Harder Proof From This Classic Book

In this video I will show you a book that I just got. It's called Basic Mathematics and it was written by the legendary Serge Lang. I have wanted this book for a very long time and I finally got it:) I will also do a problem from the book in this video. We will find the inverse of a multivariable function. I did not expect to find a problem like that in this book:)
Here is the book: amzn.to/3wvpf3q
My pencils: amzn.to/3CsiJya
My paper: amzn.to/3e0N3WB
(these are my affiliate links, so please use these if you want to support my channel)
0:00 Discussing the book
2:37 Finding the inverse of a Multivariable Function
13:16 Finishing up
If you enjoyed this video please consider liking, sharing, and subscribing.
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Thank you:)

[@fordfactor]

I have found over the years that maths books with the words "Basic" or "Introduction" in the title are generally neither basic or introductory.

[@TheMathSorcerer]

LOL yes! A great example is "Basic Algebra" by Knapp.

[@maxst6647]

The phrase "Clearly it follows..." also adorns many mathematics texts, and the hidden meaning is: grab a bottle of aspirin and be prepared to do some work, because the conclusion is anything but clear.

[@StaticBlaster]

@@maxst6647 lol.

[@tethyn]

Elementary is another term that makes me cringe.

[@larry_the]

I am almost through the entire book and have felt that it solidified a lot of the knowledge. I feel much more confident in my math ability now.

[@seanleith5312]

this a bunch of good meaning evil people destroyed the world, killed more than a hundred million totally innocent, higher intelligence people, their theory, instead of putting into trash of history, but teach it as legitimate theory. it is total insanity. Those people are criminals of the worst kind.

[@eatyourcereal7032]

How long did it take you?

[@Bradlee297]

How long did it take you?

[@NYlivinginTN]

I'm going to do that myself!

[@Peter_1986]

There is something very satisfying about constantly tackling new math topics, and applying the stuff that you have learned in previous math topics. The most familiar and traditional example of this would probably be applying arithmetic to algebra, and then you also of course have things like applying trigonometry to calculus, or the Fourier/Laplace transforms to partial differential equations.

[@TheMathSorcerer]

I totally agree!

[@DocBree13]

@@TheMathSorcerer me too!

[@RafaelToscano]

I started the book a little skeptical. I started reading the first chapter thinking “damn, it is very basic stuff”. But when it came to prove stuff, I was like: wow, it is totally natural to use it, but I never stopped to work on the proof! I’m on my second day with it and finished the exercises of the first chapter. I have to say I’m VERY excited to go through the whole book, because I just realized that basic doesn’t mean easy! Sometimes I feel “wow, I didn’t know I was so rusty” 😂. Again, I’m glad I found your channel. It is making me regain a love for mathematics.

[@ħater]

Lol, this summer for me between 8th-9th grade, I took a 6 week long Euclidean Geometry course, and I used a book written by Serge Lang. He is a very good author, and I found his explications helpful whenever the teacher didn’t explain something to my liking.

[@TheMathSorcerer]

that's so cool:)

[@mujtabaalam5907]

What wqs the name of the book?

[@schifoso]

You are a very good instructor. Your solving if the problem was clear and concise. Thank you.

[@albertkh7612]

When I saw this question I first thought about just looking at the differential and use inverse function theorem, since then it would be everywhere invertible. But I guess there is a point to using only the elementary tools, as when I took phd comps, there was this question on differential forms that asked explicitly to use integration on manifold instead of others like kunneth formula or whatever, really got stuck but that taught me to always appreciate the more fundamentals.

[@dilbertojunior]

Serge Lang is one of the best authors in mathematics.

[@TheMathSorcerer]

yes!!

[@mannydossantos9603]

Great explanation and highly enjoyable. It makes all the Math work worthwhile.

[@Felipe.Taboada.]

Gran autor, que sencillo explica todo y muy buena la editorial Springer.

[@alienmathology6251]

Preparing for Masters Entrance Examination... in between all this Channel makes me keep the balance between passion and competetion.......Thank you Proffesor🙃

[@TheMathSorcerer]

:)

[@perkinsdearborn4693]

I am working through this Basic Math book. My copy has an "Answers to Selected Exercises" section. It is the last section in the book A-3 through A-44.

[@TheMathSorcerer]

How interesting!

[@brunoborela4161]

same. coincidentally, mine does not lay flat lol

[@eri4108]

I’m half way of the book. I really like how basic stuff being defined with a precise and rigorous statement, and also like the “pretend not knowing it and try to prove it” game. I think that’s why I love mathematics .

[@Bradlee297]

How did it take you to get through it? I'm about to start tomorrow

[@azimuth4850]

Thanks for the upload!

[@math4university]

Thank you for your videos. Great material.

[@TheMathSorcerer]

thank you!!

[@danield7065]

Thank you for the vid! Very helpful! :))

[@user-eh2ec3rn6w]

Thank you. It is a nice solution.

[@Z-Diode]

Reminds me of „Why Math?“ by Rodney David Driver (* 1932, London) in Undergraduate Texts in Mathematics. A wonderful textbook experience! 🎉

[@aminramazanifar9743]

Thanks for using a paper and a pencil. It feels so refreshing to hear the pencil movements on the paper.

[@TheMathSorcerer]

❤️❤️

[@OscarGarcia-nd3go]

Thank you for solving a problem. I just found it funny how you used 2 full pages to complete 1 question and then cut back to showing the book that is called Basic Mathematics xD I loved math when I was a kid, even throughout most of high school..... but I went through a tough time (depression due to a heart problem) in my last 2 years of high school. I stopped doing my homework and just didn't care about anything. My math teacher was one of few who noticed my grades slipping. I just passed it off as me just not paying attention but It was a real focusing problem that I just couldn't control. I passed the class with a D+, just did the bare minimum to not fail the class. I just vaguely remember always feeling distracted all the time, even got called by my teacher during class and didn't realize I had been called to answer the question on the board.... 3 times, but I'd always play it off as me thinking they were talking to the other person with the same name as me.... It was pretty bad (depression), I just never told anyone due to not wanting pity. Never went to college after graduating. Don't regret not going, especially me remembering how bad my focusing was during those times... but now I'm really wanting to start going to college. I never got help but I did get better on my own, It just felt like it was a very long "thought" that lasted for years. I don't regret not getting help, it wasn't even something I thought I could get. just felt hopeless and had no idea I could even get help. A bit ago, I started walking a mile a day, slowly turning it into, several miles a day, and now I'm walking 10 miles a day. I have made this daily a walk a necessary constant that I need in my life. My cousin had a kid and he introduced me to her and wow have I never been happier to be alive. Finally feels refreshing in my head to be able to focus again. I commented this because I've been watching your vids for quite some time and truly appreciate the effort you put into teaching. Thanks again for showing us how to improve on our math.

[@va3ux]

I saw a hard cover copy of this book this week in a used book store. Didn't give it a second thought because of the title. After seeing this video, I'm going back to get it.

[@callmedeno]

Recommend to everyone, I started maths age 35, completed this my first book a couple of months ago. It is 'psychologically satisfying' throughout. My copy looks identical and has solutions in the back

[@TheMathSorcerer]

very nice!!

[@grookeywithahandgun6324]

aside from mathematics, the cover design is so good! very neat design!

[@theScholar369]

It is really nostalgic and for me the problem really felt like I was transforming my xy Cartesian plane by an angle to get new axes and then again finding a way to get the old axes. A classic example would be transforming my axes to +45 degree . 😅

[@dtonysun]

The camera position and video quality are pretty great!

[@SteveRuprecht]

"It's going to get messy now. It's going to get crazy." Sicko: "Haha YES"

[@maxmustermann-hx3fx]

There is a trick if a softcover book doesn't lay flat and you don't want to break the spine. you can lay the book with it's spine flat on a table go to a few pages from the front and from the back of the book and press down. And then repeat this for the whole book. This will soften the spine and will make it a lot easier to read softcover books.

[@ElectricTeaCup]

This is a good playlist for the book (covers the whole book): kzbin.info/www/bejne/ZmWwiKSvnalpZ8k&ab_channel=RealPhysics

[@kyleherrity8443]

I like smelling new books too. Glad I'm not the only one haha!

[@joaqogc]

Hey! I have that book. Love it! You start from the middle.

[@TheMathSorcerer]

hehe

[@Mathin3D]

LIKED because of the Interesting exercise.

[@juanmolinas]

greetings Sorcerer!, this format of you writing with a pencil is very interesting...

[@TheMathSorcerer]

Thank you! Cheers!

[@togather7032]

This was the first math book I've studied, since I decided to learn math.

[@TheMathSorcerer]

oh wow, that's cool:)

[@howuhh8960]

same

[@TheJara123]

Thanks...real thanks ..you are for real!!

[@TheJara123]

Since you are slowly turning into a unique channel for math, since the days you gave Google ads for your channel I am fan of your enthousiasm. What I would suggest is this, We need more typical problem from a class of problems and how to solve them..like your examples from basic math and induction proof for inequality..please do more problems in a systematic manner. Mainly.for advanced topics where abstract notions obscures the math, insights and the process of "bringing down the abstract" to concrete examples and the full display of the math. This way we will go all the way up again to Abstract level. So Whenever you show us how to come down from abstract to concrete, we will go from concrete to the abstract!! A real beatiful teaching cum learning cycle. I think this is what most self learners want from sorcerer like you!!

[@SequinBrain]

I know it comes with practice, but I get lost at the start when constructing the initial compositions needed to attack the problem. Knowing what works, what doesn't, or how to even form the construction once the problem has been correctly interpreted all seem to be missing when I need them present. Thanks for that run through.

[@ucchi9829]

Start with what you know.

[@SequinBrain]

@@ucchi9829 thx, but usually that's never enough to get me very far. I won't give up tho!

[@pushkarnagpure2357]

Plz review Martin liebeck's pure mathematics...thanks for your efforts and your channel!

[@stretch8390]

Hey Math Sorcerer, I have been spending a lot of time in the regression/modelling world and have found it difficult to find original publications for who developed these concepts e.g. linear mixed effect models, so I was wondering if you had any insight. Statistics seems to be a younger field than a lot of other areas of maths so I didn't think it would be so hard to find original papers and there are so many applied studies as well.

[@joaopadua7134]

Very cool

[@pietergeerkens6324]

To show that f(x,y) has an inverse it is sufficient to show that the determinant of its matrix M M = (2 -1) (1 1) has non-zero determinant. Since the determinant is 2*1 - (-1)*1 = 2+1 = 3 the function f has an inverse. Buried n the above, of course, is the subtle point that f(x,y) has a matrix precisely because it is a linear transformation - an so all the tools of linear algebra can be brought to bear. If determinants aren't available in time for this problem, then the combination of demonstrating that f is linear, and maps the basis vectors (1,0) and (0,1) to non-collinear image vectors, is also sufficient to demon- strate that f(x,y) has an inverse.

[@lugia8888]

The problem is to FIND the inverse. Determinant is useless in this case.

[@allennobody99]

@@lugia8888 if you read the original statement of the problem it says "Show that f has an inverse map." The above comment is a completely sufficient answer to the stated problem.

[@aurelienhamy7124]

Since f is a linear application, showing that Ker(f) = {0} is enough to show that f is bijective. And f(x, y) = 0 => x=y=0, so you get directly that f is bijective, and thus inversible. Of course it does not seem that at this point of the book, linear applications have been studied -- but that's interesting to see how simple arguments can do the job so fast! (I don't know if I use the good terms, I've never done maths in English before)

[@lugia8888]

@@aurelienhamy7124 Invertibility was not even the point. Your argument does not FIND the inverse.

[@aurelienhamy7124]

@@lugia8888 from the question that was asked, no, you needed to prove that the function had an inverse mapping. Finding it may be better, but was not the point here :)

[@numericalcode]

Thanks for not skipping any steps

[@TheMathSorcerer]

You're welcome!

[@user-rb5wn3xk9i]

The problem is also in chapter IV §4 of Lang’s linear algebra. I used rank-nullity to prove

[@shengjiage9923]

Hi. I found this book in hardcover and it was printed in 1971, certainly the first version of this book. I borrowed it from HKU university library and I’m certainly sure that this book was frequently used in 80s and 90s by students here. I don’t know whether there are differences between this paperback version and the old hardcover version since I didn’t buy the new one.

[@TheMathSorcerer]

awesome!

[@marcandrebighetti9882]

thats algebra I love Thanks

[@rikerion]

Good example for a proof BUT small suggestion, put a little tail on your u because its starting to look like your v. One of my first exams at uni I got the wrong answer for one question because I mixed up my 2 and my z. From that point on I have always put a horizontal stroke through my z.

[@willcox5285]

Since it only asks for the existence of an inverse, you can simply show that. To show that a mapping has an inverse, we simply need to show it is an injective function. i.e. there do exist distinct points (x1,y1) and (x2,y2) such that f(x1,y1) = f(x2,y2). To show it is true, proceed by contradiction. Basic properties of the real numbers allow you to conclude that f does indeed have an inverse mapping.

[@onazagorskiene8549]

How do I found the global minimum of this expression:(5cos(x)-2 (sin(x)) ^2+4sin(x)-3) / (6|cos(x) |+1)?

[@nerioms8249]

I like your channel ❤️ :)

[@TheMathSorcerer]

Thank you! 😊

[@alexcastrotello5165]

Esta bueno ese libro para formalizar cosas “básicas”

[@IsomorphicPhi]

Lang gets worse in other books. Not only do "Algebraic Number Theory" and "Fundamentals of Differential Geometry" lack solutions. They also lack problems. Lang is always a fantastic read, though

[@ihorb7346]

Isn’t it a bit easier to find an inverse of 2x2 matrix [[2, -1],[1,1]]?)

[@klatschpauke]

beautiful

[@TheMathSorcerer]

Thank you! Cheers!

[@rolfvantol6138]

You asked if anyone had seen a hardcover version of Serge Lang's "Basic Mathematics". Yes. Picked it up used many years ago and not even sure where. Title page has a name (initials MC, date: 2 Feb 1979, and "St.Johns" the place I imagine). From Addison-Wesley not Springer. Part of the "Addison Wesley series in introductory mathematics" 2nd printing, June 1972. Seems to be the same book. Has Library of Congress Catalog Card No. 75-132055 but no ISBN can see. Has number 4118 on the spine. And yeah has solutions for even exercises in the back. Cheers

[@rolfvantol6138]

Sorry, sometimes even, sometimes odd exercises. "Selected" I guess. And great series of videos. Especially for those of us who are focused challenged and need to go over explanations multiple times to catch the details.

[@TheMathSorcerer]

Wow so rare, thank you!!!

[@ominollo]

Thanks for your video! I read somewhere that this is a nice book but it contains many typos, can you say something about this?

[@WILDXHjordi]

Here are some alternatives: 1- Prove that f is linear, then prove that ker(f)=0, and from dimension theorem conclude that f is surjective, hence bijective, so it has an inverse. 2- In the same spirit of the linearity of f, notice that it can be associated with a matrix, calculate its determinant which is nonzero, hence f has an inverse. 3- Prove that f is injective by letting f(a)=f(b) and showing that this implies a=b, then show surjectivity by letting f(p)=q, and finding p in terms of q (which you already did in the video)

[@ntorneri]

Indeed, I did it with the determinant method (det = 4 here, hence the function has an inverse).

[@pistolinha2934]

crumbly ass pencils I'm reading discrethe mathematics with applications by susanna s, saw it in one of your videos, great book

[@saisreekar4425]

i' know this book and i used to read definitions.

[@aravartomian1]

Since This is a linear system of equations, can you just take the determinant of the matrix to show that it is invertible?

[@johnh7411]

Yes, just put it in matrix form and find the inverse, then multiply by vector (x,y) and you get the inverse function.

[@user-vz3ur3gz2n]

Any suggestions on where we could find 2nd hand books (related to maths ofc) on the internet?Greetings from a maths undergraduate from Greece!

[@TheMathSorcerer]

Woah Greece! How cool!!!! I usually search on amazon and abebooks and biblio.

[@user-vz3ur3gz2n]

Thanks for helping!

[@JamesJoyce12]

just curious - I was always taught to make my mathematical "v's" with an exaggerated line at top and my "u's" with an exaggerated tail - not done any more?

[@kani_sama]

I am at the stage of my life is where I would like to learn how to do book binding to one day replace all my paperbacks into hardbacks

[@NYlivinginTN]

I got a copy last month and it contains selected answers at the end of the book. I think it was printed on demand! Also,what page is this problem from?

[@andrewzhang5345]

Another solution: f is a linear map, and det f /= 0.

[@gmcenroe]

I think that Serge Lang books are great as well. One book that I disliked but he was a coauthor on was Geometry a High School Course. I usually look at expensive math books by getting a copy on Library Genesis, then if I like it I will try to find a used good copy. When I took Geometry in high school, from day one we were presented with theorems and axioms and taught how to do proofs. This geometry book has none of that which I could find. It was so watered down I wondered what value it had at all.

[@stevenanderson236]

After the index are answers to selected exercises

[@TheMathSorcerer]

omg they are there!!!!!!!!!

[@TheMathSorcerer]

for some reason it didn't say it in the table of contents, wow. I never bothered to look since it wasn't there!! THANK YOU!!!!

[@robwin0072]

The TOC looks quite thorough as an essential Math Book-however, are there no answers, including odd-numbered problems? Does it provide a website with a supplement of appendices with answers to the problems?

[@yourpalcal1412]

There are actually answers, it just doesnt say in the contents. He released a video correcting this

[@EpicMethGaming]

mine has answers to selected exercises at the end, maybe a difference of editions?

[@fernandocupil.6463]

Bro Podrás hacer un Review completo de Higher Algebra by Barnard Child y Higher Algebra by Hall and Knight

[@TheMathSorcerer]

buenisma idea, gracias!

[@fernandocupil.6463]

@@TheMathSorcerer GRacias bro. De verdad que ando necesitando esos libros. Saludos,

[@yossicordova2374]

Strange that the copy of the book (1971) I have, it does have answers to selected exercises.

[@andremendes4819]

Esse livro na Amazon tá só uns 358 reais, bem baratin

[@mrbaeman39lolman60]

Are the solutions available online?

[@tohirikagawa2919]

What's the best math book for reading...

[@monsterclass]

Let's build a solutions list

[@truefriend5332]

Can I use this book to learn proofs and logic?

[@ulysses_grant]

Great, I had never head about this one, just started thinking about buying it too! Does anyone have any recommendation about a good book to study mathematical logic? Thank you.

[@toanoradian]

Look up Peter Smith's "Teach Yourself Logic". He has written a good introduction to mathematical logic himself (free), but the "Teach" guide provides you with more books in case you want to go deeper into this field, from model theory, computability, modal logic, proof theory and so on. I've used the guide ever since the pandemic started and it had been an educational journey.

[@ulysses_grant]

@@toanoradian Wow, thank you so much for your recommendation! I had never head about it - or the author! - before, so I'll definitely check it out and buy it! Thank you so much!

[@raphiseguyan2680]

Thank you for your great clips. Would you pleas recommend some basic math books with solutions in it. I bought this book but as you said unfortunately there is no solution in it. And I am kind of lost with it. Once again thanks for your passion and efforts.

[@TheMathSorcerer]

If you want an older inexpensive book, try "algebra and trigonometry" by Cameron. It has answers to the odd problems. You can get it on amazon now inexpensively. I will post a video on it soon!

[@raphiseguyan2680]

@@TheMathSorcerer Great Thank you so much. Because of your clips I started to study math Gain .

[@jonathandawson3091]

Hmm. Don't see why it's difficult but yeah that new book smell though, nice :)

[@De_Bonis_Antonio]

(i) , (ii) , (iii)...What are these symbols? Does it mean "then" ? Sorry for such a stupid question. I live in Brazil. I am aware of the fact that we don't have important scientific tradition here and I do avoid reading physics and chemistry local books. Then I use to buy and study nord american and European scientific books. ( I also lived in Italy many years and like very much history of science ) Anyway, believe me, here in Brazil we have some good mathematicians and didactic books. One of them is Fields medallists, Artur Avila. About Calculus books, I read a note from aother brazilian mathematician - Geraldo Avila- ( who has been a teacher in the Courant Institute of Mathematical Sciences New York University) : "All mothern calculus books are directly or indirectly inspired by Richard Courant's book Differential and Integral Calculus" [ something like that ]. Do you agree? I bought it but I find hard to study this book.

[@TheMathSorcerer]

(i), (ii), (iii) is just a numbering convention, like 1, 2, 3. That's all. It's nice to have different types of numbers for organization. Yeah the Courant book is good but it is hard. It takes a lot of work.

[@chinesecabbagefarmer]

Random question, but do you happen to know what ancient Greeks called tangent lines?

[@TheMathSorcerer]

I just know "tangent" means "to touch" in latin according to the internet:)

[@chinesecabbagefarmer]

@@TheMathSorcerer All I could find was that meaning is from the 16th century or something. Not really sure what that all means, but the basis of why I'm wondering this is that I'm curious about how the greek root for the tangent could differ in meaning from other cultures-- thusly demonstrating the base difference in their logic. i know from some of my textbooks (Calculus: Early Transcendentals 7th Ed) that the problem of determining the equation of a tangent line to a point in a circle has been considered since 300bc greece. So given that the latin word appeared around the 16th century ad, I'm just wondering what the greeks may have called it, lol..

[@chinesecabbagefarmer]

@@TheMathSorcerer Turns out, I was thinking of the calculation of pi had been worked on since 3rd century bc. My bad

[@user-ji9pt9ox2b]

There is a solutions manual by Hannes Thorsell in a pdf format

[@DocBree13]

Awesome!

[@user-ji9pt9ox2b]

@@braynaguilar8567 youtube does not let me link pdf files in the comments

[@Maarttiin]

You should definitely buy yourself a tablet. Samsung's S7+ is top notch and is fairly cheap. BUT... I guess you must be old school and prefer paper and pencil better. I do sometimes too, but the tablet had me changing sides xd Great video!!

[@TheMathSorcerer]

Ya I use an intuous tablet sometimes. I don’t have a tablet like the one you mentioned, sounds so cool! Maybe some day👍

[@xyz.ijk.]

Reviews of This Book showed it had lots of errors. I don't know, that makes me nervous about buying a book for $50.

[@sebmata135]

Could you also just prove that f is a bijection and therefore invertible?

[@TheMathSorcerer]

Yup, he doesn’t cover that in the book though. At least not from what I saw👍

[@charlessmyth]

Boxes keep your scribbles organised.

[@badonkadonk8212]

Swag book😳

[@TheMathSorcerer]

I know I think so too lol.

[@antonshermanable]

Btw, you didn’t have to do the check g(f(x,y))=(x,y) since any inverse is an inverse from both sides

[@lilellia]

Not quite always. Take f(x) = x² and g(x) = √x, where f(g(x)) = x and g(f(x)) = |x|. Or f(x) = sin(x) and g(x) = arcsin(x) where f(g(x)) = x and g(f(x)) = x only for x on [-π/2, π/2]. It's at least not immediately obvious to me by inspection that the function in the video f(x, y) = (2x-y, y+x) is one-to-one (...the showing of which, in fact, is the goal of the problem).

[@tylerkharazi9530]

Why not write it as a matrix and compute the inverse?

[@winter8136]

Math is cool, but i dont understand it. I have knowledge in basic math, but i dont know what book shall i read next to enhance my knowledge and be able to solve complex math.

[@TheMathSorcerer]

I will make a video for this:)

[@rayanselmani480]

is this book really good ??

[@TheMathSorcerer]

So far I like it! The only thing I am not liking is that it doesn't have solutions, so far so good though!

[@rayanselmani480]

@@TheMathSorcerer thanks ❤️❤️❤️

[@raphiseguyan2680]

No Unnecessarily complicated in my opinion . And has no solutions in it.

[@alienmathology6251]

Sir I may be wrong but was it also necessary here to show that it was a bijective Function?

[@TheMathSorcerer]

no:) we just used the definition of inverse function

[@robertveith6383]

The title of this book is vague. It should add "of Undergraduate Mathematics" to the title to be clear.

[@elsareinhard5035]

i joined a discord account that literally created for people who want to learn this book.That is because the book doesn't have solutions and some answers are errors.I have seen people (like me) who eventually give up. It would be a great help if you can make udemy lessons dedicated for this book,just help to explain some stuffs.I will definitely buy that and eventually buy your more adavance math topics!.Ive been wanting to tell you about this book but i just thought that it would be too easy and basic for you to consider.

[@TheMathSorcerer]

how interesting, thank you for sharing this:)

[@ginosuinoilporcoinvasivo8216]

Dini's theorem

[@jamesmarjan5481]

Basic Mathematics is one of the hardest books of all time

[@taggosaurus]

Do an ML Khanna Review.

[@rayanselmani480]

I am looking for book review 😭😭😭

[@TheMathSorcerer]

I like it so far:)

[@Deutsch_94]

I don't know any thing about Mathematique How I learn it ?

[@hope.69]

I am a medical aspirant. My math is not even good. Will this book help me in improving calculations and all (basic things)…

[@Bradlee297]

I'm a medical aspirant too that's why I'm starting this book

[@kenneylai7932]

I am always a threat to the education process continue because they just don't give out appeals.

[@mahadevamurthi8685]

Sir Book ಯಾಲಿ ಸಿಗುತ್ತದೆ

[@abdulrazaquememon9539]

😢 i want to buy this book in my city but unfortunately i couldn't because our whole area is under water from catastrophic flooding....😭😭