Basic Mathematics by Serge Lang (PDF)

Description

This text in basic mathematics is ideal for high school or college students. It provides a firm foundation in basic principles of mathematics and thereby acts as a springboard into calculus, linear algebra and other more advanced topics. The information is clearly presented, and the author develops concepts in such a manner to show how one subject matter can relate and evolve into another.

User’s Reviews

Editorial Reviews: From the Back Cover The present book is intended as a text in basic mathematics. As such, it can have multiple use: for a one-year course in the high schools during the third or fourth year (if possible the third, so that calculus can be taken during the fourth year); for complementary reference in earlier high school grades (elementary algebra and geometry are covered); for a one-semester course at the college level, to review or to get a firm foundation in the basic mathematics to go ahead in calculus, linear algebra, or other topics.

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This book is ultimately for creating a strong foundation for calculus. “Basic” mathematics can be used lightly, as it’s still relatively challenging, but by no means difficult. You’ll spend time trying to understand proofs (by the way, he doest an excellent job of incoorporating proofs into the learning process, which is extremely necessary if you want to fully understand math) and figuring out some of the more challenging exercise problems, but the book is by no means painful to go through.I’ll be a junior in high school next year, and was recommended studying this book by a math professor in Belgium (yes, really). The biggest thing to know about math, at least in the earlier levels, is that it is cumulative. You will be absolutely and forever f’ed if you go into Calculus with hardly any knowledge of algebra. That’s why, if you find yourself struggling in math, or need to strengthen your foundations, you need to go throuh this book.It’s excellent for self studying, which is what I’m currently doing this summer. When you take this book on at your own pace, and can spend more time on each section as required, and breeze through other sections that you have a firm grasp on, you’ll find yourself enjoying the book. Serge Lang was an excellent mathematician (was… only because he is dead) and his books are well written and devoted to helping you understand the material.Remember though, that just because the book claims to be covering basic mathematics, that you won’t be challenged. It can be tough, and especially when you’re at the high school level, the mathematics won’t seem like they’re “basic”, but more like, “hey, I just learned that last year/this year/whenever.” It’s ultimately a wonderful book if you’re looking to strengthen the foundation of your mathematic ability, and Lang also includes a brief section after the first chapter about how you should logically examine math, which is invaluable.Math, to a lot of people, is pointless and not fun. It’s that way because of botched U.S. public education, a lack of good/inspiring math teachers, and partly due to a lack of motivation in students. Pick up this book, maybe make a schedule for it, and just sit down and DO math. After a while, you’ll start to see the beauty of something that is so logical and amazing, something that has been the collective work of various genius human minds, collaborating to further mathematics, that you’ll pick up a passion for math. Math is almost like a language. You can choose to go through life in ignorance, sitting by and listening to others communicate with eachother, looking on in awe, wondering what beautiful and inspiring things they could be saying, or you can sit down, get to work, and be right there with them, able to communicate in a logical and correct language that governs the universe.

⭐Overall I’m content with this book, but it has a good number of issues. If you’re using it for self study, it can be frustrating because the answers in the back are fairly sparse, and even the ones that do exist have a lot of errors. Answers will be completely wrong, or sometimes swapped (b answers a and vice versa).I think this is unacceptable for a book that was originally published in 1971 and Amazon has this version listed as 1998. They’ve had plenty of time to fix their errors for a republishing.I used this book to refresh my math knowledge after a decade away from math, and while it did an acceptable job, I did still have to reference other sources a number of times as this book will throw things at you that it hasn’t explained. As an example, simplifying roots is never covered, but they do so in their solutions, so you’ll have to learn how to do it elsewhere. This one in particular was disappointing to me, because the way exponents and roots are covered in the book was excellent, and it did set me up to understand why it works on a deeper level than any videos/articles I read on it, but it would have been way better to not have to do that.

⭐I have been going back to relearn math from essentially the ground up before I enter a PhD in computer science. This book does a great job of presenting a rigorous introduction to all the basic concepts that I took for granted in high school. For example, in my school we were just given the Pythagoras theorem and distance formula to memorize, but Lang actually includes the proofs of these.The one thing that really annoyed me were the proofs. The only way I could navigate his proofs was due to my exposure to mathematics in undergrad. Many of the proof techniques that he used seemed arbitrary, and even the solutions to proof problems in the back of the book were arbitrary and ad hoc. There is a little section in the middle of the book about proofs and mathematical logic, but it covers nowhere near the level of proofs that he uses in the text. If this is truly a text on basic mathematics, let it be basic, or fill in the reader on proof techniques.

⭐Very good as a reference book for all those subjects I studied long ago. But unlike some other books I haven’t found this one to be “fun.” It’s not the one I’d recommend if you’re looking for something you can read cover to cover. It could certainly be done, there’s just other books I’ve found to be better suited for working your way through like that (Stillwell’s Numbers and Geometry or Hardy’s Course of Pure Mathematics maybe).This one is a pleasure to dip into when you’ve got a specific topic you need to review (i.e. trig or something) but if you’re looking to buy one book to inspire you and take your mathematical knowledge and thinking to new depths, I’d go with something else.

⭐Serge Lang has written one of the better books out there for learning high-school level math well. The explanations are clear and the problems are graded from easy to hard well. It covers everything you would need to know to start higher-level math studies. Some of the problems are really hard and require a good deal of thought to solve. I would recommend this book to anyone, but especially future or current math majors. The reason that this book gets 4 stars is because several sections have few answers in the back of the book.

⭐I am very happy with this purchase. The book was delivered on time and it’s in perfect condition!

⭐Love it. And it’s getting me up to speed quickly with my old math skills

⭐I love the way this book is written! It really circumvents a lot of the dry technical material that you see in a lot of math textbooks but still gives you the most necessary information. The exercises are great for practicing the skills covered. This book is good to have on reference as well as to use for a prep for higher level math like calculus.

⭐This book covers a lot of the material needed to enter a calculus or linear algebra sequence, and approaches it by deriving important results rather than stating them without motivation. The exercises are a mixture of simple calculations and proofs of varying difficulty; however it is ostensibly possible for anyone with a basic grasp of algebra and geometry to work through this book. There are also some mediations on more comparitively abstract areas of mathematics, such as finding rational points on a circle.Having said that, the exercises in this book are frustrating. Many of the answers (odd-numbered questions only) are misprinted and INCORRECT, which can leave less confident students worrying over a problem that they may have actually solved. Furthermore, the exercises can have solutions that are not particularly “nice”, and can be worrying for people who might expect that results can be simplified further or made easier to calculate.Whilst this book is a solid training session for anyone who wants to improve their mathematical abilities, it can be punishing and frustrating at times.

⭐You couldn’t ask for a better reference/teaching tool than this book. I’m teaching myself Mathematics at the moment and I find this a valuable resource

⭐I know the author was a famous mathematician and is an acknowledged authority in number theory and this book shows it. If you have a good grounding in math and have used it regularly in your line of work you might, as a math nerd, really like this book, but, if you are not, then I think you could not find a more abstruse way to relearn math. It’s misnamed. It’s not “basic” at all. Now from the perspective of the author it may have seemed basic but it’s not. It’s number theory galore and that means its just not a good learning source for people who do not already have good math skills. I mean this guy makes addition and subtraction look hard with his approach. So if you are at the “basic” level of math as most of us are, you need to find a better book to aid your learning.

⭐it’s a nice and helpful book but unfortunately it has multiple errors. I try to refresh my mathematics skills by studying this book.Naturally I do not feel yet very secure about my skills. In this situation it is quite stressful to find equations that do not make sense. Multiple times now I was not sure, is this an error or is it just my limited maths understanding? Unfortunately, it turns out its frequently an error.For example the equation that can be used to convert radians into degrees is xdeg=pi*xrad/180 instead of the correct xdeg=180*xrad/piOverall its a nice and readable book. I learned quite a bit by studying it. But the errors can be quite confusing.I didn’t pay attention from gr8-gr10. Though the methods don’t really add up to the way my school teaches math. I did learn a lot. It’s a solid read for maybe a review over the summer or if you need extra help but it shouldn’t be used as the primary teaching method.

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