  # Introduction to Higher Mathematics

## Video Lectures

Displaying all 19 video lectures.
 Lecture 1 Play Video Problem Solving 101Welcome to Introduction to Higher Mathematics! In this video you'll see what this course will entail. You'll also learn about some problem solving techniques that will come in handy throughout the course. Enroll now at http://www.polymathlectures.org Lecture 2 Play Video Introduction to ProofsThis lecture will introduce you to the language of proofs and show you how the axioms on which you build them are important. http://www.polymathlectures.org/ Lecture 3 Play Video Propositional LogicIt's time to delve into the language of propositional logic, which turns the way we think and reason into mathematical notation. http://www.polymathlectures.org/ Lecture 4 Play Video Proof TechniquesThis lecture discusses the formation of valid arguments and then introduces a number of common proof techniques. http://www.polymathlectures.org/ Here's the link in the video: http://www.math.hmc.edu/~su/math131/good-math-writing.pdf Lecture 5 Play Video Set TheoryIn this lecture we discuss the beginnings of set theory, a topic that runs throughout almost every area of mathematics. http://www.polymathlectures.org/ Lecture 6 Play Video Predicate LogicNow we're going to "upgrade" our logic to predicate logic, which lets us have a good bit more flexibility in how we describe various situations, including the use of quantifiers. http://www.polymathlectures.org/ Lecture 7 Play Video More Proof TechniquesIt's time to apply our new predicate logic to proofs and make use of all those new quantifiers. Also introduces mathematical induction. http://www.polymathlectures.org/ Lecture 8 Play Video RelationsIn this lecture we build relationships between sets, which can represent many of the mathematical relationships we know. http://www.polymathlectures.org/ Lecture 9 Play Video FunctionsA very useful type of relation between two sets is called a function, which we explore in this lecture. This is a somewhat different and more symmetric approach to functions from the one usually taken, instead defining it in terms of uniqueness and totality relations. http://www.polymathlectures.org/ Lecture 10 Play Video Number TheoryIn this lecture we delve into number theory, one of the oldest branches of mathematics that still has unsolved problems to this day. http://www.polymathlectures.org/ Lecture 11 Play Video Modular ArithmeticThis lecture addresses a particular topic in number theory, in which we work with just the remainders when divided by a certain number. Lecture 12 Play Video InfinityNow we leave the realm of the finite and wrestle with the infinite, exploring its mysterious properties. Lecture 13 Play Video Construction of the Real NumbersTo get a more firm foundation for the real numbers, we construct them, starting with the natural numbers and working our way up. Lecture 14 Play Video TopologyHere we delve a bit into the realm of topology, adding a notion of "closeness" to our sets and seeing how that helps us understand the real numbers. Lecture 15 Play Video Sequences and FunctionsWe finish up our discussion of real numbers with a look at sequences, bringing the real numbers to completion. Lecture 16 Play Video Group TheoryNow we leave the world of real analysis and explore abstract algebra, beginning with some beautiful structures called groups that will serve to unify mathematics as a whole. Lecture 17 Play Video Rings and FieldsBuilding on the idea of groups, this lecture explores the structures called rings and fields, beginning to more closely resemble the number systems we work with every day. Lecture 18 Play Video MorphismsHold onto your seats. In this lecture we're going to explore some relationships between groups that will astound you with how interconnected they are! Lecture 19 Play Video Epilogue & Fields of MathematicsThis final lecture introduces the variety of fields of mathematics available to study, and gives an idea of why they're important. Here you'll find out about: • Linear Algebra • Calculus • Multivariable Calculus • Vector Calculus • Differential Equations • Numerical Analysis • Discrete Math • Combinatorics • Graph Theory • Discrete Calculus • Probability • Statistics • Stochastic Processes • Game Theory • Number Theory • Abstract Algebra • Real Analysis • Complex Analysis • Geometry • Topology