Geometry
This text is the fifth and final in the series of educational books written by Israel Gelfand with his colleagues for high school students. These books cover the basics of mathematics in a clear and simple format – the style Gelfand was known for internationally. Gelfand prepared these materials so as to be suitable for independent studies, thus allowing students to learn and practice the material at their own pace without a class. Geometry takes a different approach to presenting basic geometry for high-school students and others new to the subject. Rather than following the traditional axiomatic method that emphasizes formulae and logical deduction, it focuses on geometric constructions. Illustrations and problems are abundant throughout, and readers are encouraged to draw figures and “move” them in the plane, allowing them to develop and enhance their geometrical vision, imagination, and creativity. Chapters are structured so that only certain operations and the instruments to perform these operations are available for drawing objects and figures on the plane. This structure corresponds to presenting, sequentially, projective, affine, symplectic, and Euclidean geometries, all the while ensuring students have the necessary tools to follow along. Geometry is suitable for a large audience, which includes not only high school geometry students, but also teachers and anyone else interested in improving their geometrical vision and intuition, skills useful in many professions.
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What Hacker News says
- When Judea Pearl was asked how Mathematics should be taught, he answered: Chronollogicaly. [1]
Now, What do You mean to "learn Math". You have to be more specific with your goal. This is funny coming from me because I haven't able to specialize in any single topic.
Let's suppose You somehow saw The Beauty of Math somewhere and You are ready to continue your journey in such a beautiful topic.
I really believe You Will enjoy reading "Mathematics for The million" by Lancelot, I saw this recommended by Sir Anthony Hoare[2], You Will be amazed by its content (I have only ready chapter one).
What's Mathematics?
Are elementary school children doing Math when they solve addition and subtraction problems?
Is The High school student crunching mechanical procedures, plugging numbers into formulas doing Math?
Is theorem proving Math?
Daphne Koller said in an interview that we are all doing probabilistics when Thinking and making certain decisiones[3]. Are we all Mathematicians?
Do You want to study Math as a tool to help You in some other areas or study it itself as a subject?
It doesn't matter, You Will enjoy it and love it.
Trying yo answer your question. Learn some Arithmetic, just enough. Crunch some elementary algebra (in The fashion of Visualizing Algebra provided by Udacity). Read The book Geometry by Tatiana Alekseyevskaya, learn some Discrete Math (I recommend You the book on the topic by Thomas VanDrunnen), read the other four books in the gelfand series. Learn some logic. Read Introduction to Mathematical Thinking by Keith Devlin. I think at this point You Will able to read some resources o calculus or abstract Algebra.
If this seems like a Lot, don't dismay. There was this guy George Green having to take care of mill to make bread and do also found time for Mathematics, apparently DeMoivre was struggling financially. There people are just like You. Enjoy your Math journey
[1] The interview is available on Lex Fridman YouTube channel.
[2] The conversation between him and Leslie Lamport is available on YouTube
- I've been in a similar boat. Khan Academy has been helpful, as have some other resources:
* Gelfand (et al.)'s HS math book series: System of Coordinates, Functions And Graphs, Algebra, Geometry, Trigonometry
* The problem collections Challenging Problems in Geometry and Challenging Problems in Algebra, available from Dover
* The math and logic courses on Brilliant.org
* Smullyan's Introduction to Logic
* Balakrishnan's Introduction to Discrete Mathematics
* For a bracingly irreverent "skip the bullshit" perspective, George Simmons' Precalculus Mathematics In A Nutshell
* The A-Levels A* prep sequence from Imperial College London, available on edX (this is actually roughly equivalent to a first year undergrad curriculum at US unis)
General advice:
* Practice, practice, practice - solve lots of problems & push yourself to repeatedly prove and reprove things. 'Getting it' once is not enough, math is like basketball. Something all the resources above have in common: tons of exercises, with solutions
* Keep mixing up difficulty so that you get some easy wins for confidence and motivation but also challenge yourself and keep yourself humble. Occasionally dip way down to remind yourself how far you've come, and on the other hand sometimes dip into something like Concrete Mathematics to remind yourself how far you still need to climb
* Go over the same material many times from different authors/teachers/sources
* Take your time. As long as you keep challenging yourself, and keep putting in those 10-20 hours a week, every week, you'll get to a good place and then keep going
- I've been trying to learn math for years but truly have no idea where to begin. I started with Kiselev's Geometry and Gelfand's Algebra. I used to hate Math. When I read Lockharts Lament some years ago and it brought me to tears. Ever since then I've had a big appreciation for it but no real time to get into it. Gelfand's Algebra is really cool because it starts with the most basic problems ever that a 3rd grader can do, and works it's way up. It's worth checking out!
- I love introductory textbooks. Some of my favorites:
"Starting Forth" - Leo Brodie
"Computer Science, Logo-Style" series - Brian Harvey
"Algebra", "Functions and Graphs", "Methods of Coordinates", "Trigonometry" - I.M. Gelfand (does anyone know if he completed his Geometry textbook? I've never seen it. I've also never seen his "Calculus" or "Combinatorics" books, although I know they exist. I suspect his "Combinatorics" book is more advanced than he intended to produce.)
"Sage for Undergraduates" - Gregory Bard
- I like books by Ron Larson, particularly his Trigonometry and Applied Calculus books -- the applied calculus title (intended for social science and business majors) vs his "Calculus" book (intended for math, physics, and engineering majors) is much gentler for people seeing this material for the first time. Although I do highly recommend his Calculus book once you have the other book down.
Gelfand also has some nice texts on Algebra, Trig, and Geometry that are reasonably cheap, especially if used.
I'm older and went back to school later in life to study math, and these are the books I learned that material (for the first time -- I flunked math thru high school) from.
Here are the exact titles and ISBN-10s:
Ron Larson, Calculus: An Applied Approach, ISBN: 0618218696
Ron Larson, Trigonometry, ISBN: 1133954332
Israel Gelfand, Trigonometry, ISBN: 0817639144
Israel Gelfand, Geometry, ISBN:1071602977
Israel Gelfand, Algebra, ISBN: 0817636773
And as others have mentioned, Khan Academy is pretty good, although I tend to prefer patrickJMT's explanations a bit more: http://patrickjmt.com/
- Start with:
* Algebra, Gelfand
* Geometry, Kiselev
