomplex numbers,  w:Composition algebra#History and w:History of quaternions.

7 hours

## Abstract Algebra

This course is on abstract algebra (abstract algebraic systems), an advanced set of topics related to algebra, including groups, rings, ideals, fields, and more. Readers of This course are expected to have read and understood the information presented in the Linear Algebra book, or an equivalent alternative. This course shall give an introduction to the fundamental concepts of abstract algebra, such as groups, rings and ideals, and fields and Galois theory. Sources

7 hours

## Linear Algebra with Differential Equations

As the title suggests, we assume you have prior knowledge of differential equations and linear algebra separately. the course is structured into three main chapters, each with an important introduction that itself introduces material (So don't just skim over it thinking it's part of an outline) and then leads into method-heavy subsections. Other than that, enjoy!

7 hours

## Topics in Abstract Algebra

This course aims to cover algebraic structures and methods that play basic roles in other fields of mathematics such as algebraic geometry and representation theory. More precisely, the first chapter covers the rudiments of non-commutative rings and homological language that provide foundations for subsequent chapters. The second chapter covers commutative algebra, which we view as the local theory of algebraic geometry; the emphasis will be on (historical) connections to several complex variables. The third chapter is devoted to field theory, and the fourth to Linear algebra. The fifth chapter studies Lie algebra with emphasis on applications to arithmetic problems. Part I. Foundations Part II. Applications Part III. Appendix

7 hours

## Surreal Numbers and Games

Surreal numbers are a fascinating mathematical structure, built from a few simple rules but giving rise to marvellous complexity. The surreal numbers contain all the real numbers with which we are familiar, as well as an infinitude of new quantities. We will discover surreal numbers that are greater than any positive integer, and ones that are infinitesimally small. Concepts like the square root and the reciprocal of infinite quantities will not only be defined, but we will find that they show logical and beautiful behaviour. The Surreal numbers were invented by mathematician John H. Conway as part of an investigation into endgames in the game of Go, an oriental board game that also produces complex behaviour from a small set of simple rules. They were presented to the world in the form of a small novelette by Donald E. Knuth, in which a young couple on holiday discover a rock inscribed with Conway's rules and proceed to derive the entire theory. We will begin the same way, beginning with the initial axioms and working our way up to the entire vast structure. Along the way we will prove that all the familiar properties of real numbers (such as the transitive law of inequality, and the commutative law of addition) all hold. A basic familiarity with set theory is assumed; for a refresher, see Set Theory. The Beginning

7 hours

## Category Theory

This course is an introduction to category theory. It is written for those who have some understanding of one or more branches of abstract mathematics, such as group theory, analysis or topology. The course contains many examples drawn from various branches of math. If you are not familiar with some of the kinds of math mentioned, don’t worry. If all the examples are unfamiliar, it may be wise to research a few before continuing. A category is a mathematical structure, like a group or a vector space, abstractly defined by axioms. Groups were defined in this way in order to study symmetries (of physical objects and equations, among other things). Vector spaces are an abstraction of vector calculus. What makes category theory different from the study of other structures is that in a sense the concept of category is an abstraction of a kind of mathematics. (This cannot be made into a precise mathematical definition!) This makes category theory unusually self-referential and capable of treating many of the same questions that mathematical logic treats. In particular, it provides a language that unifies many concepts in different parts of math. In more detail, a category has objects and morphisms or arrows. (It is best to think of the morphisms as arrows: the word “morphism” makes you think they are set maps, and they are not always set maps. The formal definition of category is given in the chapter on categories.)

7 hours

## Number Theory

This course covers an elementary introduction to Number Theory, with an emphasis on presenting and proving a large number of theorems. No attempts will be made to derive number theory from set theory and no knowledge of Calculus will be assumed. It is important to convince yourself of the truth of each proof as you work through the course, and make sure you have a complete understanding. For those who wish to use this as a reference book, an index of theorems will be given.

7 hours

## Crystallography

Crystallography is a branch of geometry that deals with indefinitely repeating patterns. Two-dimensional crystallography can be used, for example, to describe the way tiles cover a floor. Extending the field into three dimensions allows a general description of the way atoms or molecules arrange themselves into crystals. The three-dimensional crystallography was proven to be complete over a century ago. The fact that the mathematics itself cannot be advanced without some change of its axioms has meant that it is studied less often as pure mathematics, than as a means of understanding the details of complex structures in matter. Two fields are particularly reliant on it: materials scientist use it to describe the structure of engineering materials, often with particular attention to crystallographic defects; biochemists use it to describe the structure of biopolymers (see proteomics for an example), which usually must be processed laboriously before they form crystals. In mathematical terms, a crystal is an object with translational symmetry, i.e. it can be moved some distance and remain the same. This type of symmetry is fundamentally different from the more familiar mirror symmetry (the human face) or rotational symmetry (an airplane propeller), in that objects we imagine to represent translational symmetry must be larger than ourselves. We can imagine passing Alice through the looking glass, or spinning a propeller by one blade's fraction of a rotation, while we stand still. For an experience of translational symmetry, however, we must move ourselves, and not notice the difference. This can happen in an ocean, a desert, or a large suburb, if every wave, or dune, or tract home looks exactly like the next. Just as no eye is the exact mirror of its opposite, and no propeller is perfectly balanced, no physical crystal is perfect. There will always be a boundary that gets nearer or farther after a unit of translation. Strictly speaking, any true crystal must fill the entire universe. If we imagine a "perfect" housing development (dystopian though it may be) which covers a two-dimensional plane with an infinitely-repeating pattern of homes, and want to apply crystallography to it, we can save a lot of work by eliminating all the geometric complexity of garages and sprinkler heads and such. To make things as simple as possible, we could abstract every house down to a single point, although we need to keep track of each house's orientation.

7 hours

\$1,990

#### Is learning Mathematics hard?

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