Precalculus 1: basic notions

A total of 52 hours of lectures

Due to the size of precalculus, it is divided into four parts. This page describes the first of those four parts.

Image precalculus part 1

Prerequisites

High-school maths, mainly arithmetics, some trigonometry

Curriculum

Make sure that you check with your professor what parts of the course you will need for your exam. Such things vary from country to country, from university to university, and they can even vary from year to year at the same university.

Precalculus: basic notions

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Get the outline

A detailed list of all the lectures in part 1 of the course, including which theorems will be discussed and which problems will be solved. If you are looking for a particular kind of problem or a particular concept, this is where you should look first.

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Course Objectives & Outcomes

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How to solve problems in chosen Precalculus topics (illustrated with 236 solved problems) and why these methods work.
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The basic notions will be defined in a more abstract way, and you will get an insight into their place in Calculus and some other branches of university maths.
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Arithmetic with basic rules (associativity and commutativity of addition and multiplication, distributivity) illustrated for positive integers.
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Basic concepts related to functions (domain, range, graph, surjection, injection, bijection, inverse, composition) and how to work with them.
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Geometrical operations on graphs of functions: translations, reflections, shrinking, dilating. Piecewise functions.
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Equations of straight lines in the plane: slope and intercept; first glimpse into derivative as the slope of the tangent line, and link to monotonicity.
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Logic: symbols and rules (tautologies) used for creating mathematical statements, definitions, and proofs.
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Relations: RST (equivalence) relations, order relations, functions as relations.
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Various types of proofs: direct, indirect, by contradiction, induction proof, with some examples.
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Decimal expansion of rational and irrational numbers. Density of Q and R\Q in R.
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You will also get a solid background for any future studies in Real Analysis and other courses in university mathematics.
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Repetition of chosen aspects of high school mathematics (basic notions as numbers, functions, sets, equations, and inequalities).
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By the end of this course you will be able to understand all the elements in the epsilon-delta definition of limit, both symbols and the content.
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Basic terminology for equations and inequalities, with examples of linear equations and inequalities, with and without absolute value.
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Functions: monotone (increasing, decreasing), bounded, even, odd; extremum: maximum, minimum, both local and global.
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Absolute value and its role in computing distances, with geometrical illustrations and functional approach.
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Introduction to sequences and series; short notation for sum (Sigma) and product (Pi); arithmetic, geometric, and harmonic progressions.
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Set theory: union, intersection, complement, set difference; some important rules and notation.
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Necessary and sufficient conditions: definition and examples.
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Building blocks of mathematical theories: axioms, definitions, theorems, propositions, lemmas, corollaries, etc.
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Cardinality of sets: finite sets, N, Z, Q, R; countable and uncountable sets.
Muhammad O.
Udemy student
Great teacher
Natalie G.
Udemy student
Fantastic effort and great service to everybody who struggled with maths in high school and beyond. The effort is truly appreciated. Wish more math teachers had the skill and interest to teach like this.
Wanda W.
Udemy student
Incredibly rich content. As a retired math teacher I really appreciate Hania's way of teaching and the organisation of the course. Highly recommended!
Edit: I have just finished the last part (4) of the series, and I am very impressed with the curriculum and the teaching. Hania's Precalculus series gives everybody a perfect preparation for Calculus and Real Analysis!
You get an introduction to Precalculus in the beginning of Calculus 1, too. This is my 6th course by Hania and Martin that I've completed and I can recommend all of them.
Shelton S.
Udemy student
Mind opening without going down abstract rabbit holes! Worth of every effort to grasping every word!
Sued R.
Udemy student
Finally a course that helps me to visualize and understand the math algorithms and procedures. Huge difference between most of the math courses that only teachs you the procedures without explaining what you are doing and why you are doing that!
Steven M.
Udemy student
I like the instructor's approach as she teaches the theory behind the math so you can understand the what and why as opposed to just mindlessly generating an answer.
Tetyana M.
Lecturer
This is a detailed exposition of the many basic things that have the habit of falling between the curricula of the school and those of the university. The course can readily be taken by interested school students as well as by the university students in need of grounding what they in principle should know, though perhaps do not in practice. Insufficient mathematical foundations are an old problem at university, and Hania's present course – comprehensive and immediately rewarding – addresses that problem exquisitely.
Anonymized User
Udemy student
Formatrice excellente : Hania prend le temps d'expliquer chaque notion de façon détaillée. Même si, comme moi, votre anglais n'est pas parfait, elle parle lentement et clairement si bien que vous comprenez tout ce qu'elle dit. Le seul reproche qu'on pourrait lui faire, c'est que les vidéos sont longues (au-delà de 10 minutes !!!) : il suffit alors de régler la vitesse de lecture sur 1.25, voire 1.5.Machine translation: An excellent instructor: Hania takes the time to explain each concept in detail. Even if, like me, your English isn't perfect, she speaks slowly and clearly so that you understand everything she says. The only criticism one could have is that the videos are long (over 10 minutes!!!): all you have to do is set the playback speed to 1.25 or even 1.5.
Igor V.
Udemy student
The course provides good theoretical foundation, with practical examples demonstration. It even involves concepts from abstract algebra, propositional logic, proof techniques, other concepts from foundation in mathematics. This is what you most probably don’t meet in high school, but these concepts give you deeper understanding when you start learning more advanced courses. However, it requires some maturity is school maths from you. If you have problems with basic math like arithmetic calculations, solving equations etc., I would recommend to look for another, more basic course. But if your skills are good enough, the content will be useful and surprisingly interesting.
Tee B.
Udemy student
So fantastic!