--- title: Introduction to Mathematics description: Introduction to mathematics covers foundational concepts, logical reasoning, and mathematical notation. Learners will understand how to read and write formal proofs, work with basic set theory, and transition to advanced mathematical study. category: mathematics subcategory: arithmetic-and-pre-algebra difficulty: beginner, intermediate, advanced url: /subject/introduction-to-mathematics --- # Introduction to Mathematics Introduction to mathematics covers foundational concepts, logical reasoning, and mathematical notation. Learners will understand how to read and write formal proofs, work with basic set theory, and transition to advanced mathematical study. ## Available Resources 5 Books • 2 Courses • 4 Websites ## Websites ### 1. Pauls Online Math Notes Complete lecture-note sets and worked practice problems from Paul Dawkins at Lamar University, spanning algebra, trigonometry, and calculus I through III plus differential equations. Useful for working problems step by step when a textbook explanation stalls. **Difficulty:** Beginner | **Price:** Free **Link:** https://tutorial.math.lamar.edu/ **Tags:** algebra, calculus, differential-equations, lecture-notes, practice-problems ### 2. Khan Academy Math Free video lessons and adaptive exercise sets covering arithmetic through calculus, linear algebra and statistics, with mastery tracking that identifies gaps. Suited to rebuilding school mathematics systematically before attempting university-level courses. **Difficulty:** Beginner | **Price:** Free **Link:** https://www.khanacademy.org/math **Tags:** arithmetic, algebra, calculus, statistics, video-lessons, practice-problems ### 3. Wolfram Alpha **Author:** Wolfram Research Computational knowledge engine that answers typed natural-language or symbolic queries by solving equations, factoring and simplifying expressions, and plotting functions. Students can check algebra work and inspect results, though full step-by-step solutions require a paid Pro subscription. **Difficulty:** Beginner | **Language:** English | **Price:** Free **Link:** https://www.wolframalpha.com/ **Tags:** computational-algebra, equation-solving, symbolic-computation, graphing, math-tools ### 4. artofproblemsolving.com AoPS is a comprehensive mathematics learning community offering challenging problems, step-by-step solutions, and active forums, along with online courses and textbooks for competition math. It provides problem sets, tutorials, and classroom resources covering topics from fundamentals to advanced contest math. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://artofproblemsolving.com **Tags:** websites, mathematics-statistics, mathematics ## Courses ### 1. Single Variable Calculus (MIT 18.01SC) **Author:** David Jerison MIT's self-paced 18.01SC covers differentiation, applications of derivatives, integration, techniques of integration, and infinite series, with problem sets, recitation videos, and exams with solutions. Completing it gives the standard first-semester calculus foundation. **Difficulty:** Beginner | **Price:** Free **Link:** https://ocw.mit.edu/courses/18-01sc-single-variable-calculus-fall-2010/ **Tags:** single-variable-calculus, derivatives, integrals, infinite-series ### 2. Introduction to Mathematical Thinking **Author:** Keith Devlin Stanford course by Keith Devlin on how mathematicians reason, starting with the logic of language (and, or, not, implication, quantifiers) and moving to the structure of proofs, including contradiction and induction. Learners become able to read, construct and critique simple mathematical proofs. **Difficulty:** Beginner | **Language:** English | **Duration:** Expect to require at least 10 hours of study per week to complete this course satisfactorily. | **Price:** Free **Link:** https://www.coursera.org/learn/mathematical-thinking **Tags:** mathematical-logic, proof-writing, quantifiers, mathematical-reasoning ## Podcasts ### 1. My Favorite Theorem **Author:** Kevin Knudson, Evelyn Lamb Interview podcast in which mathematicians Kevin Knudson and Evelyn Lamb ask a guest mathematician to explain a favorite theorem and why it matters. Episodes span topology, number theory, analysis, and combinatorics, giving listeners an accessible sense of research mathematics and its culture. **Difficulty:** Beginner | **Language:** en-US | **Price:** Free **Link:** https://kpknudson.com/my-favorite-theorem/ **Tags:** popular-mathematics, theorems, mathematicians, math-communication ### 2. The Joy of x **Author:** Steven Strogatz and Quanta Magazine The acclaimed mathematician and author Steven Strogatz interviews some of the world's leading scientists about their lives and work. You've learned from Quanta. Now we want to learn from you! Quanta is conducting a series of surveys to better serve our audience. Take our podcast listener survey and you will be entered to win a free Quanta book, t-shirt or tote bag: quantamag.typeform.com/podcast **Difficulty:** Beginner | **Language:** en-US | **Price:** Free **Link:** https://www.quantamagazine.org/tag/the-joy-of-x/ **Tags:** mathematics, interviews, science-communication, popular-math ## Books ### 1. Basic Mathematics - Serge Lang **Author:** Serge Lang Serge Lang rebuilds school arithmetic, algebra, geometry and trigonometry with the proofs and precision usually withheld until university. Working the exercises closes logical gaps that make later calculus and analysis courses feel unmotivated or arbitrary. **Difficulty:** Beginner | **Language:** English | **Price:** Paid **Link:** https://amzn.to/49NgOof **Tags:** algebra, geometry, trigonometry, proof, foundations ### 2. How to Solve It - George Pólya **Author:** George Polya A legendary guide on mathematical heuristics and problem-solving techniques applicable beyond mathematics. **Difficulty:** Beginner | **Language:** English | **Price:** Paid **Link:** https://amzn.to/3NJB818 **Tags:** problem-solving, heuristics, mathematical-thinking, proof ### 3. Mathematics: Its Content, Methods and Meaning **Author:** A. D. Aleksandrov, A. N. Kolmogorov, and M. A. Lavrent'ev Survey of twenty mathematical fields, from analysis and algebra to differential equations, complex variables and probability, written by leading Soviet mathematicians for readers outside the profession. It maps what each branch studies and why, without full technical machinery. **Difficulty:** Intermediate | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0880294787?tag=edmonddante07-20 **Tags:** books, mathematics-statistics, mathematics ### 4. What Is Mathematics? **Author:** Richard Courant and Herbert Robbins Courant and Robbins present number theory, geometric constructions, projective geometry, topology and calculus as connected ideas rather than techniques. Readers come away understanding what mathematical proof accomplishes and how the main branches relate to one another. **Difficulty:** Intermediate | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0195025172?tag=edmonddante07-20 **Tags:** books, mathematics-statistics, mathematics ### 5. Journey through Genius **Author:** William Dunham Twelve landmark theorems, from the lunes of Hippocrates to Cantor's transfinite numbers, presented with their original proofs and historical setting. Readers follow complete arguments and see how mathematical questions arose from the concerns of their period. **Difficulty:** Intermediate | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/014014739X?tag=edmonddante07-20 **Tags:** history-of-mathematics, famous-theorems, mathematical-proofs, greek-mathematics --- *This content is part of Dantes.io - Your Treasure Map to Knowledge* *Curated by humans at Dantes.io. Personal study use welcome; republishing this curation requires permission (team@dantes.io).* View this page online: https://dantes.io/subject/introduction-to-mathematics